pax_global_header00006660000000000000000000000064127757005530014525gustar00rootroot0000000000000052 comment=bf01bd03861f6f214be560c654abecd12909e382 sopt-2.0.0/000077500000000000000000000000001277570055300125115ustar00rootroot00000000000000sopt-2.0.0/.clang-format000066400000000000000000000045101277570055300150640ustar00rootroot00000000000000--- Language: Cpp # BasedOnStyle: LLVM AccessModifierOffset: -2 AlignAfterOpenBracket: true AlignConsecutiveAssignments: false AlignEscapedNewlinesLeft: false AlignOperands: true AlignTrailingComments: true AllowAllParametersOfDeclarationOnNextLine: true AllowShortBlocksOnASingleLine: false AllowShortCaseLabelsOnASingleLine: false AllowShortFunctionsOnASingleLine: All AllowShortIfStatementsOnASingleLine: false AllowShortLoopsOnASingleLine: false AlwaysBreakAfterDefinitionReturnType: None AlwaysBreakBeforeMultilineStrings: false AlwaysBreakTemplateDeclarations: false BinPackArguments: true BinPackParameters: true BreakBeforeBinaryOperators: All BreakBeforeBraces: Attach BreakBeforeTernaryOperators: false BreakConstructorInitializersBeforeComma: false ColumnLimit: 100 CommentPragmas: '^ IWYU pragma:' ConstructorInitializerAllOnOneLineOrOnePerLine: false ConstructorInitializerIndentWidth: 4 ContinuationIndentWidth: 4 Cpp11BracedListStyle: true DerivePointerAlignment: false DisableFormat: false ExperimentalAutoDetectBinPacking: false ForEachMacros: [ foreach, Q_FOREACH, BOOST_FOREACH ] IndentCaseLabels: false IndentWidth: 2 IndentWrappedFunctionNames: false KeepEmptyLinesAtTheStartOfBlocks: true MacroBlockBegin: '' MacroBlockEnd: '' MaxEmptyLinesToKeep: 1 NamespaceIndentation: None ObjCBlockIndentWidth: 2 ObjCSpaceAfterProperty: false ObjCSpaceBeforeProtocolList: true PenaltyBreakBeforeFirstCallParameter: 19 PenaltyBreakComment: 300 PenaltyBreakFirstLessLess: 120 PenaltyBreakString: 1000 PenaltyExcessCharacter: 1000000 PenaltyReturnTypeOnItsOwnLine: 10 PointerAlignment: Right SpaceAfterCStyleCast: false SpaceBeforeAssignmentOperators: true SpaceBeforeParens: false SpaceInEmptyParentheses: false SpacesBeforeTrailingComments: 1 SpacesInAngles: false SpacesInContainerLiterals: true SpacesInCStyleCastParentheses: false SpacesInParentheses: false SpacesInSquareBrackets: false Standard: Cpp11 TabWidth: 8 UseTab: Never IncludeCategories: - Regex: '^(<|")sopt/config.h(>|")$' Priority: 2 - Regex: '^(<|")sopt/wavelets/' Priority: 10 - Regex: '^(<|")sopt/' Priority: 9 - Regex: '^(<.*/.*>)' Priority: 7 - Regex: '^(<.*>)' Priority: 6 - Regex: '".\*"' Priority: 8 ... sopt-2.0.0/.gitignore000066400000000000000000000027011277570055300145010ustar00rootroot00000000000000doc/c/bdwn.png doc/c/dir_*.html doc/c/dir_*.html doc/c/dir_*.html doc/c/dynsections.js doc/c/form_0.png doc/c/form_1.png doc/c/form_10.png doc/c/form_11.png doc/c/form_12.png doc/c/form_13.png doc/c/form_14.png doc/c/form_15.png doc/c/form_16.png doc/c/form_17.png doc/c/form_18.png doc/c/form_19.png doc/c/form_2.png doc/c/form_20.png doc/c/form_21.png doc/c/form_22.png doc/c/form_23.png doc/c/form_24.png doc/c/form_25.png doc/c/form_26.png doc/c/form_27.png doc/c/form_28.png doc/c/form_29.png doc/c/form_3.png doc/c/form_30.png doc/c/form_31.png doc/c/form_32.png doc/c/form_33.png doc/c/form_34.png doc/c/form_35.png doc/c/form_36.png doc/c/form_37.png doc/c/form_38.png doc/c/form_39.png doc/c/form_4.png doc/c/form_40.png doc/c/form_41.png doc/c/form_42.png doc/c/form_43.png doc/c/form_44.png doc/c/form_45.png doc/c/form_46.png doc/c/form_5.png doc/c/form_6.png doc/c/form_7.png doc/c/form_8.png doc/c/form_9.png doc/c/formula.repository doc/c/ftv2blank.png doc/c/ftv2cl.png doc/c/ftv2doc.png doc/c/ftv2folderclosed.png doc/c/ftv2folderopen.png doc/c/ftv2lastnode.png doc/c/ftv2link.png doc/c/ftv2mlastnode.png doc/c/ftv2mnode.png doc/c/ftv2mo.png doc/c/ftv2node.png doc/c/ftv2ns.png doc/c/ftv2plastnode.png doc/c/ftv2pnode.png doc/c/ftv2splitbar.png doc/c/ftv2vertline.png doc/c/nav_g.png doc/c/sopt__about_8c.html doc/c/sopt__about_8c_source.html doc/c/sync_off.png doc/c/sync_on.png images/wavedec.tiff lib/*.a build/ .*.swp .settings python/tests/__pycache__ sopt-2.0.0/CMakeLists.txt000066400000000000000000000035331277570055300152550ustar00rootroot00000000000000cmake_minimum_required(VERSION 2.8) project(Sopt CXX) # Location of extra cmake includes for the project list(APPEND CMAKE_MODULE_PATH ${PROJECT_SOURCE_DIR}/cmake_files) # Downloads and installs GreatCMakeCookOff # It contains a number of cmake recipes include(LookUp-GreatCMakeCookOff) # Version and git hash id include(VersionAndGitRef) set_version(2.0.0) get_gitref() option(tests "Enable testing" on) option(python "Enable python" on) option(benchmarks "Enable benchmarking" on) option(examples "Enable Examples" on) option(logging "Enable logging" on) option(regressions "Enable regressions" on) option(openmp "Enable OpenMP" on) option(archer "Compiling on ARCHER" off) if(regressions) enable_language(C) endif() if(tests) enable_testing() endif() # Add c++11 stuff include(AddCPP11Flags) include(CheckCXX11Features) if(tests AND python) include(AddPyTest) endif() cxx11_feature_check(REQUIRED unique_ptr nullptr override constructor_delegate) include(compilations) # search/install dependencies include(dependencies) # sets rpath policy explicitly if(CMAKE_VERSION VERSION_LESS 3.0) set_property(GLOBAL PROPERTY MACOSX_RPATH ON) else() cmake_policy(SET CMP0042 NEW) endif() if(tests OR examples) enable_testing() endif() if(tests) include(AddCatchTest) endif() if(examples) include(AddExample) endif() if(benchmarks) include(AddBenchmark) endif() if(regressions) include(AddRegression) endif() if(python) include(python_dependencies) include(PythonModule) endif() if(python) add_subdirectory(python) endif() add_subdirectory(cpp) # Exports all Sopt so other packages can access it include(export_sopt) sopt-2.0.0/LICENSE.txt000066400000000000000000000354231277570055300143430ustar00rootroot00000000000000 GNU GENERAL PUBLIC LICENSE Version 2, June 1991 Copyright (C) 1989, 1991 Free Software Foundation, Inc. 59 Temple Place, Suite 330, Boston, MA 02111-1307 USA Everyone is permitted to copy and distribute verbatim copies of this license document, but changing it is not allowed. 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Prototype Matlab implementations of various algorithms are also included. ## Contributors SOPT was initially created by Rafael Carrillo, Jason McEwen and Yves Wiaux but major contributions have since been made by a number of others. The full list of contributors is as follows: * [Rafael E. Carrillo](http://people.epfl.ch/rafael.carrillo) * [Jason D. McEwen](http://www.jasonmcewen.org) * [Yves Wiaux](http://basp.eps.hw.ac.uk) * [Vijay Kartik](https://people.epfl.ch/vijay.kartik) * [Mayeul d'Avezac](https://github.com/mdavezac) * [Luke Pratley](https://about.me/luke.pratley) * [David Perez-Suarez](https://dpshelio.github.io) ## References When referencing this code, please cite our related papers: 1. R. E. Carrillo, J. D. McEwen, D. Van De Ville, J.-P. Thiran, and Y. Wiaux. "Sparsity averaging for compressive imaging", IEEE Signal Processing Letters, 20(6):591-594, 2013, [arXiv:1208.2330](http://arxiv.org/abs/arXiv:1208.2330) 1. A. Onose, R. E. Carrillo, A. Repetti, J. D. McEwen, J.-P. Thiran, J.-C. Pesquet, and Y. Wiaux. "Scalable splitting algorithms for big-data interferometric imaging in the SKA era". Mon. Not. Roy. Astron. Soc., 462(4):4314-4335, 2016, [arXiv:1601.04026](http://arxiv.org/abs/arXiv:1601.04026) ## Webpage http://basp-group.github.io/sopt/ ## Installation ### C++ pre-requisites and dependencies - [CMake](http://www.cmake.org/): a free software that allows cross-platform compilation - [tiff](http://www.libtiff.org/): Tag Image File Format library - [OpenMP](http://openmp.org/wp/): Optional. Speeds up some of the operations. - [UCL/GreatCMakeCookOff](https://github.com/UCL/GreatCMakeCookOff): Collection of cmake recipes. Downloaded automatically if absent. - [Eigen 3](http://eigen.tuxfamily.org/index.php?title=Main_Page): Modern C++ linear algebra. Downloaded automatically if absent. - [spdlog](https://github.com/gabime/spdlog): Optional. Logging library. Downloaded automatically if absent. - [philsquared/Catch](https://github.com/philsquared/Catch): Optional - only for testing. A C++ unit-testing framework. Downloaded automatically if absent. - [google/benchmark](https://github.com/google/benchmar): Optional - only for benchmarks. A C++ micro-benchmarking framework. Downloaded automatically if absent. ### Python pre-requisites and dependencies - [numpy](http://www.numpy.org/): Fundamental package for scientific computing with Python - [scipy](https://www.scipy.org/): User-friendly and efficient numerical routines such as routines for numerical integration and optimization - [pandas](http://pandas.pydata.org/): library providing high-performance, easy-to-use data structures and data analysis tools - [cython](http://cython.org/): Makes writing C extensions for Python as easy as Python itself. Downloaded automatically if absent. - [pytest](http://doc.pytest.org/en/latest/): Optional - for testing only. Unit-testing framework for python. Downloaded automatically if absent and testing is not disabled. ### Installing Sopt Once the dependencies are present, the program can be built with: ``` cd /path/to/code mkdir build cd build cmake -DCMAKE_BUILD_TYPE=Release .. make ``` To test everything went all right: ``` cd /path/to/code/build ctest . ``` To install in directory `/X`, with libraries going to `X/lib`, python modules to `X/lib/pythonA.B/site-packages/sopt`, etc, do: ``` cd /path/to/code/build cmake -DCMAKE_INSTALL_PREFIX=/X .. make install ``` ## Support If you have any questions or comments, feel free to contact Rafael Carrillo or Jason McEwen, or add an issue in the [issue tracker](https://github.com/basp-group/sopt/issues). ## Notes The code is given for educational purpose. For the matlab version of the code see the folder matlab. ## License SOPT: Sparse OPTimisation package Copyright (C) 2013 Rafael Carrillo, Jason McEwen, Yves Wiaux This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details (LICENSE.txt). You should have received a copy of the GNU General Public License along with this program; if not, write to the Free Software Foundation, Inc., 51 Franklin Street, Fifth Floor, Boston, MA 02110-1301, USA. sopt-2.0.0/cmake_files/000077500000000000000000000000001277570055300147535ustar00rootroot00000000000000sopt-2.0.0/cmake_files/AddExample.cmake000066400000000000000000000027111277570055300177620ustar00rootroot00000000000000if(NOT TARGET examples) add_custom_target(examples COMMAND ctest -L examples ${PROJECT_BINARY_DIR}) endif() function(add_example targetname) cmake_parse_arguments(example "NOTEST" "WORKING_DIRECTORY" "LIBRARIES;LABELS;DEPENDS" ${ARGN}) # Source deduce from targetname if possible unset(source) if(EXISTS "${CMAKE_CURRENT_SOURCE_DIR}/${targetname}.cc") set(source ${targetname}.cc) elseif("${example_UNPARSED_ARGUMENTS}" STREQUAL "") message(FATAL_ERROR "No source given or found for ${targetname}") endif() add_executable(example_${targetname} ${source} ${example_UNPARSED_ARGUMENTS}) set_target_properties(example_${targetname} PROPERTIES OUTPUT_NAME ${targetname}) if(example_LIBRARIES) target_link_libraries(example_${targetname} ${example_LIBRARIES}) endif() add_dependencies(examples example_${targetname}) if(example_DEPENDS) add_dependencies(example_${targetname} ${example_DEPENDS}) endif() if(TARGET lookup_dependencies) add_dependencies(example_${targetname} lookup_dependencies) endif() # Add to tests if(NOT example_NOTEST) unset(EXTRA_ARGS) if(example_WORKING_DIRECTORY) set(EXTRA_ARGS WORKING_DIRECTORY ${example_WORKING_DIRECTORY}) endif() add_test(NAME test_example_${targetname} COMMAND example_${targetname} ${EXTRA_ARGS}) list(APPEND example_LABELS examples) set_tests_properties(test_example_${targetname} PROPERTIES LABELS "${example_LABELS}") endif() endfunction() sopt-2.0.0/cmake_files/AddRegression.cmake000066400000000000000000000006741277570055300205150ustar00rootroot00000000000000if(NOT TARGET regressions) add_custom_target(regressions COMMAND ctest -L regressions ${PROJECT_BINARY_DIR}) endif() include(AddCatchTest) function(add_regression targetname) cmake_parse_arguments(regr "" "" "LIBRARIES;LABELS;INCLUDES" ${ARGN}) add_catch_test(${targetname} ${regr_UNPARSED_ARGUMENTS} LIBRARIES ${Sopt_LIBRARIES} ${regr_LIBRARIES} LABELS ${regr_LABELS} "regression" INCLUDES ${regr_INCLUDE} ) endfunction() sopt-2.0.0/cmake_files/LookUp-GreatCMakeCookOff.cmake000066400000000000000000000037741277570055300224110ustar00rootroot00000000000000# This file is part of the GreatCMakeCookOff package and distributed under the MIT Licences. # Upon inclusion in a cmake file, it will download the GreatCMakeCookOff and make itself known to # CMake. It should be added explicitely to build systems that make use of recipes from the cook-off. # And it should be included prior to using cook-off recipes: # # ```{CMake} # include(LookUp-GreatCMakeCookOff) # ``` # First attempts to find the package set(COOKOFF_DOWNLOAD_DIR "${PROJECT_BINARY_DIR}/external/src/GreatCMakeCookOff") find_package(GreatCMakeCookOff NO_MODULE PATHS "${COOKOFF_DOWNLOAD_DIR}" QUIET) # Otherwise attempts to download it. # Does not use ExternalProject_Add to avoid doing a recursive cmake step. if(NOT GreatCMakeCookOff_FOUND) message(STATUS "[GreatCMakeCookOff] not found. Will attempt to clone it.") # Need git for cloning. find_package(Git) if(NOT GIT_FOUND) message(FATAL_ERROR "[Git] not found. Cannot download GreatCMakeCookOff") endif() # Remove GreatCMakeCookOff directory if it exists if(EXISTS "${COOKOFF_DOWNLOAD_DIR}") execute_process( COMMAND ${CMAKE_COMMAND} -E remove_directory "${COOKOFF_DOWNLOAD_DIR}" OUTPUT_QUIET ) endif() if(NOT COOKOFF_GITREPO) set(COOKOFF_GITREPO https://github.com/UCL/GreatCMakeCookOff.git) endif() execute_process( COMMAND ${GIT_EXECUTABLE} clone "${COOKOFF_GITREPO}" "${COOKOFF_DOWNLOAD_DIR}" RESULT_VARIABLE CLONING_COOKOFF OUTPUT_QUIET ERROR_VARIABLE CLONING_ERROR ) if(NOT ${CLONING_COOKOFF} EQUAL 0) message(STATUS "${CLONING_ERROR}") message(FATAL_ERROR "[GreatCMakeCookOff] git cloning failed.") else() message(STATUS "[GreatCMakeCookOff] downloaded to ${COOKOFF_DOWNLOAD_DIR}") find_package(GreatCMakeCookOff NO_MODULE PATHS "${COOKOFF_DOWNLOAD_DIR}" QUIET) endif() set(GreatCMakeCookOff_DIR "${COOKOFF_DOWNLOAD_DIR}/cmake") set(GreatCMakeCookOff_FOUND TRUE) endif() unset(COOKOFF_DOWNLOAD_DIR) # Adds GreatCMakeCookOff to module paths initialize_cookoff() sopt-2.0.0/cmake_files/LookUpSopt.cmake000066400000000000000000000032011277570055300200300ustar00rootroot00000000000000# Looks up [Sopt](http://basp-group.github.io/sopt/) # # - GIT_REPOSITORY: defaults to https://github.com/UCL/sopt.git # - GIT_TAG: defaults to master # - BUILD_TYPE: defaults to Release # if(Sopt_ARGUMENTS) cmake_parse_arguments(Sopt "" "GIT_REPOSITORY;GIT_TAG;BUILD_TYPE" "" ${Sopt_ARGUMENTS}) endif() if(NOT Sopt_GIT_REPOSITORY) set(Sopt_GIT_REPOSITORY https://github.com/UCL/sopt.git) endif() if(NOT Sopt_GIT_TAG) set(Sopt_GIT_TAG master) endif() if(NOT Sopt_BUILD_TYPE) set(Sopt_BUILD_TYPE Release) endif() # write subset of variables to cache for sopt to use include(PassonVariables) passon_variables(Lookup-Sopt FILENAME "${EXTERNAL_ROOT}/src/SoptVariables.cmake" PATTERNS "CMAKE_[^_]*_R?PATH" "CMAKE_C_.*" "BLAS_.*" "FFTW3_.*" "TIFF_.*" "GreatCMakeCookOff_DIR" ALSOADD "\nset(CMAKE_INSTALL_PREFIX \"${EXTERNAL_ROOT}\" CACHE STRING \"\")\n" ) ExternalProject_Add( Lookup-Sopt PREFIX ${EXTERNAL_ROOT} GIT_REPOSITORY ${Sopt_GIT_REPOSITORY} GIT_TAG ${Sopt_GIT_TAG} CMAKE_ARGS -C "${EXTERNAL_ROOT}/src/SoptVariables.cmake" -DBUILD_SHARED_LIBS=OFF -DCMAKE_BUILD_TYPE=${Sopt_BUILD_TYPE} -DNOEXPORT=TRUE -Dtests=FALSE -Dpython=FALSE -Dexamples=FALSE -Dlogging=FALSE -Dregressions=FALSE -Dopenmp=FALSE -Dcpp=FALSE -DCLIBS_ONLY=FALSE INSTALL_DIR ${EXTERNAL_ROOT} LOG_DOWNLOAD ON LOG_CONFIGURE ON LOG_BUILD ON ) add_recursive_cmake_step(Lookup-Sopt DEPENDEES install) foreach(dep Eigen3 spdlog) lookup_package(${dep}) if(TARGET ${dep}) add_dependencies(Lookup-Sopt ${dep}) endif() endforeach() sopt-2.0.0/cmake_files/SoptConfig.in.cmake000066400000000000000000000012571277570055300204420ustar00rootroot00000000000000get_filename_component(Sopt_CMAKE_DIR "${CMAKE_CURRENT_LIST_FILE}" PATH) message(STATUS "Linking to sopt package in ${Sopt_CMAKE_DIR}") set(Sopt_INCLUDE_DIRS "@ALL_INCLUDE_DIRS@") if(NOT TARGET libsopt AND EXISTS "${Sopt_CMAKE_DIR}/SoptCTargets.cmake") include("${Sopt_CMAKE_DIR}/SoptCTargets.cmake") endif() if(NOT TARGET sopt AND EXISTS "${Sopt_CMAKE_DIR}/SoptCPPTargets.cmake") include("${Sopt_CMAKE_DIR}/SoptCPPTargets.cmake") endif() unset(Sopt_LIBRARIES) if(TARGET sopt) list(APPEND Sopt_LIBRARIES sopt) set(Sopt_CPP_LIBRARY sopt) endif() if(TARGET libsopt) list(APPEND Sopt_LIBRARIES libsopt) set(Sopt_C_LIBRARY libsopt) endif() set(Sopt_ABOUT_EXECUTABLE sopt_about) sopt-2.0.0/cmake_files/SoptConfigVersion.in.cmake000066400000000000000000000005021277570055300220000ustar00rootroot00000000000000set(PACKAGE_VERSION "@Sopt_VERSION@") if("${PACKAGE_VERSION}" VERSION_LESS "${PACKAGE_FIND_VERSION}") set(PACKAGE_VERSION_COMPATIBLE FALSE) else() set(PACKAGE_VERSION_COMPATIBLE TRUE) if("${PACKAGE_VERSION}" VERSION_EQUAL "${PACKAGE_FIND_VERSION}") set(PACKAGE_VERSION_EXACT FALSE) endif() endif() sopt-2.0.0/cmake_files/compilations.cmake000066400000000000000000000001511277570055300204530ustar00rootroot00000000000000try_compile(SOPT_HAS_USING "${CMAKE_CURRENT_BINARY_DIR}" "${PROJECT_SOURCE_DIR}/cmake_files/using.cc") sopt-2.0.0/cmake_files/dependencies.cmake000066400000000000000000000024241277570055300204050ustar00rootroot00000000000000include(PackageLookup) # check for existence, or install external projects lookup_package(Eigen3 ARGUMENTS HG_REPOSITORY https://bitbucket.org/LukePratley/eigen) if(logging) lookup_package(spdlog REQUIRED) endif() find_package(TIFF) if(examples OR regression) if(NOT TIFF_FOUND) message(FATAL_ERROR "Examples and regressions require TIFF") endif() endif() if(regressions) find_package(FFTW3 REQUIRED DOUBLE) set(REGRESSION_ORACLE_ID "last_of_c" CACHE STRING "Commmit/tag/branch againts which to run regressions") lookup_package(Sopt REQUIRED DOWNLOAD_BY_DEFAULT PATHS "${EXTERNAL_ROOT}" NO_DEFAULT_PATH ARGUMENTS GIT_REPOSITORY "https://www.github.com/basp-group/sopt.git" GIT_TAG ${REGRESSION_ORACLE_ID} BUILD_TYPE Release ) endif() if(openmp) find_package(OpenMP) if(OPENMP_FOUND) set(SOPT_DEFAULT_OPENMP_THREADS 4 CACHE STRING "Number of threads used in testing") set(SOPT_OPENMP TRUE) add_library(openmp::openmp INTERFACE IMPORTED GLOBAL) set_target_properties(openmp::openmp PROPERTIES INTERFACE_COMPILE_OPTIONS "${OpenMP_CXX_FLAGS}" INTERFACE_LINK_LIBRARIES "${OpenMP_CXX_FLAGS}") else() message(STATUS "Could not find OpenMP. Compiling without.") set(SOPT_OPENMP FALSE) endif() endif() sopt-2.0.0/cmake_files/export_sopt.cmake000066400000000000000000000021661277570055300203500ustar00rootroot00000000000000# Exports Sopt so other packages can access it export(TARGETS sopt FILE "${PROJECT_BINARY_DIR}/SoptCPPTargets.cmake") # Avoids creating an entry in the cmake registry. if(NOT NOEXPORT) export(PACKAGE Sopt) endif() # First in binary dir set(ALL_INCLUDE_DIRS "${PROJECT_SOURCE_DIR}/cpp" "${PROJECT_BINARY_DIR}/include") configure_File(cmake_files/SoptConfig.in.cmake "${PROJECT_BINARY_DIR}/SoptConfig.cmake" @ONLY ) configure_File(cmake_files/SoptConfigVersion.in.cmake "${PROJECT_BINARY_DIR}/SoptConfigVersion.cmake" @ONLY ) # Then for installation tree file(RELATIVE_PATH REL_INCLUDE_DIR "${CMAKE_INSTALL_PREFIX}/share/cmake/sopt" "${CMAKE_INSTALL_PREFIX}/include" ) set(ALL_INCLUDE_DIRS "\${Sopt_CMAKE_DIR}/${REL_INCLUDE_DIR}") configure_file(cmake_files/SoptConfig.in.cmake "${PROJECT_BINARY_DIR}/CMakeFiles/SoptConfig.cmake" @ONLY ) # Finally install all files install(FILES "${PROJECT_BINARY_DIR}/CMakeFiles/SoptConfig.cmake" "${PROJECT_BINARY_DIR}/SoptConfigVersion.cmake" DESTINATION share/cmake/sopt COMPONENT dev ) install(EXPORT SoptCPPTargets DESTINATION share/cmake/sopt COMPONENT dev) sopt-2.0.0/cmake_files/python_dependencies.cmake000066400000000000000000000033161277570055300220070ustar00rootroot00000000000000# Scripts to run purify from build directory. Good for testing/debuggin. include(EnvironmentScript) # Function to install python files in python path ${PYTHON_PKG_DIR} include(PythonInstall) # Ability to find installed python packages include(PythonPackage) # Installs python packages that are missing. # We choose to do this only for packages that are required to build purify. # Leave it to the user to install packages that are needed for running # purify. include(PythonPackageLookup) # Creates script for running python with the bempp package available # Also makes python packages and selected directories available to the # build system add_to_python_path("${PROJECT_BINARY_DIR}/python") add_to_python_path("${EXTERNAL_ROOT}/python") add_python_eggs("${PROJECT_SOURCE_DIR}" EXCLUDE "${PROJECT_SOURCE_DIR}/purify*egg" "${PROJECT_BINARY_DIR}/Purify*egg" ) set(LOCAL_PYTHON_EXECUTABLE "${PROJECT_BINARY_DIR}/localpython.sh") create_environment_script( EXECUTABLE "${PYTHON_EXECUTABLE}" PATH "${LOCAL_PYTHON_EXECUTABLE}" PYTHON ) # Python interpreter + libraries find_package(CoherentPython) # Only required for building if(NOT cython_EXECUTABLE) lookup_python_package(Cython REQUIRED PATH "${EXTERNAL_ROOT}/python") endif() # Also required for production find_python_package(numpy) find_python_package(scipy) # Finds additional info, like libraries, include dirs... # no need check numpy features, it's all handled by cython. set(no_numpy_feature_tests TRUE) find_package(Numpy REQUIRED) if(tests) include(AddPyTest) setup_pytest("${EXTERNAL_ROOT}/python" "${PROJECT_BINARY_DIR}/py.test.sh") lookup_python_package(pywt PIPNAME PyWavelets REQUIRED PATH "${EXTERNAL_ROOT}/python") endif() sopt-2.0.0/cmake_files/using.cc000066400000000000000000000003441277570055300164100ustar00rootroot00000000000000#include //! Sparsity Averaging Reweighted Analysis class SARA : public std::vector { public: // Constructors using std::vector::vector; }; int main(int, char const**) { SARA s = {0, 0}; return 0; } sopt-2.0.0/cpp/000077500000000000000000000000001277570055300132735ustar00rootroot00000000000000sopt-2.0.0/cpp/CMakeLists.txt000066400000000000000000000020401277570055300160270ustar00rootroot00000000000000# Setup logging set(SOPT_LOGGER_NAME "sopt" CACHE STRING "NAME of the logger") set(SOPT_COLOR_LOGGING true CACHE BOOL "Whether to add color to the log") if(logging) set(SOPT_DO_LOGGING 1) set(SOPT_TEST_LOG_LEVEL critical CACHE STRING "Level when logging tests") set_property(CACHE SOPT_TEST_LOG_LEVEL PROPERTY STRINGS off critical error warn info debug trace) else() unset(SOPT_DO_LOGGING) set(SOPT_TEST_LOG_LEVEL off) endif() set(version ${Sopt_VERSION}) string(REGEX REPLACE "\\." ";" version "${Sopt_VERSION}") list(GET version 0 Sopt_VERSION_MAJOR) list(GET version 1 Sopt_VERSION_MINOR) list(GET version 2 Sopt_VERSION_PATCH) configure_file(sopt/config.in.h "${PROJECT_BINARY_DIR}/include/sopt/config.h") add_subdirectory(sopt) if(regressions OR examples) # Tiff wrappers and whatnot add_subdirectory(tools_for_tests) endif() if(tests) add_subdirectory(tests) endif() if(examples) add_subdirectory(examples) endif() if(benchmarks) add_subdirectory(benchmarks) endif() if(regressions) add_subdirectory(regressions) endif() sopt-2.0.0/cpp/benchmarks/000077500000000000000000000000001277570055300154105ustar00rootroot00000000000000sopt-2.0.0/cpp/benchmarks/CMakeLists.txt000066400000000000000000000002021277570055300201420ustar00rootroot00000000000000add_benchmark(wavelets LIBRARIES sopt) add_benchmark(conjugate_gradient LIBRARIES sopt) add_benchmark(l1_proximal LIBRARIES sopt) sopt-2.0.0/cpp/benchmarks/conjugate_gradient.cc000066400000000000000000000035051277570055300215560ustar00rootroot00000000000000#include "sopt/conjugate_gradient.h" #include #include template void matrix_cg(benchmark::State &state) { auto const N = state.range_x(); auto const epsilon = std::pow(10, -state.range_y()); auto const A = sopt::Image::Random(N, N).eval(); auto const b = sopt::Array::Random(N).eval(); auto const AhA = A.matrix().transpose().conjugate() * A.matrix(); auto const Ahb = A.matrix().transpose().conjugate() * b.matrix(); auto output = sopt::Vector::Zero(N).eval(); sopt::ConjugateGradient cg(0, epsilon); while(state.KeepRunning()) cg(output, AhA, Ahb); state.SetBytesProcessed(int64_t(state.iterations()) * int64_t(N) * sizeof(TYPE)); } template void function_cg(benchmark::State &state) { auto const N = state.range_x(); auto const epsilon = std::pow(10, -state.range_y()); auto const A = sopt::Image::Random(N, N).eval(); auto const b = sopt::Array::Random(N).eval(); auto const AhA = A.matrix().transpose().conjugate() * A.matrix(); auto const Ahb = A.matrix().transpose().conjugate() * b.matrix(); typedef sopt::Vector t_Vector; auto func = [&AhA](t_Vector &out, t_Vector const &input) { out = AhA * input; }; auto output = sopt::Vector::Zero(N).eval(); sopt::ConjugateGradient cg(0, epsilon); while(state.KeepRunning()) cg(output, func, Ahb); state.SetBytesProcessed(int64_t(state.iterations()) * int64_t(N) * sizeof(TYPE)); } BENCHMARK_TEMPLATE(matrix_cg, sopt::t_complex)->RangePair(1, 256, 4, 12)->UseRealTime(); BENCHMARK_TEMPLATE(matrix_cg, sopt::t_real)->RangePair(1, 256, 4, 12)->UseRealTime(); BENCHMARK_TEMPLATE(function_cg, sopt::t_complex)->RangePair(1, 256, 4, 12)->UseRealTime(); BENCHMARK_TEMPLATE(function_cg, sopt::t_real)->RangePair(1, 256, 4, 12)->UseRealTime(); BENCHMARK_MAIN() sopt-2.0.0/cpp/benchmarks/l1_proximal.cc000066400000000000000000000024721277570055300201530ustar00rootroot00000000000000#include #include #include #include #include template void function_l1p(benchmark::State &state) { typedef typename sopt::real_type::type Real; auto const N = state.range_x(); auto const input = sopt::Vector::Random(N).eval(); auto const Psi = sopt::Matrix::Random(input.size(), input.size() * 10).eval(); sopt::Vector const weights = sopt::Vector::Random(Psi.cols()).normalized().array().abs(); auto const l1 = sopt::proximal::L1() .tolerance(std::pow(10, -state.range_y())) .itermax(100) .fista_mixing(true) .positivity_constraint(true) .nu(1) .Psi(Psi) .weights(weights); Real const gamma = 1e-2 / Psi.array().abs().sum(); auto output = sopt::Vector::Zero(N).eval(); while(state.KeepRunning()) l1(output, gamma, input); state.SetBytesProcessed(int64_t(state.iterations()) * int64_t(N) * sizeof(TYPE)); } BENCHMARK_TEMPLATE(function_l1p, sopt::t_complex)->RangePair(1, 256, 4, 12)->UseRealTime(); BENCHMARK_TEMPLATE(function_l1p, sopt::t_real)->RangePair(1, 256, 4, 12)->UseRealTime(); BENCHMARK_MAIN() sopt-2.0.0/cpp/benchmarks/wavelets.cc000066400000000000000000000073411277570055300175560ustar00rootroot00000000000000#include #include #include unsigned get_size(unsigned requested, unsigned levels) { auto const N = (1u << levels); return requested % N == 0 ? requested : requested + N - requested % N; } std::string get_name(unsigned db) { std::ostringstream sstr; sstr << "DB" << db; return sstr.str(); } template void direct_matrix(benchmark::State &state) { auto const Nx = get_size(state.range_x(), LEVEL); auto const Ny = get_size(state.range_y(), LEVEL); auto const input = sopt::Image::Random(Nx, Ny).eval(); auto output = sopt::Image::Zero(Nx, Ny).eval(); auto const wavelet = sopt::wavelets::factory(get_name(DB), LEVEL); while(state.KeepRunning()) wavelet.direct(output, input); state.SetBytesProcessed(int64_t(state.iterations()) * int64_t(Nx) * int64_t(Ny) * sizeof(TYPE)); } template void indirect_matrix(benchmark::State &state) { auto const Nx = get_size(state.range_x(), LEVEL); auto const Ny = get_size(state.range_y(), LEVEL); auto const input = sopt::Image::Random(Nx, Ny).eval(); auto output = sopt::Image::Zero(Nx, Ny).eval(); auto const wavelet = sopt::wavelets::factory(get_name(DB), LEVEL); while(state.KeepRunning()) wavelet.indirect(input, output); state.SetBytesProcessed(int64_t(state.iterations()) * int64_t(Nx) * int64_t(Ny) * sizeof(TYPE)); } template void direct_vector(benchmark::State &state) { auto const Nx = get_size(state.range_x(), LEVEL); auto const input = sopt::Array::Random(Nx).eval(); auto output = sopt::Array::Zero(Nx).eval(); auto const wavelet = sopt::wavelets::factory(get_name(DB), LEVEL); while(state.KeepRunning()) wavelet.direct(output, input); state.SetBytesProcessed(int64_t(state.iterations()) * int64_t(Nx) * sizeof(TYPE)); } template void indirect_vector(benchmark::State &state) { auto const Nx = get_size(state.range_x(), LEVEL); auto const input = sopt::Array::Random(Nx).eval(); auto output = sopt::Array::Zero(Nx).eval(); auto const wavelet = sopt::wavelets::factory(get_name(DB), LEVEL); while(state.KeepRunning()) wavelet.indirect(input, output); state.SetBytesProcessed(int64_t(state.iterations()) * int64_t(Nx) * sizeof(TYPE)); } auto const n = 64; auto const N = 256 * 3; BENCHMARK_TEMPLATE(direct_matrix, sopt::t_complex, 1, 1)->RangePair(n, N, n, N)->UseRealTime(); BENCHMARK_TEMPLATE(direct_matrix, sopt::t_real, 1, 1)->RangePair(n, N, n, N)->UseRealTime(); BENCHMARK_TEMPLATE(direct_matrix, sopt::t_complex, 10, 1)->RangePair(n, N, n, N)->UseRealTime(); BENCHMARK_TEMPLATE(direct_vector, sopt::t_complex, 1, 1)->Range(n, N)->UseRealTime(); BENCHMARK_TEMPLATE(direct_vector, sopt::t_complex, 10, 1)->Range(n, N)->UseRealTime(); BENCHMARK_TEMPLATE(direct_vector, sopt::t_complex, 1, 2)->Range(n, N)->UseRealTime(); BENCHMARK_TEMPLATE(direct_vector, sopt::t_real, 1, 1)->Range(n, N)->UseRealTime(); BENCHMARK_TEMPLATE(indirect_matrix, sopt::t_complex, 1, 1)->RangePair(n, N, n, N)->UseRealTime(); BENCHMARK_TEMPLATE(indirect_matrix, sopt::t_real, 1, 1)->RangePair(n, N, n, N)->UseRealTime(); BENCHMARK_TEMPLATE(indirect_matrix, sopt::t_complex, 10, 1)->RangePair(n, N, n, N)->UseRealTime(); BENCHMARK_TEMPLATE(indirect_vector, sopt::t_complex, 1, 1)->Range(n, N)->UseRealTime(); BENCHMARK_TEMPLATE(indirect_vector, sopt::t_complex, 10, 1)->Range(n, N)->UseRealTime(); BENCHMARK_TEMPLATE(indirect_vector, sopt::t_complex, 1, 2)->Range(n, N)->UseRealTime(); BENCHMARK_TEMPLATE(indirect_vector, sopt::t_real, 1, 1)->Range(n, N)->UseRealTime(); BENCHMARK_MAIN() sopt-2.0.0/cpp/examples/000077500000000000000000000000001277570055300151115ustar00rootroot00000000000000sopt-2.0.0/cpp/examples/CMakeLists.txt000066400000000000000000000006611277570055300176540ustar00rootroot00000000000000add_example(wavelets LIBRARIES sopt LABELS wavelets) add_example(sara LIBRARIES sopt LABELS wavelets) add_example(positive_quadrant_projection LIBRARIES sopt LABELS proximals) add_example(l1_norm LIBRARIES sopt LABELS proximals) add_example(soft_threshhold LIBRARIES sopt LABELS proximals) add_example(conjugate_gradient LIBRARIES sopt) add_example(l1_proximal LIBRARIES sopt) add_subdirectory(sdmm) add_subdirectory(proximal_admm) sopt-2.0.0/cpp/examples/conjugate_gradient.cc000066400000000000000000000037271277570055300212650ustar00rootroot00000000000000#include #include #include int main(int, char const **) { // Conjugate-gradient solves Ax=b, where A is positive definite. // The input to our conjugate gradient can be a matrix or a function // Lets try both approaches. // Creates the input. typedef sopt::Vector t_Vector; typedef sopt::Matrix t_Matrix; t_Vector const b = t_Vector::Random(8); t_Matrix const A = t_Matrix::Random(b.size(), b.size()); // Transform to solvable problem A^hA x = A^hb, where A^h is the conjugate transpose t_Matrix const AhA = A.conjugate().transpose() * A; t_Vector const Ahb = A.conjugate().transpose() * b; // The same transform can be realised using a function, where out = A^h * A * input. // This will recompute AhA every time the function is applied by the conjugate gradient. It is not // optmial for this case. But the function interface means A could be an FFT. auto aha_function = [&A](t_Vector &out, t_Vector const &input) { out = A.conjugate().transpose() * A * input; }; // Conjugate gradient with unlimited iterations and a convergence criteria of 1e-12 sopt::ConjugateGradient cg(std::numeric_limits::max(), 1e-12); // Call conjugate gradient using both approaches auto as_matrix = cg(AhA, Ahb); auto as_function = cg(aha_function, Ahb); // Check result if(not(as_matrix.good and as_function.good)) throw std::runtime_error("Expected convergence"); if(as_matrix.niters != as_function.niters) throw std::runtime_error("Expected same number of iterations"); if(as_matrix.residual > cg.tolerance() or as_function.residual > cg.tolerance()) throw std::runtime_error("Expected better convergence"); if(not as_matrix.result.isApprox(as_function.result, 1e-6)) throw std::runtime_error("Expected same result"); if(not(A * as_matrix.result).isApprox(b, 1e-6)) throw std::runtime_error("Expected solution to Ax=b"); return 0; } sopt-2.0.0/cpp/examples/l1_norm.cc000066400000000000000000000004601277570055300167670ustar00rootroot00000000000000#include #include int main(int, char const **) { sopt::Image> input(2, 2), weights(2, 2); input << 1, -2, 3, -4; weights << 5, 6, 7, 8; if(sopt::l1_norm(input, weights) != 1 * 5 + 2 * 6 + 3 * 7 + 4 * 8) throw std::exception(); return 0; } sopt-2.0.0/cpp/examples/l1_proximal.cc000066400000000000000000000032551277570055300176540ustar00rootroot00000000000000#include #include #include int main(int, char const **) { sopt::logging::initialize(); typedef sopt::t_complex Scalar; typedef sopt::real_type::type Real; auto const input = sopt::Vector::Random(10).eval(); auto const Psi = sopt::Matrix::Random(input.size(), input.size() * 10).eval(); sopt::Vector const weights = sopt::Vector::Random(Psi.cols()).normalized().array().abs(); auto const l1 = sopt::proximal::L1() .tolerance(1e-12) .itermax(100) .fista_mixing(true) .positivity_constraint(true) .nu(1) .Psi(Psi) .weights(weights); // gamma should be sufficiently small. Or is it nu should not be 1? // In any case, this seems to work. Real const gamma = 1e-2 / Psi.array().abs().sum(); auto const result = l1(gamma, input); if(not result.good) SOPT_THROW("Did not converge"); // Check the proximal is a minimum in any allowed direction (positivity constraint) Real const eps = 1e-4; for(size_t i(0); i < 10; ++i) { sopt::Vector const dir = sopt::Vector::Random(input.size()).normalized() * eps; sopt::Vector const position = sopt::positive_quadrant(result.proximal + dir); Real const dobj = l1.objective(input, position, gamma); // Fuzzy logic if(dobj < result.objective - 1e-8) SOPT_THROW("This is not the minimum we are looking for: ") << dobj << " <~ " << result.objective; } return 0; } sopt-2.0.0/cpp/examples/positive_quadrant_projection.cc000066400000000000000000000012761277570055300234230ustar00rootroot00000000000000#include #include int main(int, char const **) { // Create a matrix with a single negative real numbers typedef sopt::Image> t_Matrix; t_Matrix input = 2 * t_Matrix::Ones(5, 5) + t_Matrix::Random(5, 5); // Apply projection t_Matrix posquad = sopt::positive_quadrant(input); // imaginary part and negative real part becomes zero if((posquad.array().imag() != 0).any()) throw std::runtime_error("Imaginary part not zero"); // positive real part unchanged posquad.real()(2, 3) = input.real()(2, 3); if((posquad.array().real() != input.array().real()).all()) throw std::runtime_error("Real part was modified"); return 0; } sopt-2.0.0/cpp/examples/proximal_admm/000077500000000000000000000000001277570055300177425ustar00rootroot00000000000000sopt-2.0.0/cpp/examples/proximal_admm/CMakeLists.txt000066400000000000000000000007441277570055300225070ustar00rootroot00000000000000add_example(padmm_euclidian_norm euclidian_norm.cc LIBRARIES sopt LABELS padmm) set_target_properties(example_padmm_euclidian_norm PROPERTIES OUTPUT_NAME euclidian_norm) add_example(padmm_inpainting inpainting.cc LIBRARIES sopt tools_for_tests) set_target_properties(example_padmm_inpainting PROPERTIES OUTPUT_NAME inpainting) add_example(padmm_reweighted reweighted.cc LIBRARIES sopt tools_for_tests) set_target_properties(example_padmm_reweighted PROPERTIES OUTPUT_NAME reweighted) sopt-2.0.0/cpp/examples/proximal_admm/euclidian_norm.cc000066400000000000000000000046611277570055300232500ustar00rootroot00000000000000#include #include #include #include #include // We will minimize ||x - x_0|| + ||x - x_1||, ||.|| the euclidian norm int main(int, char const **) { // Initializes and sets logger (if compiled with logging) // See set_level function for levels. sopt::logging::initialize(); // Some typedefs for simplicity typedef sopt::t_real t_Scalar; typedef sopt::Vector t_Vector; typedef sopt::Matrix t_Matrix; // Creates the target vectors auto const N = 5; t_Vector const target0 = t_Vector::Random(N); t_Vector const target1 = t_Vector::Random(N) * 4; // Creates the resulting proximal // In practice g_0 and g_1 are any functions with the signature // void(t_Vector &output, t_Vector::Scalar gamma, t_Vector const &input) // They are the proximal of ||x - x_0|| and ||x - x_1|| auto prox_g0 = sopt::proximal::translate(sopt::proximal::EuclidianNorm(), -target0); auto prox_g1 = sopt::proximal::translate(sopt::proximal::EuclidianNorm(), -target1); auto padmm = sopt::algorithm::ProximalADMM(prox_g0, prox_g1, t_Vector::Zero(N)) .itermax(5000) .is_converged(sopt::RelativeVariation(1e-12)) .gamma(0.01) // Phi == -1, so that we can minimize f(x) + g(x), as per problem definition in // padmm. .Phi(-t_Matrix::Identity(N, N)); // Alternatively, padmm can be called with a tuple (x, residual) as argument // Here, we default to (Φ^Ty/ν, ΦΦ^Ty/ν - y) auto const diagnostic = padmm(); // diagnostic should tell us the function converged // it also contains diagnostic.niters - the number of iterations, and cg_diagnostic - the // diagnostic from the last call to the conjugate gradient. if(not diagnostic.good) throw std::runtime_error("Did not converge!"); // x should be any point on the segment linking x_0 and x_1 t_Vector const segment = (target1 - target0).normalized(); t_Scalar const alpha = (diagnostic.x - target0).transpose() * segment; if((target1 - target0).transpose() * segment < alpha) throw std::runtime_error("Point beyond x_1 plane"); if(alpha < 0e0) throw std::runtime_error("Point before x_0 plane"); if((diagnostic.x - target0 - alpha * segment).stableNorm() > 1e-8) throw std::runtime_error("Point not on (x_0, x_1) line"); return 0; } sopt-2.0.0/cpp/examples/proximal_admm/inpainting.cc000066400000000000000000000114151277570055300224130ustar00rootroot00000000000000#include #include #include #include #include #include #include #include #include #include #include #include #include #include // This header is not part of the installed sopt interface // It is only present in tests #include #include // \min_{x} ||\Psi^Tx||_1 \quad \mbox{s.t.} \quad ||y - Ax||_2 < \epsilon and x \geq 0 int main(int argc, char const **argv) { // Some typedefs for simplicity typedef double Scalar; // Column vector - linear algebra - A * x is a matrix-vector multiplication // type expected by ProximalADMM typedef sopt::Vector Vector; // Matrix - linear algebra - A * x is a matrix-vector multiplication // type expected by ProximalADMM typedef sopt::Matrix Matrix; // Image - 2D array - A * x is a coefficient-wise multiplication // Type expected by wavelets and image write/read functions typedef sopt::Image Image; std::string const input = argc >= 2 ? argv[1] : "cameraman256"; std::string const output = argc == 3 ? argv[2] : "none"; if(argc > 3) { std::cout << "Usage:\n" "$ " << argv[0] << " [input [output]]\n\n" "- input: path to the image to clean (or name of standard SOPT image)\n" "- output: filename pattern for output image\n"; exit(0); } // Set up random numbers for C and C++ auto const seed = std::time(0); std::srand((unsigned int)seed); std::mt19937 mersenne(std::time(0)); // Initializes and sets logger (if compiled with logging) // See set_level function for levels. sopt::logging::initialize(); SOPT_HIGH_LOG("Read input file {}", input); Image const image = sopt::notinstalled::read_standard_tiff(input); SOPT_HIGH_LOG("Initializing sensing operator"); sopt::t_uint nmeasure = 0.33 * image.size(); auto const sampling = sopt::linear_transform(sopt::Sampling(image.size(), nmeasure, mersenne)); SOPT_HIGH_LOG("Initializing wavelets"); auto const wavelet = sopt::wavelets::factory("DB4", 4); auto const psi = sopt::linear_transform(wavelet, image.rows(), image.cols()); SOPT_HIGH_LOG("Computing proximal-ADMM parameters"); Vector const y0 = sampling * Vector::Map(image.data(), image.size()); auto const snr = 30.0; auto const sigma = y0.stableNorm() / std::sqrt(y0.size()) * std::pow(10.0, -(snr / 20.0)); auto const epsilon = std::sqrt(nmeasure + 2 * std::sqrt(y0.size())) * sigma; SOPT_HIGH_LOG("Create dirty vector"); std::normal_distribution<> gaussian_dist(0, sigma); Vector y(y0.size()); for(sopt::t_int i = 0; i < y0.size(); i++) y(i) = y0(i) + gaussian_dist(mersenne); // Write dirty imagte to file if(output != "none") { Vector const dirty = sampling.adjoint() * y; sopt::utilities::write_tiff(Matrix::Map(dirty.data(), image.rows(), image.cols()), "dirty_" + output + ".tiff"); } SOPT_HIGH_LOG("Creating proximal-ADMM Functor"); auto const padmm = sopt::algorithm::ImagingProximalADMM(y) .itermax(500) .gamma(1e-1) .relative_variation(5e-4) .l2ball_proximal_epsilon(epsilon) .tight_frame(false) .l1_proximal_tolerance(1e-2) .l1_proximal_nu(1) .l1_proximal_itermax(50) .l1_proximal_positivity_constraint(true) .l1_proximal_real_constraint(true) .residual_convergence(epsilon * 1.001) .lagrange_update_scale(0.9) .nu(1e0) .Psi(psi) .Phi(sampling); SOPT_HIGH_LOG("Starting proximal-ADMM"); // Alternatively, padmm can be called with a tuple (x, residual) as argument // Here, we default to (Φ^Ty/ν, ΦΦ^Ty/ν - y) auto const diagnostic = padmm(); SOPT_HIGH_LOG("proximal-ADMM returned {}", diagnostic.good); // diagnostic should tell us the function converged // it also contains diagnostic.niters - the number of iterations, and cg_diagnostic - the // diagnostic from the last call to the conjugate gradient. if(not diagnostic.good) throw std::runtime_error("Did not converge!"); SOPT_HIGH_LOG("SOPT-proximal-ADMM converged in {} iterations", diagnostic.niters); if(output != "none") sopt::utilities::write_tiff(Matrix::Map(diagnostic.x.data(), image.rows(), image.cols()), output + ".tiff"); return 0; } sopt-2.0.0/cpp/examples/proximal_admm/reweighted.cc000066400000000000000000000123031277570055300223770ustar00rootroot00000000000000#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include // This header is not part of the installed sopt interface // It is only present in tests #include #include // \min_{x} ||\Psi^Tx||_1 \quad \mbox{s.t.} \quad ||y - Ax||_2 < \epsilon and x \geq 0 int main(int argc, char const **argv) { // Some typedefs for simplicity typedef double Scalar; // Column vector - linear algebra - A * x is a matrix-vector multiplication // type expected by ProximalADMM typedef sopt::Vector Vector; // Matrix - linear algebra - A * x is a matrix-vector multiplication // type expected by ProximalADMM typedef sopt::Matrix Matrix; // Image - 2D array - A * x is a coefficient-wise multiplication // Type expected by wavelets and image write/read functions typedef sopt::Image Image; std::string const input = argc >= 2 ? argv[1] : "cameraman256"; std::string const output = argc == 3 ? argv[2] : "none"; if(argc > 3) { std::cout << "Usage:\n" "$ " << argv[0] << " [input [output]]\n\n" "- input: path to the image to clean (or name of standard SOPT image)\n" "- output: filename pattern for output image\n"; exit(0); } // Set up random numbers for C and C++ auto const seed = std::time(0); std::srand((unsigned int)seed); std::mt19937 mersenne(std::time(0)); // Initializes and sets logger (if compiled with logging) // See set_level function for levels. sopt::logging::initialize(); SOPT_MEDIUM_LOG("Read input file {}", input); Image const image = sopt::notinstalled::read_standard_tiff(input); SOPT_MEDIUM_LOG("Initializing sensing operator"); sopt::t_uint nmeasure = 0.33 * image.size(); auto const sampling = sopt::linear_transform(sopt::Sampling(image.size(), nmeasure, mersenne)); SOPT_MEDIUM_LOG("Initializing wavelets"); sopt::wavelets::SARA const sara{std::make_tuple(std::string{"DB3"}, 1u), std::make_tuple(std::string{"DB1"}, 2u), std::make_tuple(std::string{"DB1"}, 3u)}; auto const psi = sopt::linear_transform(sara, image.rows(), image.cols()); SOPT_MEDIUM_LOG("Computing proximal-ADMM parameters"); Vector const y0 = sampling * Vector::Map(image.data(), image.size()); auto const snr = 30.0; auto const sigma = y0.stableNorm() / std::sqrt(y0.size()) * std::pow(10.0, -(snr / 20.0)); auto const epsilon = std::sqrt(nmeasure + 2 * std::sqrt(y0.size())) * sigma; SOPT_MEDIUM_LOG("Create dirty vector"); std::normal_distribution<> gaussian_dist(0, sigma); Vector y(y0.size()); for(sopt::t_int i = 0; i < y0.size(); i++) y(i) = y0(i) + gaussian_dist(mersenne); // Write dirty imagte to file if(output != "none") { Vector const dirty = sampling.adjoint() * y; sopt::utilities::write_tiff(Matrix::Map(dirty.data(), image.rows(), image.cols()), "dirty_" + output + ".tiff"); } SOPT_MEDIUM_LOG("Creating proximal-ADMM Functor"); auto const padmm = sopt::algorithm::ImagingProximalADMM(y) .itermax(500) .gamma(1e-1) .relative_variation(5e-4) .l2ball_proximal_epsilon(epsilon) .tight_frame(false) .l1_proximal_tolerance(1e-2) .l1_proximal_nu(1) .l1_proximal_itermax(50) .l1_proximal_positivity_constraint(true) .l1_proximal_real_constraint(true) .residual_convergence(epsilon * 1.001) .lagrange_update_scale(0.9) .nu(1e0) .Psi(psi) .Phi(sampling); SOPT_MEDIUM_LOG("Creating the reweighted algorithm"); // positive_quadrant projects the result of PADMM on the positive quadrant. // This follows the reweighted algorithm for SDMM auto const min_delta = sigma * std::sqrt(y.size()) / std::sqrt(8 * image.size()); auto const reweighted = sopt::algorithm::reweighted(padmm).itermax(5).min_delta(min_delta).is_converged( sopt::RelativeVariation(1e-3)); SOPT_MEDIUM_LOG("Starting proximal-ADMM"); // Alternatively, padmm can be called with a tuple (x, residual) as argument // Here, we default to (Φ^Ty/ν, ΦΦ^Ty/ν - y) auto const diagnostic = reweighted(); SOPT_MEDIUM_LOG("proximal-ADMM returned {}", diagnostic.good); SOPT_MEDIUM_LOG("SOPT-proximal-ADMM converged in {} iterations", diagnostic.niters); if(output != "none") sopt::utilities::write_tiff(Matrix::Map(diagnostic.algo.x.data(), image.rows(), image.cols()), output + ".tiff"); return 0; } sopt-2.0.0/cpp/examples/sara.cc000066400000000000000000000021501277570055300163440ustar00rootroot00000000000000#include int main(int, char const **) { // Creates SARA with two wavelets typedef std::tuple t_i; sopt::wavelets::SARA sara{t_i{"DB4", 5}, t_i{"DB8", 2}}; // Then another one for good measure sara.emplace_back("DB3", 7); // Creates a random signal sopt::Image input = sopt::Image::Random(128, 128); // Now gets its coefficients auto coefficients = sara.direct(input); // And transform back. We pass a pre-defined matrix explicitly to illustrate that API. // But we could just store the return value as above. sopt::Image recover; // This matrix will be resized if necessary sara.indirect(coefficients, recover); // Check the reconstruction is corrrect if(not input.isApprox(recover)) throw std::exception(); // The coefficient for each wavelet basis is stored alongs columns: sopt::Image const DB3_coeffs = sara[2].direct(input) / std::sqrt(sara.size()); if(not coefficients.rightCols(input.cols()).isApprox(DB3_coeffs)) throw std::exception(); return 0; } sopt-2.0.0/cpp/examples/sdmm/000077500000000000000000000000001277570055300160515ustar00rootroot00000000000000sopt-2.0.0/cpp/examples/sdmm/CMakeLists.txt000066400000000000000000000002751277570055300206150ustar00rootroot00000000000000add_example(euclidian_norm LIBRARIES sopt LABELS sdmm) add_example(inpainting LIBRARIES sopt tools_for_tests LABELS sdmm) add_example(reweighted LIBRARIES sopt tools_for_tests LABELS sdmm) sopt-2.0.0/cpp/examples/sdmm/euclidian_norm.cc000066400000000000000000000076721277570055300213640ustar00rootroot00000000000000#include #include #include // We will minimize ||L_0 x - x_0|| + ||L_1 x - x_1||, ||.|| the euclidian norm int main(int, char const **) { // Initializes and sets logger (if compiled with logging) // See set_level function for levels. sopt::logging::initialize(); // Some typedefs for simplicity typedef sopt::t_complex t_Scalar; typedef sopt::Vector t_Vector; typedef sopt::Matrix t_Matrix; // Creates the transformation matrices auto const N = 10; t_Matrix const L0 = t_Matrix::Random(N, N) * 2; t_Matrix const L1 = t_Matrix::Random(N, N) * 4; // L1_direct and L1_adjoint are used to demonstrate that we can define L_i in SDMM both directly // as matrices, or as a pair of functions that apply a linear operator and its transpose. auto L1_direct = [&L1](t_Vector &out, t_Vector const &input) { out = L1 * input; }; auto L1_adjoint = [&L1](t_Vector &out, t_Vector const &input) { out = L1.adjoint() * input; }; // Creates the target vectors t_Vector const target0 = t_Vector::Random(N); t_Vector const target1 = t_Vector::Random(N); // Creates the resulting proximal // In practice g_0 and g_1 are any functions with the signature // void(t_Vector &output, t_Vector::Scalar gamma, t_Vector const &input) auto prox_g0 = sopt::proximal::translate(sopt::proximal::EuclidianNorm(), -target0); auto prox_g1 = sopt::proximal::translate(sopt::proximal::EuclidianNorm(), -target1); // This function is called at every iteration. It should return true when convergence is achieved. // Otherwise the convex optimizer will trudge on until the requisite number of iterations have // been achieved. // It takes the convex minimizer and the current candidate output vector as arguments. // The example below assumes convergence when the candidate vector does not change anymore. std::shared_ptr previous; auto relative = [&previous](t_Vector const &candidate) { if(not previous) { previous = std::make_shared(candidate); return false; } auto const norm = (*previous - candidate).stableNorm(); SOPT_TRACE(" - Checking convergence {}", norm); auto const result = norm < 1e-8 * candidate.size(); *previous = candidate; return result; }; // Now we can create the sdmm convex minimizer // Its parameters are set by calling member functions with appropriate names. auto sdmm = sopt::algorithm::SDMM() .itermax(500) // maximum number of iterations .gamma(1) .conjugate_gradient(std::numeric_limits::max(), 1e-12) .is_converged(relative) // Any number of (proximal g_i, L_i) pairs can be added // L_i can be a matrix .append(prox_g0, L0) // L_i can be a pair of functions applying a linear transform and its adjoint .append(prox_g1, L1_direct, L1_adjoint); t_Vector result; t_Vector const input = t_Vector::Random(N); auto const diagnostic = sdmm(result, input); // diagnostic should tell us the function converged // it also contains diagnostic.niters - the number of iterations, and cg_diagnostic - the // diagnostic from the last call to the conjugate gradient. if(not diagnostic.good) throw std::runtime_error("Did not converge!"); // Lets test we are at a minimum by recreating the objective function // and checking that stepping in any direction raises its value auto const objective = [&target0, &target1, &L0, &L1](t_Vector const &x) { return (L0 * x - target0).stableNorm() + (L1 * x - target1).stableNorm(); }; auto const minimum = objective(result); for(int i(0); i < N; ++i) { t_Vector epsilon = t_Vector::Zero(N); epsilon(i) = 1e-4; auto const at_x_plus_epsilon = objective(input + epsilon); if(minimum >= at_x_plus_epsilon) throw std::runtime_error("That's no minimum!"); } return 0; } sopt-2.0.0/cpp/examples/sdmm/inpainting.cc000066400000000000000000000117201277570055300205210ustar00rootroot00000000000000#include #include #include #include #include #include #include #include #include #include #include #include #include #include // This header is not part of the installed sopt interface // It is only present in tests #include #include // \min_{x} ||\Psi^Tx||_1 \quad \mbox{s.t.} \quad ||y - Ax||_2 < \epsilon and x \geq 0 int main(int argc, char const **argv) { // Some typedefs for simplicity typedef double Scalar; // Column vector - linear algebra - A * x is a matrix-vector multiplication // type expected by SDMM typedef sopt::Vector Vector; // Matrix - linear algebra - A * x is a matrix-vector multiplication // type expected by SDMM typedef sopt::Matrix Matrix; // Image - 2D array - A * x is a coefficient-wise multiplication // Type expected by wavelets and image write/read functions typedef sopt::Image Image; std::string const input = argc >= 2 ? argv[1] : "cameraman256"; std::string const output = argc == 3 ? argv[2] : "none"; if(argc > 3) { std::cout << "Usage:\n" "$ " << argv[0] << " [input [output]]\n\n" "- input: path to the image to clean (or name of standard SOPT image)\n" "- output: filename pattern for output image\n"; exit(0); } // Set up random numbers for C and C++ auto const seed = std::time(0); std::srand((unsigned int)seed); std::mt19937 mersenne(std::time(0)); // Initializes and sets logger (if compiled with logging) // See set_level function for levels. sopt::logging::initialize(); SOPT_HIGH_LOG("Read input file {}", input); Image const image = sopt::notinstalled::read_standard_tiff(input); SOPT_HIGH_LOG("Initializing sensing operator"); sopt::t_uint nmeasure = 0.33 * image.size(); auto const sampling = sopt::linear_transform(sopt::Sampling(image.size(), nmeasure, mersenne)); SOPT_HIGH_LOG("Initializing wavelets"); auto const wavelet = sopt::wavelets::factory("DB4", 4); auto const psi = sopt::linear_transform(wavelet, image.rows(), image.cols()); SOPT_HIGH_LOG("Computing sdmm parameters"); Vector const y0 = sampling * Vector::Map(image.data(), image.size()); auto const snr = 30.0; auto const sigma = y0.stableNorm() / std::sqrt(y0.size()) * std::pow(10.0, -(snr / 20.0)); auto const epsilon = std::sqrt(nmeasure + 2 * std::sqrt(y0.size())) * sigma; SOPT_HIGH_LOG("Create dirty vector"); std::normal_distribution<> gaussian_dist(0, sigma); Vector y(y0.size()); for(sopt::t_int i = 0; i < y0.size(); i++) y(i) = y0(i) + gaussian_dist(mersenne); // Write dirty imagte to file if(output != "none") { Vector const dirty = sampling.adjoint() * y; sopt::utilities::write_tiff(Matrix::Map(dirty.data(), image.rows(), image.cols()), "dirty_" + output + ".tiff"); } SOPT_HIGH_LOG("Initializing convergence function"); auto relvar = sopt::RelativeVariation(5e-2); auto convergence = [&y, &sampling, &psi, &relvar](sopt::Vector const &x) -> bool { SOPT_MEDIUM_LOG("||x - y||_2: {}", (y - sampling * x).stableNorm()); SOPT_MEDIUM_LOG("||Psi^Tx||_1: {}", sopt::l1_norm(psi.adjoint() * x)); SOPT_MEDIUM_LOG("||abs(x) - x||_2: {}", (x.array().abs().matrix() - x).stableNorm()); return relvar(x); }; SOPT_HIGH_LOG("Creating SDMM Functor"); auto const sdmm = sopt::algorithm::SDMM() .itermax(3000) .gamma(0.1) .conjugate_gradient(200, 1e-8) .is_converged(convergence) // Any number of (proximal g_i, L_i) pairs can be added // ||Psi^dagger x||_1 .append(sopt::proximal::l1_norm, psi.adjoint(), psi) // ||y - A x|| < epsilon .append(sopt::proximal::translate(sopt::proximal::L2Ball(epsilon), -y), sampling) // x in positive quadrant .append(sopt::proximal::positive_quadrant); SOPT_HIGH_LOG("Allocating result vector"); Vector result(image.size()); SOPT_HIGH_LOG("Starting SDMM"); auto const diagnostic = sdmm(result, Vector::Zero(image.size())); SOPT_HIGH_LOG("SDMM returned {}", diagnostic.good); // diagnostic should tell us the function converged // it also contains diagnostic.niters - the number of iterations, and cg_diagnostic - the // diagnostic from the last call to the conjugate gradient. if(not diagnostic.good) throw std::runtime_error("Did not converge!"); SOPT_HIGH_LOG("SOPT-SDMM converged in {} iterations", diagnostic.niters); if(output != "none") sopt::utilities::write_tiff(Matrix::Map(result.data(), image.rows(), image.cols()), output + ".tiff"); return 0; } sopt-2.0.0/cpp/examples/sdmm/reweighted.cc000066400000000000000000000144341277570055300205150ustar00rootroot00000000000000#include #include #include #include #include #include #include #include #include #include #include #include #include #include #include #include // This header is not part of the installed sopt interface // It is only present in tests #include #include // \min_{x} ||W_j\Psi^Tx||_1 \quad \mbox{s.t.} \quad ||y - Ax||_2 < \epsilon and x \geq 0 // with W_j = ||\Psi^Tx_{j-1}||_1 // By iterating this algorithm, we can approximate L0 from L1. int main(int argc, char const **argv) { // Some typedefs for simplicity typedef double Scalar; // Column vector - linear algebra - A * x is a matrix-vector multiplication // type expected by SDMM typedef sopt::Vector Vector; // Matrix - linear algebra - A * x is a matrix-vector multiplication // type expected by SDMM typedef sopt::Matrix Matrix; // Image - 2D array - A * x is a coefficient-wise multiplication // Type expected by wavelets and image write/read functions typedef sopt::Image Image; std::string const input = argc >= 2 ? argv[1] : "cameraman256"; std::string const output = argc == 3 ? argv[2] : "none"; if(argc > 3) { std::cout << "Usage:\n" "$ " << argv[0] << " [input [output]]\n\n" "- input: path to the image to clean (or name of standard SOPT image)\n" "- output: filename pattern for output image\n"; exit(0); } // Set up random numbers for C and C++ auto const seed = std::time(0); std::srand((unsigned int)seed); std::mt19937 mersenne(std::time(0)); // Initializes and sets logger (if compiled with logging) // See set_level function for levels. sopt::logging::initialize(); SOPT_HIGH_LOG("Read input file {}", input); Image const image = sopt::notinstalled::read_standard_tiff(input); SOPT_HIGH_LOG("Initializing sensing operator"); sopt::t_uint nmeasure = 0.33 * image.size(); auto const sampling = sopt::linear_transform(sopt::Sampling(image.size(), nmeasure, mersenne)); SOPT_HIGH_LOG("Initializing wavelets"); auto const wavelet = sopt::wavelets::factory("DB4", 4); auto const psi = sopt::linear_transform(wavelet, image.rows(), image.cols()); SOPT_HIGH_LOG("Computing sdmm parameters"); Vector const y0 = sampling * Vector::Map(image.data(), image.size()); auto const snr = 30.0; auto const sigma = y0.stableNorm() / std::sqrt(y0.size()) * std::pow(10.0, -(snr / 20.0)); auto const epsilon = std::sqrt(nmeasure + 2 * std::sqrt(y0.size())) * sigma; SOPT_HIGH_LOG("Create dirty vector"); std::normal_distribution<> gaussian_dist(0, sigma); Vector y(y0.size()); for(sopt::t_int i = 0; i < y0.size(); i++) y(i) = y0(i) + gaussian_dist(mersenne); // Write dirty imagte to file if(output != "none") { Vector const dirty = sampling.adjoint() * y; sopt::utilities::write_tiff(Matrix::Map(dirty.data(), image.rows(), image.cols()), "dirty_" + output + ".tiff"); } SOPT_HIGH_LOG("Initializing convergence function"); auto relvar = sopt::RelativeVariation(5e-2); auto convergence = [&y, &sampling, &psi, &relvar](sopt::Vector const &x) -> bool { SOPT_MEDIUM_LOG("||x - y||_2: {}", (y - sampling * x).stableNorm()); SOPT_MEDIUM_LOG("||Psi^Tx||_1: {}", sopt::l1_norm(psi.adjoint() * x)); SOPT_MEDIUM_LOG("||abs(x) - x||_2: {}", (x.array().abs().matrix() - x).stableNorm()); return relvar(x); }; SOPT_HIGH_LOG("Creating SDMM Functor"); auto const sdmm = sopt::algorithm::SDMM() .itermax(3000) .gamma(0.1) .conjugate_gradient(200, 1e-8) .is_converged(convergence) // Any number of (proximal g_i, L_i) pairs can be added // ||Psi^dagger x||_1 .append(sopt::proximal::l1_norm, psi.adjoint(), psi) // ||y - A x|| < epsilon .append(sopt::proximal::translate(sopt::proximal::L2Ball(epsilon), -y), sampling) // x in positive quadrant .append(sopt::proximal::positive_quadrant); SOPT_HIGH_LOG("Creating the reweighted algorithm"); // positive_quadrant projects the result of SDMM on the positive quadrant. // This follows the reweighted algorithm in the original C implementation. auto const posq = positive_quadrant(sdmm); typedef std::remove_const::type t_PosQuadSDMM; auto const min_delta = sigma * std::sqrt(y.size()) / std::sqrt(8 * image.size()); // Sets weight after each sdmm iteration. // In practice, this means replacing the proximal of the l1 objective function. auto set_weights = [](t_PosQuadSDMM &sdmm, Vector const &weights) { sdmm.algorithm().proximals(0) = [weights](Vector &out, Scalar gamma, Vector const &x) { out = sopt::soft_threshhold(x, gamma * weights); }; }; auto call_PsiT = [&psi](t_PosQuadSDMM const &, Vector const &x) -> Vector { return psi.adjoint() * x; }; auto const reweighted = sopt::algorithm::reweighted(posq, set_weights, call_PsiT) .itermax(5) .min_delta(min_delta) .is_converged(sopt::RelativeVariation(1e-3)); SOPT_HIGH_LOG("Computing warm-start SDMM"); auto warm_start = sdmm(Vector::Zero(image.size())); warm_start.x = sopt::positive_quadrant(warm_start.x); SOPT_HIGH_LOG("SDMM returned {}", warm_start.good); SOPT_HIGH_LOG("Computing warm-start SDMM"); auto const result = reweighted(warm_start); // result should tell us the function converged // it also contains result.niters - the number of iterations, and cg_diagnostic - the // result from the last call to the conjugate gradient. if(not result.good) throw std::runtime_error("Did not converge!"); SOPT_HIGH_LOG("SOPT-SDMM converged in {} iterations", result.niters); if(output != "none") sopt::utilities::write_tiff(Matrix::Map(result.algo.x.data(), image.rows(), image.cols()), output + ".tiff"); return 0; } sopt-2.0.0/cpp/examples/soft_threshhold.cc000066400000000000000000000005641277570055300206240ustar00rootroot00000000000000#include #include int main(int, char const **) { sopt::Array<> input(6); input << 1e1, 2e1, 3e1, 4e1, 1e4, 2e4; if(not(sopt::soft_threshhold(input, 2.5e1).head(2).array() < 1e-8).all()) throw std::exception(); if(not(sopt::soft_threshhold(input, 2.5e1).tail(4).array() > 1e-8).all()) throw std::exception(); return 0; } sopt-2.0.0/cpp/examples/wavelets.cc000066400000000000000000000022411277570055300172510ustar00rootroot00000000000000#include int main(int, char const **) { // Creates Daubechies 4 wavelet, with 5 levels auto const wavelets = sopt::wavelets::factory("DB4", 5); // Creates a random signal sopt::Image input = sopt::Image::Random(128, 128); // Now gets its coefficients auto coefficients = wavelets.direct(input); // And transform back auto recover = wavelets.indirect(coefficients); // Check the reconstruction is corrrect if(not input.isApprox(recover)) throw std::exception(); // To save on memory allocation we could also use a pre-allocated matrix; coefficients.fill(0); recover.fill(0); wavelets.direct(coefficients, input); wavelets.indirect(coefficients, recover); if(not input.isApprox(recover)) throw std::exception(); // Finally, it is possible to use expressions // For instance, we can do a 1d transform on a single row wavelets.direct(coefficients.row(2).transpose(), input.row(2).transpose() * 2); wavelets.indirect(coefficients.row(2).transpose(), recover.row(2).transpose()); if(not input.row(2).isApprox(recover.row(2) * 0.5)) throw std::exception(); return 0; } sopt-2.0.0/cpp/regressions/000077500000000000000000000000001277570055300156365ustar00rootroot00000000000000sopt-2.0.0/cpp/regressions/CMakeLists.txt000066400000000000000000000004561277570055300204030ustar00rootroot00000000000000add_regression(sdmm_inpainting LABELS sdmm LIBRARIES tools_for_tests sopt) add_regression(l1_proximal LABELS sdmm LIBRARIES sopt tools_for_tests sopt) add_regression(padmm_inpainting LABELS sdmm LIBRARIES tools_for_tests sopt) add_regression(reweighted_sdmm LABELS sdmm LIBRARIES tools_for_tests sopt) sopt-2.0.0/cpp/regressions/l1_proximal.cc000066400000000000000000000102611277570055300203740ustar00rootroot00000000000000#include #include #include #include #include #include #include "sopt/l1_proximal.h" #include "sopt_prox.h" #include "tools_for_tests/cdata.h" std::random_device rd; std::default_random_engine rengine(rd()); template sopt::Matrix concatenated_permutations(sopt::t_uint i, sopt::t_uint j) { std::vector cols(j); std::iota(cols.begin(), cols.end(), 0); std::shuffle(cols.begin(), cols.end(), rengine); assert(j % i == 0); auto const N = j / i; auto const elem = 1e0 / std::sqrt(static_cast::type>(N)); sopt::Matrix result = sopt::Matrix::Zero(i, cols.size()); for(typename sopt::Matrix::Index k(0); k < result.cols(); ++k) result(cols[k] / N, k) = elem; return result; } template sopt::Vector c_proximal(sopt::proximal::L1 const &l1, typename sopt::real_type::type gamma, sopt::Vector const &x, bool pos = false, bool tf = false) { using namespace sopt; typedef typename sopt::real_type::type Real; int const nr = (l1.Psi().adjoint() * x).size(); Vector weights = l1.weights(); if(l1.weights().size() == 1) weights = l1.weights()(1) * Vector::Ones(nr); CData const psi_data{nr, x.size(), l1.Psi(), 0, 0}; sopt_prox_l1param params = { 0, // verbosity static_cast(l1.itermax()), // max iter l1.tolerance(), // relative change l1.nu(), // nu tf ? 1 : 0, // tight frame pos ? 1 : 0 // Positivity constraints }; Vector xin = x; Vector result = Vector::Zero(x.size()); Vector dummy = Vector::Zero(nr); Vector sol = Vector::Zero(x.size()); Vector u = Vector::Zero(nr); Vector v = Vector::Zero(nr); sopt_prox_l1(static_cast(result.data()), static_cast(xin.data()), x.size(), nr, &direct_transform, (void **)&psi_data, &adjoint_transform, (void **)&psi_data, weights.data(), gamma, std::is_same::value ? 1 : 0, params, static_cast(dummy.data()), static_cast(sol.data()), static_cast(u.data()), static_cast(v.data())); return result; } TEST_CASE("Compare L1 proximals", "") { using namespace sopt; typedef std::complex Scalar; auto l1 = proximal::L1(); Vector const input = Vector::Random(15); SECTION("Check tight frame") { auto const Psi = concatenated_permutations(input.size(), input.size() * 10); auto const weights = Vector::Random(Psi.cols()).array().abs().matrix().eval(); auto const gamma = 1e0 / static_cast(weights.size()); l1.Psi(Psi).weights(weights).tolerance(1e-12).itermax(10000); auto const cpp = l1.tight_frame(gamma, input); auto const c = c_proximal(l1, gamma, input, false, true); CHECK(cpp.isApprox(c)); } SECTION("Check general case") { auto const Psi = Matrix::Random(input.size(), input.size() * 10).eval(); auto const weights = Vector::Random(Psi.cols()).array().abs().matrix().eval(); auto const gamma = 1e0 / static_cast(weights.size()); SECTION("No constraints") { for(auto i : {2, 10, 25, 500, 1000}) { l1.Psi(Psi).weights(weights).tolerance(1e-12).itermax(i - 1); auto const c = c_proximal(l1, gamma, input, false, false); auto const cpp = l1.itermax(i)(gamma, input); CHECK(cpp.proximal.isApprox(c, 1e-8 * c.array().abs().maxCoeff())); } } SECTION("Positivity constraints") { for(auto i : {2, 10, 25, 500, 1000}) { l1.Psi(Psi).weights(weights).tolerance(1e-12).itermax(i - 1).positivity_constraint(true); auto const c = c_proximal(l1, gamma, input, true, false); auto const cpp = l1.itermax(i)(gamma, input); CAPTURE(c.transpose()); CAPTURE(cpp.proximal.transpose()); CHECK(cpp.proximal.isApprox(c, 1e-8 * c.array().abs().maxCoeff())); } } } } sopt-2.0.0/cpp/regressions/padmm_inpainting.cc000066400000000000000000000101051277570055300214600ustar00rootroot00000000000000#include #include #include #include #include "sopt/imaging_padmm.h" #include "sopt/logging.h" #include "sopt/proximal.h" #include "sopt/relative_variation.h" #include "sopt/sampling.h" #include "sopt/wavelets.h" #include "tools_for_tests/cdata.h" #include "tools_for_tests/directories.h" #include "tools_for_tests/inpainting.h" #include "tools_for_tests/tiffwrappers.h" #include "sopt_l1.h" typedef double Scalar; typedef sopt::Vector t_Vector; typedef sopt::Matrix t_Matrix; sopt::algorithm::ImagingProximalADMM create_admm(sopt::LinearTransform const &phi, sopt::LinearTransform const &psi, sopt_l1_param_padmm const ¶ms, t_Vector const &target) { using namespace sopt; return algorithm::ImagingProximalADMM(target) .itermax(params.max_iter + 1) .gamma(params.gamma) .relative_variation(params.rel_obj) .l2ball_proximal_epsilon(params.epsilon) .tight_frame(params.paraml1.tight == 1) .l1_proximal_tolerance(params.paraml1.rel_obj) .l1_proximal_nu(params.paraml1.nu) .l1_proximal_itermax(params.paraml1.max_iter) .l1_proximal_positivity_constraint(params.paraml1.pos == 1) .l1_proximal_real_constraint(params.real_out) .residual_convergence(params.epsilon * params.epsilon_tol_scale) .lagrange_update_scale(params.lagrange_update_scale) .nu(params.nu) .Psi(psi) .Phi(phi) // just for show, 1 is the default value, so these calls do not do anything .l2ball_proximal_weights(Vector::Ones(1)) .l1_proximal_weights(Vector::Ones(1)); } TEST_CASE("Compare ADMM C++ and C", "") { using namespace sopt; // Read image and create target vector y Image<> const image = notinstalled::read_standard_tiff("cameraman256"); t_uint const nmeasure = 0.5 * image.size(); extern std::unique_ptr mersenne; auto const sampling = linear_transform(Sampling(image.size(), nmeasure, *mersenne)); auto const y = dirty(sampling, image, *mersenne); sopt_l1_param_padmm params{ 0, // verbosity 200, // max iter 0.1, // gamma 0.0005, // relative change epsilon(sampling, image), // radius of the l2ball 1, // real out 1, // real meas { 0, // verbose = 1; 8, // max_iter = 50; 0.01, // rel_obj = 0.01; 1, // nu = 1; 0, // tight = 0; 1, // pos = 1; }, 1.001, // epsilon tol scale 0.9, // lagrange_update_scale 1.0, // nu }; // Create c++ SDMM auto const wavelet = wavelets::factory("DB2", 2); auto const psi = linear_transform(wavelet, image.rows(), image.cols()); // Create C bindings for C++ operators CData const sampling_data{image.size(), y.size(), sampling, 0, 0}; CData const psi_data{image.size(), image.size(), psi, 0, 0}; // Try increasing number of iterations and check output of c and c++ algorithms are the same for(t_uint i : {1, 5, 10}) { SECTION(fmt::format("With {} iterations", i)) { sopt_l1_param_padmm c_params = params; c_params.max_iter = i; auto admm = ::create_admm(sampling, psi, c_params, y); auto const cpp = admm(); t_Vector c = t_Vector::Zero(image.size()); t_Vector l1_weights = t_Vector::Ones(image.size()); t_Vector l2_weights = t_Vector::Ones((sampling * image).size()); sopt_l1_solver_padmm( (void *)c.data(), c.size(), &direct_transform, (void **)&sampling_data, &adjoint_transform, (void **)&sampling_data, &direct_transform, (void **)&psi_data, &adjoint_transform, (void **)&psi_data, // synthesis c.size(), (void *)y.data(), y.size(), l1_weights.data(), l2_weights.data(), c_params); CAPTURE(cpp.x.head(5).transpose()); CAPTURE(c.head(5).transpose()); CHECK(cpp.x.isApprox(c)); } }; } sopt-2.0.0/cpp/regressions/reweighted_sdmm.cc000066400000000000000000000117361277570055300213240ustar00rootroot00000000000000#include #include #include #include #include #include "sopt/logging.h" #include "sopt/positive_quadrant.h" #include "sopt/proximal.h" #include "sopt/relative_variation.h" #include "sopt/reweighted.h" #include "sopt/sampling.h" #include "sopt/sdmm.h" #include "sopt/wavelets.h" #include "tools_for_tests/cdata.h" #include "tools_for_tests/directories.h" #include "tools_for_tests/inpainting.h" #include "tools_for_tests/tiffwrappers.h" #include "sopt_l1.h" typedef double Scalar; typedef sopt::Vector t_Vector; typedef sopt::Matrix t_Matrix; sopt::algorithm::SDMM create_sdmm(sopt::LinearTransform const &sampling, sopt::LinearTransform const &psi, t_Vector const &y, sopt_l1_sdmmparam const ¶ms) { using namespace sopt; return algorithm::SDMM() .itermax(params.max_iter) .gamma(params.gamma) .conjugate_gradient(params.cg_max_iter, params.cg_tol) .is_converged(RelativeVariation(params.rel_obj)) .append(proximal::l1_norm, psi.adjoint(), psi) .append(proximal::translate(proximal::L2Ball(params.epsilon), -y), sampling) .append(proximal::positive_quadrant); } sopt::algorithm::Reweighted>> create_rwsdmm(sopt::algorithm::SDMM const &sdmm, sopt::LinearTransform const &psi, sopt_l1_rwparam const ¶ms) { using namespace sopt::algorithm; auto set_weights = [](PositiveQuadrant> &sdmm, t_Vector const &weights) { sdmm.algorithm().proximals(0) = [weights](t_Vector &out, Scalar gamma, t_Vector const &x) { out = sopt::soft_threshhold(x, gamma * weights); }; }; auto call_PsiT = [&psi](PositiveQuadrant> const &, t_Vector const &x) -> t_Vector { return psi.adjoint() * x; }; auto const pq = positive_quadrant(sdmm); return reweighted(pq, set_weights, call_PsiT) .itermax(params.max_iter) .min_delta(params.sigma) .is_converged(sopt::RelativeVariation(params.rel_var)); } TEST_CASE("Compare Reweighted SDMMS", "") { using namespace sopt; // Read image and create target vector y Image<> const image = notinstalled::read_standard_tiff("cameraman256"); t_uint const nmeasure = 0.5 * image.size(); extern std::unique_ptr mersenne; auto const sampling = linear_transform(Sampling(image.size(), nmeasure, *mersenne)); auto const y = dirty(sampling, image, *mersenne); sopt_l1_sdmmparam params = { 10, // verbosity 2, // max iter 0.1, // gamma 0.01, // relative change epsilon(sampling, image), // radius of the l2ball 1e-3, // Relative tolerance on epsilon std::is_same::type>::value ? 1 : 0, // real data 200, // cg max iter 1e-8, // cg tol }; sopt_l1_rwparam rwparam = { 0, 5, 0.001, sigma(sampling, image) * std::sqrt(y.size()) / std::sqrt(8 * image.size()), 0}; // Create c++ SDMM auto const wavelet = wavelets::factory("DB2", 2); auto const psi = linear_transform(wavelet, image.rows(), image.cols()); // Create C bindings for C++ operators CData const sampling_data{image.size(), y.size(), sampling, 0, 0}; CData const psi_data{image.size(), image.size(), psi, 0, 0}; auto sdmm = ::create_sdmm(sampling, psi, y, params); auto warm_start = sdmm(t_Vector::Zero(image.size())); warm_start.x = positive_quadrant(warm_start.x); // Try increasing number of iterations and check output of c and c++ algorithms are the same for(t_uint i : {0, 1, 2, 5}) { SECTION(fmt::format("With {} iterations", i)) { sopt_l1_rwparam c_params = rwparam; c_params.max_iter = i; auto rwsdmm = ::create_rwsdmm(sdmm, psi, c_params); auto const cpp = rwsdmm(warm_start); CHECK(image.size() == (psi.adjoint() * t_Vector::Zero(image.size())).size()); t_Vector c = t_Vector::Zero(image.size()); t_Vector weights = t_Vector::Ones((psi.adjoint() * c).size()); sopt_l1_rwsdmm((void *)c.data(), c.size(), &direct_transform, (void **)&sampling_data, &adjoint_transform, (void **)&sampling_data, &direct_transform, (void **)&psi_data, &adjoint_transform, (void **)&psi_data, // synthesis (psi.adjoint() * c).size(), (void *)y.data(), y.size(), params, c_params); CAPTURE(cpp.algo.x.head(25).tail(3).transpose()); CAPTURE(c.head(25).tail(3).transpose()); CAPTURE((cpp.algo.x - c).head(25).tail(3).transpose()); CHECK(cpp.algo.x.isApprox(c)); } }; } sopt-2.0.0/cpp/regressions/sdmm_inpainting.cc000066400000000000000000000076711277570055300213400ustar00rootroot00000000000000#include #include #include #include #include #include "sopt/logging.h" #include "sopt/proximal.h" #include "sopt/relative_variation.h" #include "sopt/sampling.h" #include "sopt/sdmm.h" #include "sopt/wavelets.h" #include "tools_for_tests/cdata.h" #include "tools_for_tests/directories.h" #include "tools_for_tests/inpainting.h" #include "tools_for_tests/tiffwrappers.h" #include "sopt_l1.h" typedef double Scalar; typedef sopt::Vector t_Vector; typedef sopt::Matrix t_Matrix; sopt::algorithm::SDMM create_sdmm(sopt::LinearTransform const &sampling, sopt::LinearTransform const &psi, t_Vector const &y, t_Vector const &weights, sopt_l1_sdmmparam const ¶ms) { using namespace sopt; auto weighted_l1_norm = [&weights](t_Vector &out, real_type::type gamma, t_Vector const &x) { assert(gamma > 1e-12); assert((weights.array() > 1e-12).all()); out = soft_threshhold(x, gamma * weights); }; return algorithm::SDMM() .itermax(params.max_iter) .gamma(params.gamma) .conjugate_gradient(params.cg_max_iter, params.cg_tol) .is_converged(RelativeVariation(params.rel_obj)) .append(weighted_l1_norm, psi.adjoint(), psi) .append(proximal::translate(proximal::L2Ball(params.epsilon), -y), sampling) .append(proximal::positive_quadrant); } TEST_CASE("Compare SDMMS", "") { using namespace sopt; // Read image and create target vector y Image<> const image = notinstalled::read_standard_tiff("cameraman256"); t_uint const nmeasure = 0.5 * image.size(); extern std::unique_ptr mersenne; auto const sampling = linear_transform(Sampling(image.size(), nmeasure, *mersenne)); auto const y = dirty(sampling, image, *mersenne); sopt_l1_sdmmparam params = { 0, // verbosity 50, // max iter 0.1, // gamma 0.01, // relative change epsilon(sampling, image), // radius of the l2ball 1e-3, // Relative tolerance on epsilon std::is_same::type>::value ? 1 : 0, // real data 200, // cg max iter 1e-8, // cg tol }; // Create c++ SDMM auto const wavelet = wavelets::factory("DB2", 2); auto const psi = linear_transform(wavelet, image.rows(), image.cols()); // Create C bindings for C++ operators CData const sampling_data{image.size(), y.size(), sampling, 0, 0}; CData const psi_data{image.size(), image.size(), psi, 0, 0}; t_Vector const weights = t_Vector::Random((psi * t_Vector::Zero(image.size())).size()).array().abs(); // Try increasing number of iterations and check output of c and c++ algorithms are the same for(t_uint i : {1, 2, 5, 10}) { SECTION(fmt::format("With {} iterations", i)) { t_Vector const input = t_Vector::Random(image.size()); sopt_l1_sdmmparam c_params = params; c_params.max_iter = i; auto sdmm = ::create_sdmm(sampling, psi, y, weights, c_params); t_Vector cpp(image.size()); sdmm(cpp, positive_quadrant(input)); t_Vector c = input; sopt_l1_sdmm((void *)c.data(), c.size(), &direct_transform, (void **)&sampling_data, &adjoint_transform, (void **)&sampling_data, &direct_transform, (void **)&psi_data, &adjoint_transform, (void **)&psi_data, // synthesis weights.size(), (void *)y.data(), y.size(), const_cast(weights.data()), c_params); CHECK(cpp.isApprox(c)); } }; } sopt-2.0.0/cpp/sopt/000077500000000000000000000000001277570055300142605ustar00rootroot00000000000000sopt-2.0.0/cpp/sopt/CMakeLists.txt000066400000000000000000000043611277570055300170240ustar00rootroot00000000000000# list of headers set(headers imaging_padmm.h logging.disabled.h maths.h proximal.h relative_variation.h sdmm.h wavelets.h conjugate_gradient.h l1_proximal.h logging.enabled.h padmm.h proximal_expression.h reweighted.h types.h wrapper.h exception.h linear_transform.h logging.h positive_quadrant.h real_type.h sampling.h ${PROJECT_BINARY_DIR}/include/sopt/config.h ) set(wavelet_headers wavelets/direct.h wavelets/indirect.h wavelets/innards.impl.h wavelets/sara.h wavelets/wavelet_data.h wavelets/wavelets.h ) set(sources wavelets/wavelets.cc wavelets/wavelet_data.cc) if(TIFF_FOUND) list(APPEND sources utilities.cc) list(APPEND headers utilities.h) endif() add_library(sopt SHARED ${sources}) add_dependencies(sopt lookup_dependencies) set(version "${Sopt_VERSION_MAJOR}.${Sopt_VERSION_MINOR}.${Sopt_VERSION_PATCH}") set_target_properties(sopt PROPERTIES VERSION ${version} SOVERSION ${version}) target_include_directories(sopt PUBLIC $ $ $) if(TIFF_FOUND) target_link_libraries(sopt ${TIFF_LIBRARY}) target_include_directories(sopt SYSTEM PUBLIC ${TIFF_INCLUDE_DIR}) endif() # Add spdlog as direct dependency if not downloaded if(SPDLOG_INCLUDE_DIR AND NOT spdlog_BUILT_AS_EXTERNAL_PROJECT) target_include_directories(sopt SYSTEM PUBLIC ${SPDLOG_INCLUDE_DIR}) elseif(SPDLOG_INCLUDE_DIR AND spdlog_BUILT_AS_EXTERNAL_PROJECT) target_include_directories(sopt SYSTEM PUBLIC $) endif() # Add eigen as direct dependency if not downloaded if(EIGEN3_INCLUDE_DIR AND NOT Eigen3_BUILT_AS_EXTERNAL_PROJECT) target_include_directories(sopt SYSTEM PUBLIC ${EIGEN3_INCLUDE_DIR}) elseif(EIGEN3_INCLUDE_DIR AND Eigen3_BUILT_AS_EXTERNAL_PROJECT) target_include_directories(sopt SYSTEM PUBLIC $) endif() if(TARGET openmp::openmp) target_link_libraries(sopt openmp::openmp) endif() install(FILES ${headers} DESTINATION include/sopt) install(FILES ${wavelet_headers} DESTINATION include/sopt/wavelets) install(TARGETS sopt EXPORT SoptCPPTargets DESTINATION share/cmake/sopt LIBRARY DESTINATION lib ARCHIVE DESTINATION lib INCLUDES DESTINATION include ) sopt-2.0.0/cpp/sopt/config.in.h000066400000000000000000000023271277570055300163070ustar00rootroot00000000000000#ifndef SOPT_CPP_CONFIG_H #define SOPT_CPP_CONFIG_H //! Problems with using and constructors #cmakedefine SOPT_HAS_USING #ifndef SOPT_HAS_USING #define SOPT_HAS_NOT_USING #endif //! True if using OPENMP #cmakedefine SOPT_OPENMP //! Macro to start logging or not #cmakedefine SOPT_DO_LOGGING #include #include namespace sopt { //! Returns library version inline std::string version() { return "@Sopt_VERSION@"; } //! Returns library version inline std::tuple version_tuple() { return std::tuple( @Sopt_VERSION_MAJOR@, @Sopt_VERSION_MINOR@, @Sopt_VERSION_PATCH@); } //! Returns library git reference, if known inline std::string gitref() { return "@Sopt_GITREF@"; } //! Default logging level inline std::string default_logging_level() { return "@SOPT_TEST_LOG_LEVEL@"; } //! Default logger name inline std::string default_logger_name() { return "@SOPT_LOGGER_NAME@"; } //! Wether to add color to the logger inline constexpr bool color_logger() { return @SOPT_COLOR_LOGGING@; } # ifdef SOPT_OPENMP //! Number of threads used during testing inline constexpr std::size_t number_of_threads_in_tests() { return @SOPT_DEFAULT_OPENMP_THREADS@; } # endif } #endif sopt-2.0.0/cpp/sopt/conjugate_gradient.h000066400000000000000000000121101277570055300202600ustar00rootroot00000000000000#ifndef SOPT_CONJUGATE_GRADIENT #define SOPT_CONJUGATE_GRADIENT #include "sopt/config.h" #include #include #include "sopt/logging.h" #include "sopt/types.h" #include "sopt/maths.h" #include "sopt/wrapper.h" namespace sopt { //! Solves $Ax = b$ for $x$, given $A$ and $b$. class ConjugateGradient { //! \brief Wraps around a matrix to fake a functor //! \details xout = A * xin becomes apply_matrix_instance(xout, xin); template class ApplyMatrix; public: //! Values indicating how the algorithm ran struct Diagnostic { //! Number of iterations t_uint niters; //! Residual t_real residual; //! Wether convergence was achieved bool good; }; //! Values indicating how the algorithm ran and its result; template struct DiagnosticAndResult : public Diagnostic { Vector result; }; //! \brief Creates conjugate gradient operator //! \param[in] itermax: Maximum number of iterations. 0 means algorithm breaks only if //! convergence is reached. //! \param[in] tolerance: Convergence criteria ConjugateGradient(t_uint itermax = std::numeric_limits::max(), t_real tolerance = 1e-8) : tolerance_(tolerance), itermax_(itermax) {} virtual ~ConjugateGradient() {} //! \brief Computes $x$ for $Ax=b$ //! \details Specialization that converts A from a matrix to a functor. //! This convertion is only so we write the conjugate-gradient algorithm only once for //! A as a matrix and A as a functor. A as a functor means A can be a complex operation, e.g. an //! FFT or two. template Diagnostic operator()(VECTOR &x, Eigen::MatrixBase const &A, Eigen::MatrixBase const &b) const { return implementation(x, A, b); } //! \brief Computes $x$ for $Ax=b$ //! \details Specialisation where A is a functor and b and x are matrix-like objects. This is //! the innermost specialization. template typename std::enable_if, T1>::value, Diagnostic>::type operator()(VECTOR &x, T1 const &A, Eigen::MatrixBase const &b) const { return implementation(x, details::wrap(A), b); } //! \brief Computes $x$ for $Ax=b$ //! \details Specialisation where x is constructed during call and returned. And x is a matrix //! rather than an array. template DiagnosticAndResult operator()(A_TYPE const &A, Eigen::MatrixBase const &b) const { DiagnosticAndResult result; result.result = Vector::Zero(b.size()); *static_cast(&result) = operator()(result.result, A, b); return result; } //! \brief Maximum number of iterations //! \details 0 means algorithm breaks only if convergence is reached. t_uint itermax() const { return itermax_; } //! \brief Sets maximum number of iterations //! \details 0 means algorithm breaks only if convergence is reached. void itermax(t_uint const &itermax) { itermax_ = itermax; } //! Tolerance criteria t_real tolerance() const { return tolerance_; } //! Sets tolerance criteria void tolerance(t_real const &tolerance) { if(tolerance <= 0e0) throw std::domain_error("Incorrect tolerance input"); tolerance_ = tolerance; } protected: //! Tolerance criteria t_real tolerance_; //! Maximum number of iteration t_uint itermax_; //! \details Work array to hold v Image<> work_v; //! Work array to hold r Image<> work_r; //! Work array to hold p Image<> work_p; private: //! \brief Just one implementation for all types //! \note This is a template function, to avoid repetition, but it is not declared in the //! header. template Diagnostic implementation(VECTOR &x, MATRIXLIKE const &A, Eigen::MatrixBase const &b) const; }; template ConjugateGradient::Diagnostic ConjugateGradient::implementation(VECTOR &x, MATRIXLIKE const &A, Eigen::MatrixBase const &b) const { typedef typename T1::Scalar Scalar; typedef typename real_type::type Real; x.resize(b.size()); if(std::abs((b.transpose().conjugate() * b)(0)) < tolerance()) { x.fill(0); return {0, 0, 1}; } Vector Ap(b.size()); x = b; Vector residuals = b - A * x; Vector p = residuals; Real residual = std::abs((residuals.transpose().conjugate() * residuals)(0)); t_uint i(0); for(; i < itermax(); ++i) { Ap = A * p; Scalar const alpha = residual / (p.transpose().conjugate() * Ap)(0); x += alpha * p; residuals -= alpha * Ap; Real new_residual = std::abs((residuals.transpose().conjugate() * residuals)(0)); SOPT_LOW_LOG("CG iteration {} - residuals: {}", i, new_residual); if(std::abs(new_residual) < tolerance()) { residual = new_residual; break; } p = residuals + new_residual / residual * p; residual = new_residual; } return {i, residual, residual < tolerance()}; } } /* sopt */ #endif sopt-2.0.0/cpp/sopt/exception.h000066400000000000000000000023671277570055300164370ustar00rootroot00000000000000#ifndef SOPT_EXCEPTION #define SOPT_EXCEPTION #include "sopt/config.h" #include #include #include namespace sopt { //! Root exception for sopt class Exception : public std::exception { protected: //! Constructor for derived classes Exception(std::string const &name, std::string const &filename, size_t lineno) : std::exception(), message(header(name, filename, lineno)) {} public: //! Creates exception Exception(std::string const &filename, size_t lineno) : Exception("sopt::Exception", filename, lineno) {} //! Creates message const char *what() const noexcept override { return message.c_str(); } //! Header of the message static std::string header(std::string const &name, std::string const &filename, size_t lineno) { std::ostringstream header; header << name << " at " << filename << ":" << lineno; return header.str(); } //! Adds to message template Exception &operator<<(OBJECT const &object) { std::ostringstream msg; msg << message << object; message = msg.str(); return *this; } private: //! Message to issue std::string message; }; #define SOPT_THROW(MSG) throw(sopt::Exception(__FILE__, __LINE__) << "\n" << MSG) } /* sopt */ #endif sopt-2.0.0/cpp/sopt/imaging_padmm.h000066400000000000000000000365061277570055300172340ustar00rootroot00000000000000#ifndef SOPT_L1_PROXIMAL_ADMM_H #define SOPT_L1_PROXIMAL_ADMM_H #include "sopt/config.h" #include #include #include #include "sopt/exception.h" #include "sopt/l1_proximal.h" #include "sopt/linear_transform.h" #include "sopt/logging.h" #include "sopt/padmm.h" #include "sopt/proximal.h" #include "sopt/types.h" namespace sopt { namespace algorithm { //! \brief Specialization of Proximal ADMM for Purify //! \details \f$\min_{x, z} f(x) + h(z)\f$ subject to \f$Φx + z = y\f$, where \f$f(x) = //! ||Ψ^Hx||_1 + i_C(x)\f$ and \f$h(x) = i_B(z)\f$ with \f$C = R^N_{+}\f$ and \f$B = {z \in R^M: //! ||z||_2 \leq \epsilon}\f$ template class ImagingProximalADMM : private ProximalADMM { //! Defines convergence behaviour struct Breaker; public: //! Scalar type typedef typename ProximalADMM::value_type value_type; typedef typename ProximalADMM::Scalar Scalar; typedef typename ProximalADMM::Real Real; typedef typename ProximalADMM::t_Vector t_Vector; typedef typename ProximalADMM::t_LinearTransform t_LinearTransform; typedef typename ProximalADMM::t_Proximal t_Proximal; typedef typename ProximalADMM::t_IsConverged t_IsConverged; using ProximalADMM::initial_guess; //! Values indicating how the algorithm ran struct Diagnostic : public ProximalADMM::Diagnostic { //! Diagnostic from calling L1 proximal typename proximal::L1::Diagnostic l1_diag; Diagnostic(t_uint niters = 0u, bool good = false, typename proximal::L1::Diagnostic const &l1diag = typename proximal::L1::Diagnostic()) : ProximalADMM::Diagnostic(niters, good), l1_diag(l1diag) {} Diagnostic(t_uint niters, bool good, typename proximal::L1::Diagnostic const &l1diag, t_Vector &&residual) : ProximalADMM::Diagnostic(niters, good, std::move(residual)), l1_diag(l1diag) {} }; //! Holds result vector as well struct DiagnosticAndResult : public Diagnostic { //! Output x t_Vector x; }; template ImagingProximalADMM(Eigen::MatrixBase const &target) : ProximalADMM(nullptr, nullptr, target), l1_proximal_(), l2ball_proximal_(1e0), tight_frame_(false), relative_variation_(1e-4), residual_convergence_(1e-4) { set_f_and_g_proximal_to_members_of_this(); } ImagingProximalADMM(ImagingProximalADMM const &c) : ProximalADMM(c), l1_proximal_(c.l1_proximal_), l2ball_proximal_(c.l2ball_proximal_), tight_frame_(c.tight_frame_), relative_variation_(c.relative_variation_), residual_convergence_(c.residual_convergence_) { set_f_and_g_proximal_to_members_of_this(); } ImagingProximalADMM(ImagingProximalADMM &&c) : ProximalADMM(std::move(c)), l1_proximal_(std::move(c.l1_proximal_)), l2ball_proximal_(std::move(c.l2ball_proximal_)), tight_frame_(c.tight_frame_), relative_variation_(c.relative_variation_), residual_convergence_(c.residual_convergence_) { set_f_and_g_proximal_to_members_of_this(); } void operator=(ImagingProximalADMM const &c) { ProximalADMM::operator=(c); l1_proximal_ = c.l1_proximal_; l2ball_proximal_ = c.l2ball_proximal_; tight_frame_ = c.tight_frame_; set_f_and_g_proximal_to_members_of_this(); } void operator=(ImagingProximalADMM &&c) { ProximalADMM::operator=(std::move(c)); l1_proximal_ = std::move(c.l1_proximal_); l2ball_proximal_ = std::move(c.l2ball_proximal_); tight_frame_ = std::move(c.tight_frame_); set_f_and_g_proximal_to_members_of_this(); } virtual ~ImagingProximalADMM() {} // Macro helps define properties that can be initialized as in // auto sdmm = ProximalADMM().prop0(value).prop1(value); #define SOPT_MACRO(NAME, TYPE, CODE) \ TYPE const &NAME() const { return NAME##_; } \ ImagingProximalADMM &NAME(TYPE const &NAME) { \ NAME##_ = NAME; \ CODE; \ return *this; \ } \ \ protected: \ TYPE NAME##_; \ \ public: //! The L1 proximal functioning as f SOPT_MACRO(l1_proximal, proximal::L1, set_f_and_g_proximal_to_members_of_this()); //! The weighted L2 proximal functioning as g SOPT_MACRO(l2ball_proximal, proximal::WeightedL2Ball, set_f_and_g_proximal_to_members_of_this()); //! Whether Ψ is a tight-frame or not SOPT_MACRO(tight_frame, bool, ); //! \brief Convergence of the relative variation of the objective functions //! \details If negative, this convergence criteria is disabled. SOPT_MACRO(relative_variation, Real, ); //! \brief Convergence of the residuals //! \details If negative, this convergence criteria is disabled. SOPT_MACRO(residual_convergence, Real, ); #undef SOPT_MACRO //! \brief Analysis operator Ψ //! \details Under-the-hood, the object is actually owned by the L1 proximal. t_LinearTransform const &Psi() const { return l1_proximal().Psi(); } //! Analysis operator Ψ template typename std::enable_if= 1, ImagingProximalADMM &>::type Psi(ARGS &&... args) { l1_proximal().Psi(std::forward(args)...); return *this; } //! \brief Analysis operator Φ t_LinearTransform const &Phi() const { return ProximalADMM::Phi(); } //! Φ initialized via some call to \ref linear_transform template typename std::enable_if= 1, ImagingProximalADMM &>::type Phi(ARGS &&... args) { ProximalADMM::Phi(std::forward(args)...); return *this; } //! target measurements t_Vector const &target() const { return ProximalADMM::target(); } //! target measurements template ImagingProximalADMM &target(Eigen::MatrixBase const &target) const { ProximalADMM::target(target); return *this; } //! \brief L1 proximal used during calculation //! \details Non-const version to setup the object. proximal::L1 &l1_proximal() { return l1_proximal_; } //! \brief Proximal of the L2 ball //! \details Non-const version to setup the object. proximal::WeightedL2Ball &l2ball_proximal() { return l2ball_proximal_; } //! Type-erased version of the l1 proximal t_Proximal const &f_proximal() { return ProximalADMM::f_proximal(); } //! Type-erased version of the l2 proximal t_Proximal const &g_proximal() { return ProximalADMM::g_proximal(); } // Forwards get/setters to L1 and L2Ball proximals // In practice, we end up with a bunch of functions that make it simpler to set or get values // associated with the two proximal operators. // E.g.: `paddm.l1_proximal_itermax(100).l2ball_epsilon(1e-2).l1_proximal_tolerance(1e-4)`. // ~~~ #define SOPT_MACRO(VAR, NAME, PROXIMAL) \ /** \brief Forwards to l1_proximal **/ \ decltype(std::declval const>().VAR()) NAME##_proximal_##VAR() const { \ return NAME##_proximal().VAR(); \ } \ /** \brief Forwards to l1_proximal **/ \ ImagingProximalADMM &NAME##_proximal_##VAR( \ decltype(std::declval const>().VAR()) VAR) { \ NAME##_proximal().VAR(VAR); \ return *this; \ } SOPT_MACRO(itermax, l1, L1); SOPT_MACRO(tolerance, l1, L1); SOPT_MACRO(positivity_constraint, l1, L1); SOPT_MACRO(real_constraint, l1, L1); SOPT_MACRO(fista_mixing, l1, L1); SOPT_MACRO(nu, l1, L1); SOPT_MACRO(weights, l1, L1); SOPT_MACRO(epsilon, l2ball, WeightedL2Ball); SOPT_MACRO(weights, l2ball, WeightedL2Ball); #undef SOPT_MACRO // Includes getters and redefines setters to return this object #define SOPT_MACRO(NAME) \ using ProximalADMM::NAME; \ /** \brief Forwards to ProximalADMM base class **/ \ ImagingProximalADMM &NAME(decltype(std::declval>().NAME()) NAME) { \ ProximalADMM::NAME(NAME); \ return *this; \ } SOPT_MACRO(itermax); SOPT_MACRO(gamma); SOPT_MACRO(nu); SOPT_MACRO(lagrange_update_scale); SOPT_MACRO(is_converged); #undef SOPT_MACRO //! Calls l1 proximal operator, checking for real constraints and tight frame template typename proximal::L1::Diagnostic l1_proximal(Eigen::MatrixBase &out, Real gamma, Eigen::MatrixBase const &x) const { return l1_proximal_real_constraint() ? call_l1_proximal(out, gamma, x.real().template cast()) : call_l1_proximal(out, gamma, x); } //! Forwards call to weighted L2 ball proximal template auto l2ball_proximal(Eigen::MatrixBase const &x) const -> decltype(std::declval const>()(Real(0), x)) { return l2ball_proximal()(Real(0), x); } //! \brief Call Proximal ADMM for L1 and L2 ball //! \param[out] out: Output x vector //! \param[in] guess: for both x and the residuals Diagnostic operator()(t_Vector &out, std::tuple const &guess) const { return operator()(out, std::get<0>(guess), std::get<1>(guess)); } //! \brief Call Proximal ADMM for L1 and L2 ball //! \param[out] out: Output x vector Diagnostic operator()(t_Vector &out) const { return operator()(out, initial_guess()); } //! \brief Calls Proximal ADMM for L1 and L2 ball //! \param[in] guess: for both x and the residuals DiagnosticAndResult operator()(std::tuple const &guess) const { DiagnosticAndResult result; static_cast(result) = operator()(result.x, guess); return result; } //! \brief Calls Proximal ADMM for L1 and L2 ball //! \param[in] warm_start: uses result from previous run to restart the calculations DiagnosticAndResult operator()(DiagnosticAndResult const &warm_start) const { DiagnosticAndResult result; static_cast(result) = operator()(result.x, warm_start.x, warm_start.residual); return result; } //! \brief Calls Proximal ADMM for L1 and L2 ball //! \param[in] warm_start: uses result from previous run to restart the calculations DiagnosticAndResult operator()() const { DiagnosticAndResult result; static_cast(result) = operator()(result.x); return result; } protected: //! Keeps track of the last call to the L1 proximal mutable typename proximal::L1::Diagnostic l1_diagnostic; //! Calls l1 proximal operator, checking for thight frame template typename proximal::L1::Diagnostic call_l1_proximal(Eigen::MatrixBase &out, Real gamma, Eigen::MatrixBase const &x) const { if(tight_frame()) { l1_proximal().tight_frame(out, gamma, x); return {0, 0, l1_proximal().objective(x, out, gamma), true}; } return l1_proximal()(out, gamma, x); } //! Sets the result from this call to L1 proximal so it can be used later void erased_f_proximal(t_Vector &out, Real gamma, t_Vector const &x) const { l1_diagnostic = l1_proximal(out, gamma, x); } //! References void set_f_and_g_proximal_to_members_of_this() { using namespace std::placeholders; ProximalADMM::f_proximal( std::bind(&ImagingProximalADMM::erased_f_proximal, this, _1, _2, _3)); ProximalADMM::g_proximal(std::cref(l2ball_proximal())); } //! \brief Call Proximal ADMM for L1 and L2 ball //! \param[out] out: Output x vector //! \param[in] guess: initial guess for the image //! \param[in] res: initial guess for the residual Diagnostic operator()(t_Vector &out, t_Vector const &guess, t_Vector const &res) const; }; template typename ImagingProximalADMM::Diagnostic ImagingProximalADMM:: operator()(t_Vector &out, t_Vector const &x_guess, t_Vector const &res_guess) const { SOPT_HIGH_LOG("Performing Proximal ADMM with L1 and L2 operators"); ProximalADMM::sanity_check(x_guess, res_guess); t_Vector lambda = t_Vector::Zero(target().size()); t_Vector z = t_Vector::Zero(target().size()); t_Vector residual = res_guess; out = x_guess; bool const has_user_convergence = static_cast(is_converged()); l1_diagnostic = {0, 0, 0, false}; SOPT_TRACE(" - Initialization"); std::pair objectives{sopt::l1_norm(Psi().adjoint() * out, l1_proximal_weights()), 0}; bool converged = false; for(t_uint niters(0); niters < itermax(); ++niters) { SOPT_LOW_LOG(" - Iteration {}/{}. ", niters, itermax()); ProximalADMM::iteration_step(out, residual, lambda, z); // print-out stuff objectives.second = objectives.first; objectives.first = sopt::l1_norm(Psi().adjoint() * out, l1_proximal_weights()); t_real const relative_objective = std::abs(objectives.first - objectives.second) / objectives.first; auto const residual_norm = sopt::l2_norm(residual, l2ball_proximal_weights()); SOPT_LOW_LOG(" - objective: obj value = {}, rel obj = {}", objectives.first, relative_objective); SOPT_LOW_LOG(" - Residuals: epsilon = {}, residual norm = {}", l2ball_proximal_epsilon(), residual_norm); // convergence stuff auto const user = (not has_user_convergence) or is_converged(out); auto const res = residual_convergence() <= 0e0 or residual_norm < residual_convergence(); auto const rel = relative_variation() <= 0e0 or relative_objective < relative_variation(); converged = user and rel and res; if(converged) { SOPT_MEDIUM_LOG(" - converged in {} of {} iterations", niters, itermax()); break; } } if(not converged) SOPT_ERROR(" - did not converge within {} iterations", itermax()); return {itermax(), converged, l1_diagnostic, std::move(residual)}; } } } /* sopt::algorithm */ #endif sopt-2.0.0/cpp/sopt/l1_proximal.h000066400000000000000000000400701277570055300166610ustar00rootroot00000000000000#ifndef SOPT_L1_PROXIMAL_H #define SOPT_L1_PROXIMAL_H #include "sopt/config.h" #include #include #include #include "sopt/linear_transform.h" #include "sopt/maths.h" #include "sopt/proximal_expression.h" namespace sopt { namespace proximal { //! \brief L1 proximal, including linear transform //! \details This function computes the prox operator of the l1 //! norm for the input vector \f$x\f$. It solves the problem: //! \f[ min_{z} 0.5||x - z||_2^2 + γ ||Ψ^† z||_w1 \f] //! where \f$Ψ \in C^{N_x \times N_r} \f$ is the sparsifying operator, and \f[|| ||_w1\f] is the //! weighted L1 norm. template class L1TightFrame { public: //! Underlying scalar type typedef SCALAR Scalar; //! Underlying real scalar type typedef typename real_type::type Real; L1TightFrame() : Psi_(linear_transform_identity()), nu_(1e0), weights_(Vector::Ones(1)) {} #define SOPT_MACRO(NAME, TYPE) \ TYPE const &NAME() const { return NAME##_; } \ L1TightFrame &NAME(TYPE const &NAME) { \ NAME##_ = NAME; \ return *this; \ } \ \ protected: \ TYPE NAME##_; \ \ public: //! Linear transform applied to input prior to L1 norm SOPT_MACRO(Psi, LinearTransform>); //! Bound on the squared norm of the operator Ψ SOPT_MACRO(nu, Real); #undef SOPT_MACRO //! Weights of the l1 norm Vector const &weights() const { return weights_; } //! Weights of the l1 norm template L1TightFrame &weights(Eigen::MatrixBase const &w) { if((w.array() < 0e0).any()) SOPT_THROW("Weights cannot be negative"); if(w.stableNorm() < 1e-12) SOPT_THROW("Weights cannot be null"); weights_ = w; return *this; } //! Set weights to a single value L1TightFrame &weights(Real const &value) { if(value <= 0e0) SOPT_THROW("Weight cannot be negative or null"); weights_ = Vector::Ones(1) * value; return *this; } //! Set Ψ and Ψ^† using arguments that sopt::linear_transform understands template typename std::enable_if= 1, L1TightFrame &>::type Psi(ARGS &&... args) { Psi_ = linear_transform(std::forward(args)...); return *this; } //! Computes proximal for given γ template typename std::enable_if::value == is_complex::value and is_complex::value == is_complex::value>::type operator()(Eigen::MatrixBase &out, Real gamma, Eigen::MatrixBase const &x) const; //! Lazy version template ProximalExpression const &, T0> operator()(Real const &gamma, Eigen::MatrixBase const &x) const { return {*this, gamma, x}; } //! \f[ 0.5||x - z||_2^2 + γ||Ψ^† z||_w1 \f] template typename std::enable_if::value == is_complex::value and is_complex::value == is_complex::value, Real>::type objective(Eigen::MatrixBase const &x, Eigen::MatrixBase const &z, Real const &gamma) const; protected: //! Weights associated with the l1-norm Vector weights_; }; template template typename std::enable_if::value == is_complex::value and is_complex::value == is_complex::value>::type L1TightFrame:: operator()(Eigen::MatrixBase &out, Real gamma, Eigen::MatrixBase const &x) const { Vector const psit_x = Psi().adjoint() * x; if(weights().size() == 1) out = Psi() * (soft_threshhold(psit_x, nu() * gamma * weights()(0)) - psit_x) / nu() + x; else out = Psi() * (soft_threshhold(psit_x, nu() * gamma * weights()) - psit_x) / nu() + x; SOPT_LOW_LOG("Prox L1: objective = {}", objective(x, out, gamma)); } template template typename std::enable_if::value == is_complex::value and is_complex::value == is_complex::value, typename real_type::type>::type L1TightFrame::objective(Eigen::MatrixBase const &x, Eigen::MatrixBase const &z, Real const &gamma) const { return 0.5 * (x - z).squaredNorm() + gamma * sopt::l1_norm(Psi().adjoint() * z, weights()); } //! \brief L1 proximal, including linear transform //! \details This function computes the prox operator of the l1 //! norm for the input vector \f$x\f$. It solves the problem: //! \f[ min_{z} 0.5||x - z||_2^2 + γ ||Ψ^† z||_w1 \f] //! where \f$Ψ \in C^{N_x \times N_r} \f$ is the sparsifying operator, and \f[|| ||_w1\f] is the //! weighted L1 norm. template class L1 : protected L1TightFrame { public: //! Functor to do fista mixing class FistaMixing; //! Functor to do no mixing class NoMixing; //! Functor to check convergence and cycling class Breaker; using L1TightFrame::objective; //! Underlying scalar type typedef typename L1TightFrame::Scalar Scalar; //! Underlying real scalar type typedef typename L1TightFrame::Real Real; //! How did calling L1 go? struct Diagnostic { //! Number of iterations t_uint niters; //! Relative variation of the objective function Real relative_variation; //! Value of the objective function Real objective; //! Wether convergence was achieved bool good; Diagnostic(t_uint niters = 0, Real relative_variation = 0, Real objective = 0, bool good = false) : niters(niters), relative_variation(relative_variation), objective(objective), good(good) { } }; //! Result from calling L1 struct DiagnosticAndResult : public Diagnostic { //! The proximal value Vector proximal; }; //! Computes proximal for given γ template Diagnostic operator()(Eigen::MatrixBase &out, Real gamma, Vector const &x) const { // Note that we *must* call eval on x, in case it is an expression involving out if(fista_mixing()) return operator()(out, gamma, x, FistaMixing()); else return operator()(out, gamma, x, NoMixing()); } //! Lazy version template DiagnosticAndResult operator()(Real const &gamma, Eigen::MatrixBase const &x) const { DiagnosticAndResult result; static_cast(result) = operator()(result.proximal, gamma, x); return result; } L1() : L1TightFrame(), itermax_(0), tolerance_(1e-8), positivity_constraint_(false), real_constraint_(false), fista_mixing_(true) {} #define SOPT_MACRO(NAME, TYPE) \ TYPE const &NAME() const { return NAME##_; } \ L1 &NAME(TYPE const &NAME) { \ NAME##_ = NAME; \ return *this; \ } \ \ protected: \ TYPE NAME##_; \ \ public: //! \brief Maximum number of iterations before bailing out //! \details 0 means algorithm breaks only if convergence is reached. SOPT_MACRO(itermax, t_uint); //! Tolerance criteria SOPT_MACRO(tolerance, Real); //! Whether to apply positivity constraints SOPT_MACRO(positivity_constraint, bool); //! Whether the output should be constrained to be real SOPT_MACRO(real_constraint, bool); //! Whether to do fista mixing or not SOPT_MACRO(fista_mixing, bool); #undef SOPT_MACRO //! Weights of the l1 norm Vector const &weights() const { return L1TightFrame::weights(); } //! Set weights to an array of values template L1 &weights(Eigen::MatrixBase const &w) { L1TightFrame::weights(w); return *this; } //! Set weights to a single value L1 &weights(Real const &w) { L1TightFrame::weights(w); return this; } //! Bounds on the squared norm of the operator Ψ Real nu() const { return L1TightFrame::nu(); } //! Sets the bound on the squared norm of the operator Ψ L1 &nu(Real const &nu) { L1TightFrame::nu(nu); return *this; } //! Linear transform applied to input prior to L1 norm LinearTransform> const &Psi() const { return L1TightFrame::Psi(); } //! Set Ψ and Ψ^† using a matrix template typename std::enable_if= 1, L1 &>::type Psi(ARGS &&... args) { L1TightFrame::Psi(std::forward(args)...); return *this; } //! \brief Special case if Ψ ia a tight frame. //! \see L1TightFrame template auto tight_frame(T &&... args) const -> decltype(this->L1TightFrame::operator()(std::forward(args)...)) { return this->L1TightFrame::operator()(std::forward(args)...); } protected: //! Applies one or another soft-threshhold, depending on weight template Vector apply_soft_threshhold(Real gamma, Eigen::MatrixBase const &x) const; //! Applies constraints to input expression template void apply_constraints(Eigen::MatrixBase &out, Eigen::MatrixBase const &x) const; //! Operation with explicit mixing step template Diagnostic operator()(Eigen::MatrixBase &out, Real gamma, Vector const &x, MIXING mixing) const; }; //! Computes proximal for given γ template template typename L1::Diagnostic L1:: operator()(Eigen::MatrixBase &out, Real gamma, Vector const &x, MIXING mixing) const { SOPT_MEDIUM_LOG(" Starting Proximal L1 operator:"); t_uint niters = 0; out = x; Breaker breaker(objective(out, x, gamma), tolerance(), false); // not fista_mixing()); SOPT_LOW_LOG(" - iter {}, prox_fval = {}", niters, breaker.current()); Vector const res = Psi().adjoint() * out; Vector u_l1 = 1e0 / nu() * (res - apply_soft_threshhold(gamma, res)); apply_constraints(out, x - Psi() * u_l1); // Move on to other iterations for(++niters; niters < itermax() or itermax() == 0; ++niters) { auto const do_break = breaker(objective(x, out, gamma)); SOPT_LOW_LOG(" - iter {}, prox_fval = {}, rel_fval = {}", niters, breaker.current(), breaker.relative_variation()); if(do_break) break; Vector const res = u_l1 * nu() + Psi().adjoint() * out; mixing(u_l1, 1e0 / nu() * (res - apply_soft_threshhold(gamma, res)), niters); apply_constraints(out, x - Psi() * u_l1); } if(breaker.two_cycle()) SOPT_WARN("Two-cycle detected when computing L1"); if(breaker.converged()) { SOPT_LOW_LOG(" Proximal L1 operator converged at {} in {} iterations", breaker.current(), niters); } else SOPT_ERROR(" Proximal L1 operator did not converge after {} iterations", niters); return {niters, breaker.relative_variation(), breaker.current(), breaker.converged()}; } template template Vector L1::apply_soft_threshhold(Real gamma, Eigen::MatrixBase const &x) const { if(weights().size() == 1) return soft_threshhold(x, gamma * weights()(0)); else return soft_threshhold(x, gamma * weights()); } template template void L1::apply_constraints(Eigen::MatrixBase &out, Eigen::MatrixBase const &x) const { if(positivity_constraint()) out = sopt::positive_quadrant(x); else if(real_constraint()) out = x.real().template cast(); else out = x; } template class L1::FistaMixing { public: typedef typename real_type::type Real; FistaMixing() : t(1){}; template void operator()(Vector &previous, Eigen::MatrixBase const &unmixed, t_uint iter) { // reset if(iter == 0) { previous = unmixed; return; } if(iter <= 1) t = next(1); auto const prior_t = t; t = next(t); auto const alpha = (prior_t - 1) / t; previous = (1e0 + alpha) * unmixed.derived() - alpha * previous; } static Real next(Real t) { return 0.5 + 0.5 * std::sqrt(1e0 + 4e0 * t * t); } private: Real t; }; template class L1::NoMixing { public: template void operator()(Vector &previous, Eigen::MatrixBase const &unmixed, t_uint) { previous = unmixed; } }; template class L1::Breaker { public: typedef typename real_type::type Real; //! Constructs a breaker object //! \param[in] objective: the first objective function //! \param[in] tolerance: Convergence criteria for convergence //! \param[in] do_two_cycle: Whether to enable two cycle detections. Only necessary when mixing //! is not enabled. Breaker(Real objective, Real tolerance = 1e-8, bool do_two_cycle = true) : tolerance_(tolerance), iter(0), objectives({{objective, 0, 0, 0}}), do_two_cycle(do_two_cycle) {} //! True if we should break out of loop bool operator()(Real objective) { ++iter; objectives = {{objective, objectives[0], objectives[1], objectives[2]}}; return converged() or two_cycle(); } //! Current objective Real current() const { return objectives[0]; } //! Current objective Real previous() const { return objectives[1]; } //! Variation in the objective function Real relative_variation() const { return std::abs((current() - previous()) / current()); } //! \brief Whether we have a cycle of period two //! \details Cycling is prone to happen without mixing, it seems. bool two_cycle() const { return do_two_cycle and iter > 3 and std::abs(objectives[0] - objectives[2]) < tolerance() and std::abs(objectives[1] - objectives[3]) < tolerance(); } //! True if relative variation smaller than tolerance bool converged() const { // If current ~ 0, then defaults to absolute convergence // This is mainly to avoid a division by zero if(std::abs(current() * 1000) < tolerance()) return std::abs(previous() * 1000) < tolerance(); return relative_variation() < tolerance(); } //! Tolerance criteria Real tolerance() const { return tolerance_; } //! Tolerance criteria L1::Breaker &tolerance(Real tol) const { tolerance_ = tol; return *this; } protected: Real tolerance_; t_uint iter; std::array objectives; bool do_two_cycle; }; } } /* sopt::proximal */ #endif sopt-2.0.0/cpp/sopt/linear_transform.h000066400000000000000000000230111277570055300177730ustar00rootroot00000000000000#ifndef SOPT_OPERATORS_H #define SOPT_OPERATORS_H #include "sopt/config.h" #include #include #include #include #include "sopt/logging.h" #include "sopt/types.h" #include "sopt/maths.h" #include "sopt/wrapper.h" namespace sopt { namespace details { //! \brief Wraps a matrix into a function and its conjugate transpose //! \details This class helps to wrap matrices into functions, such that we can use and store them //! such that SDMM algorithms can refer to them. template class MatrixToLinearTransform; //! Wraps a tranposed matrix into a function and its conjugate transpose template class MatrixAdjointToLinearTransform; } //! Joins together direct and indirect operators template class LinearTransform : public details::WrapFunction { public: //! Type of the wrapped functions typedef OperatorFunction t_Function; //! Constructor //! \param[in] direct: function with signature void(VECTOR&, VECTOR const&) which applies a //! linear operator to a vector. //! \param[in] indirect: function with signature void(VECTOR&, VECTOR const&) which applies a //! the conjugate transpose linear operator to a vector. //! \param[in] sizes: 3 integer elements (a, b, c) such that if the input to linear operator is //! of size N, then the output is of size (a * N) / b + c. A similar quantity is deduced for //! the indirect operator. LinearTransform(t_Function const &direct, t_Function const &indirect, std::array sizes = {{1, 1, 0}}) : LinearTransform( direct, sizes, indirect, {{sizes[1], sizes[0], sizes[0] == 0 ? 0 : -(sizes[2] * sizes[1]) / sizes[0]}}) { assert(sizes[0] != 0); } //! Constructor //! \param[in] direct: function with signature void(VECTOR&, VECTOR const&) which applies a //! linear operator to a vector. //! \param[in] dsizes: 3 integer elements (a, b, c) such that if the input to the linear //! operator is of size N, then the output is of size (a * N) / b + c. //! \param[in] indirect: function with signature void(VECTOR&, VECTOR const&) which applies a //! the conjugate transpose linear operator to a vector. //! \param[in] dsizes: 3 integer elements (a, b, c) such that if the input to the indirect //! linear operator is of size N, then the output is of size (a * N) / b + c. LinearTransform(t_Function const &direct, std::array dsizes, t_Function const &indirect, std::array isizes) : LinearTransform(details::wrap(direct, dsizes), details::wrap(indirect, isizes)) {} LinearTransform(details::WrapFunction const &direct, details::WrapFunction const &indirect) : details::WrapFunction(direct), indirect_(indirect) {} LinearTransform(LinearTransform const &c) : details::WrapFunction(c), indirect_(c.indirect_) {} LinearTransform(LinearTransform &&c) : details::WrapFunction(std::move(c)), indirect_(std::move(c.indirect_)) {} void operator=(LinearTransform const &c) { details::WrapFunction::operator=(c); indirect_ = c.indirect_; } void operator=(LinearTransform &&c) { details::WrapFunction::operator=(std::move(c)); indirect_ = std::move(c.indirect_); } //! Indirect transform LinearTransform adjoint() const { return {indirect_, static_cast const &>(*this)}; } using details::WrapFunction::operator*; using details::WrapFunction::sizes; using details::WrapFunction::rows; private: //! Function applying conjugate transpose operator details::WrapFunction indirect_; }; //! Helper function to creates a function operator //! \param[in] direct: function with signature void(VECTOR&, VECTOR const&) which applies a //! linear operator to a vector. //! \param[in] indirect: function with signature void(VECTOR&, VECTOR const&) which applies a //! the conjugate transpose linear operator to a vector. //! \param[in] sizes: 3 integer elements (a, b, c) such that if the input to linear operator is //! of size N, then the output is of size (a * N) / b + c. A similar quantity is deduced for //! the indirect operator. template LinearTransform linear_transform(OperatorFunction const &direct, OperatorFunction const &indirect, std::array const &sizes = {{1, 1, 0}}) { return {direct, indirect, sizes}; } //! Helper function to creates a function operator //! \param[in] direct: function with signature void(VECTOR&, VECTOR const&) which applies a //! linear operator to a vector. //! \param[in] dsizes: 3 integer elements (a, b, c) such that if the input to the linear //! operator is of size N, then the output is of size (a * N) / b + c. //! \param[in] indirect: function with signature void(VECTOR&, VECTOR const&) which applies a //! the conjugate transpose linear operator to a vector. //! \param[in] dsizes: 3 integer elements (a, b, c) such that if the input to the indirect //! linear operator is of size N, then the output is of size (a * N) / b + c. template LinearTransform linear_transform(OperatorFunction const &direct, std::array const &dsizes, OperatorFunction const &indirect, std::array const &isizes) { return {direct, dsizes, indirect, isizes}; } //! Convenience no-op function template LinearTransform &linear_transform(LinearTransform &passthrough) { return passthrough; } //! Creates a linear transform from a pair of wrappers template LinearTransform linear_transform(details::WrapFunction const &direct, details::WrapFunction const &adjoint) { return {direct, adjoint}; } namespace details { template class MatrixToLinearTransform { //! The underlying raw matrix type typedef typename std::remove_const::type>::type Raw; //! The matrix underlying the expression typedef typename Raw::PlainObject PlainMatrix; public: //! The output type typedef typename std::conditional, PlainMatrix>::value, Vector, Array>::type PlainObject; //! \brief Creates from an expression //! \details Expression is evaluated and the result stored internally. This object owns a //! copy of the matrix. It might share it with a few friendly neighbors. template MatrixToLinearTransform(Eigen::MatrixBase const &A) : matrix(std::make_shared(A)) {} //! Creates from a shared matrix. MatrixToLinearTransform(std::shared_ptr const &x) : matrix(x){}; //! Performs operation void operator()(PlainObject &out, PlainObject const &x) const { #ifndef NDEBUG if((*matrix).cols() != x.size()) SOPT_THROW("Input vector and matrix do not match: ") << out.cols() << " columns for " << x.size() << " elements."; #endif out = (*matrix) * x; } //! \brief Returns conjugate transpose operator //! \details The matrix is shared. MatrixAdjointToLinearTransform adjoint() const { return MatrixAdjointToLinearTransform(matrix); } private: //! Wrapped matrix std::shared_ptr matrix; }; template class MatrixAdjointToLinearTransform { public: typedef typename MatrixToLinearTransform::PlainObject PlainObject; //! \brief Creates from an expression //! \details Expression is evaluated and the result stored internally. This object owns a //! copy of the matrix. It might share it with a few friendly neighbors. template MatrixAdjointToLinearTransform(Eigen::MatrixBase const &A) : matrix(std::make_shared(A)) {} //! Creates from a shared matrix. MatrixAdjointToLinearTransform(std::shared_ptr const &x) : matrix(x){}; //! Performs operation void operator()(PlainObject &out, PlainObject const &x) const { #ifndef NDEBUG if((*matrix).rows() != x.size()) SOPT_THROW("Input vector and matrix adjoint do not match: ") << out.cols() << " rows for " << x.size() << " elements."; #endif out = matrix->adjoint() * x; } //! \brief Returns adjoint operator //! \details The matrix is shared. MatrixToLinearTransform adjoint() const { return MatrixToLinearTransform(matrix); } private: std::shared_ptr matrix; }; } //! Helper function to creates a function operator template LinearTransform> linear_transform(Eigen::MatrixBase const &A) { details::MatrixToLinearTransform> const matrix(A); if(A.rows() == A.cols()) return {matrix, matrix.adjoint()}; else { t_int const gcd = details::gcd(A.cols(), A.rows()); t_int const a = A.cols() / gcd; t_int const b = A.rows() / gcd; return {matrix, matrix.adjoint(), {{b, a, 0}}}; } } //! Helper function to create a linear transform that's just the identity template LinearTransform> linear_transform_identity() { return {[](Vector &out, Vector const &in) { out = in; }, [](Vector &out, Vector const &in) { out = in; }}; } } #endif sopt-2.0.0/cpp/sopt/logging.disabled.h000066400000000000000000000013771277570055300176350ustar00rootroot00000000000000#ifndef SOPT_LOGGING_DISABLED_H #define SOPT_LOGGING_DISABLED_H #include "sopt/config.h" #include #include namespace sopt { namespace logging { //! Name of the sopt logger const std::string name_prefix = "sopt::"; inline std::shared_ptr initialize(std::string const &) { return nullptr; } inline std::shared_ptr initialize() { return nullptr; } inline std::shared_ptr get(std::string const &) { return nullptr; } inline std::shared_ptr get() { return nullptr; } inline void set_level(std::string const &, std::string const &){}; inline void set_level(std::string const &){}; inline bool has_level(std::string const &, std::string const &) { return false; } } } //! \macro For internal use only #define SOPT_LOG_(...) #endif sopt-2.0.0/cpp/sopt/logging.enabled.h000066400000000000000000000051351277570055300174540ustar00rootroot00000000000000#ifndef SOPT_LOGGING_ENABLED_H #define SOPT_LOGGING_ENABLED_H #include "sopt/config.h" #include #include #include "sopt/exception.h" namespace sopt { namespace logging { void set_level(std::string const &level, std::string const &name = ""); //! \brief Initializes a logger. //! \details Logger only exists as long as return is kept alive. inline std::shared_ptr initialize(std::string const &name = "") { auto const result = spdlog::stdout_logger_mt(default_logger_name() + name, color_logger()); set_level(default_logging_level(), name); return result; } //! Returns shared pointer to logger or null if it does not exist inline std::shared_ptr get(std::string const &name = "") { return spdlog::get(default_logger_name() + name); } //! \brief Sets loggin level //! \details Levels can be one of //! - "trace" //! - "debug" //! - "info" //! - "warn" //! - "err" //! - "critical" //! - "off" inline void set_level(std::string const &level, std::string const &name) { auto const logger = get(name); if(not logger) SOPT_THROW("No logger by the name of ") << name << ".\n"; #define SOPT_MACRO(LEVEL) \ if(level == #LEVEL) \ logger->set_level(spdlog::level::LEVEL) SOPT_MACRO(trace); else SOPT_MACRO(debug); else SOPT_MACRO(info); else SOPT_MACRO(warn); else SOPT_MACRO(err); else SOPT_MACRO(critical); else SOPT_MACRO(off); #undef SOPT_MACRO else SOPT_THROW("Unknown logging level ") << level << "\n"; } inline bool has_level(std::string const &level, std::string const &name = "") { auto const logger = get(name); if(not logger) return false; #define SOPT_MACRO(LEVEL) \ if(level == #LEVEL) \ return logger->level() >= spdlog::level::LEVEL SOPT_MACRO(trace); else SOPT_MACRO(debug); else SOPT_MACRO(info); else SOPT_MACRO(warn); else SOPT_MACRO(err); else SOPT_MACRO(critical); else SOPT_MACRO(off); #undef SOPT_MACRO else SOPT_THROW("Unknown logging level ") << level << "\n"; } } } //! \macro For internal use only #define SOPT_LOG_(NAME, TYPE, ...) \ if(auto sopt_logging_##__func__##_##__LINE__ = sopt::logging::get(NAME)) \ sopt_logging_##__func__##_##__LINE__->TYPE(__VA_ARGS__) #endif sopt-2.0.0/cpp/sopt/logging.h000066400000000000000000000021661277570055300160640ustar00rootroot00000000000000#ifndef SOPT_LOGGING_H #define SOPT_LOGGING_H #include "sopt/config.h" #ifdef SOPT_DO_LOGGING #include "sopt/logging.enabled.h" #else #include "sopt/logging.disabled.h" #endif //! \macro Normal but significant condition or critical error #define SOPT_NOTICE(...) SOPT_LOG_(, critical, __VA_ARGS__) //! \macro Something is definitely wrong, algorithm exits #define SOPT_ERROR(...) SOPT_LOG_(, error, __VA_ARGS__) //! \macro Something might be going wrong #define SOPT_WARN(...) SOPT_LOG_(, warn, __VA_ARGS__) //! \macro Verbose informational message about normal condition #define SOPT_INFO(...) SOPT_LOG_(, info, __VA_ARGS__) //! \macro Output some debugging #define SOPT_DEBUG(...) SOPT_LOG_(, debug, __VA_ARGS__) //! \macro Output internal values of no interest to anyone //! \details Except maybe when debugging. #define SOPT_TRACE(...) SOPT_LOG_(, trace, __VA_ARGS__) //! High priority message #define SOPT_HIGH_LOG(...) SOPT_LOG_(, critical, __VA_ARGS__) //! Medium priority message #define SOPT_MEDIUM_LOG(...) SOPT_LOG_(, info, __VA_ARGS__) //! Low priority message #define SOPT_LOW_LOG(...) SOPT_LOG_(, debug, __VA_ARGS__) #endif sopt-2.0.0/cpp/sopt/maths.h000066400000000000000000000153721277570055300155550ustar00rootroot00000000000000#ifndef SOPT_MATHS_H #define SOPT_MATHS_H #include "sopt/config.h" #include #include #include #include #include "sopt/exception.h" #include "sopt/types.h" namespace sopt { //! Computes the standard deviation of a vector template typename real_type::type standard_deviation(Eigen::ArrayBase const &x) { return (x - x.mean()).matrix().stableNorm() / std::sqrt(x.size()); } //! Computes the standard deviation of a vector template typename real_type::type standard_deviation(Eigen::MatrixBase const &x) { return standard_deviation(x.array()); } //! abs(x) < threshhold ? 0: x - sgn(x) * threshhold template typename std::enable_if::value or is_complex::value, SCALAR>::type soft_threshhold(SCALAR const &x, typename real_type::type const &threshhold) { auto const normalized = std::abs(x); return normalized < threshhold ? SCALAR(0) : (x * (SCALAR(1) - threshhold / normalized)); } namespace details { //! Expression to create projection onto positive orthant template class ProjectPositiveQuadrant { public: SCALAR operator()(const SCALAR &value) const { return std::max(value, SCALAR(0)); } }; //! Specialization for complex numbers template class ProjectPositiveQuadrant> { public: SCALAR operator()(SCALAR const &value) const { return std::max(value, SCALAR(0)); } std::complex operator()(std::complex const &value) const { return {operator()(value.real()), SCALAR(0)}; } }; //! Helper template typedef to instantiate soft_threshhold that takes an Eigen object template using SoftThreshhold = decltype( std::bind(soft_threshhold, std::placeholders::_1, typename real_type::type(1))); } //! Expression to create projection onto positive quadrant template Eigen::CwiseUnaryOp, const T> positive_quadrant(Eigen::DenseBase const &input) { typedef details::ProjectPositiveQuadrant Projector; typedef Eigen::CwiseUnaryOp UnaryOp; return UnaryOp(input.derived(), Projector()); } //! Expression to create soft-threshhold template Eigen::CwiseUnaryOp, const T> soft_threshhold(Eigen::DenseBase const &input, typename real_type::type const &threshhold) { typedef typename T::Scalar Scalar; typedef typename real_type::type Real; return Eigen::CwiseUnaryOp, const T> {input.derived(), std::bind(soft_threshhold, std::placeholders::_1, Real(threshhold))}; } //! \brief Expression to create soft-threshhold with multiple parameters //! \details Operates over a vector of threshholds: ``out(i) = soft_threshhold(x(i), h(i))`` //! Threshhold and input vectors must have the same size and type. The latter condition is enforced //! by CwiseBinaryOp, unfortunately. template typename std::enable_if::value and std::is_arithmetic::value, Eigen::CwiseBinaryOp>::type soft_threshhold(Eigen::DenseBase const &input, Eigen::DenseBase const &threshhold) { if(input.size() != threshhold.size()) SOPT_THROW("Threshhold and input should have the same size"); return {input.derived(), threshhold.derived(), soft_threshhold}; } //! \brief Expression to create soft-threshhold with multiple parameters //! \details Operates over a vector of threshholds: ``out(i) = soft_threshhold(x(i), h(i))`` //! Threshhold and input vectors must have the same size and type. The latter condition is enforced //! by CwiseBinaryOp, unfortunately. So we cast threshhold from real to complex and back. template typename std:: enable_if::value and std::is_arithmetic::value, Eigen::CwiseBinaryOp< typename T0::Scalar (*)(typename T0::Scalar const &, typename T0::Scalar const &), const T0, decltype(std::declval().template cast())>>::type soft_threshhold(Eigen::DenseBase const &input, Eigen::DenseBase const &threshhold) { if(input.size() != threshhold.size()) SOPT_THROW("Threshhold and input should have the same size: ") << threshhold.size() << " vs " << input.size(); typedef typename T0::Scalar Complex; auto const func = [](Complex const &x, Complex const &t) -> Complex { return soft_threshhold(x, t.real()); }; return {input.derived(), threshhold.derived().template cast(), func}; } //! Computes weighted L1 norm template typename real_type::type l1_norm(Eigen::ArrayBase const &input, Eigen::ArrayBase const &weights) { if(weights.size() == 1) return input.cwiseAbs().sum() * std::abs(weights(0)); return (input.cwiseAbs() * weights).real().sum(); } //! Computes weighted L1 norm template typename real_type::type l1_norm(Eigen::MatrixBase const &input, Eigen::MatrixBase const &weight) { return l1_norm(input.array(), weight.array()); } //! Computes L1 norm template typename real_type::type l1_norm(Eigen::ArrayBase const &input) { return input.cwiseAbs().sum(); } //! Computes L1 norm template typename real_type::type l1_norm(Eigen::MatrixBase const &input) { return l1_norm(input.array()); } //! Computes weighted L2 norm template typename real_type::type l2_norm(Eigen::ArrayBase const &input, Eigen::ArrayBase const &weights) { if(weights.size() == 1) return input.matrix().stableNorm() * std::abs(weights(0)); return (input * weights).matrix().stableNorm(); } //! Computes weighted L2 norm template typename real_type::type l2_norm(Eigen::MatrixBase const &input, Eigen::MatrixBase const &weights) { return l2_norm(input.derived().array(), weights.derived().array()); } namespace details { //! Greatest common divisor inline t_int gcd(t_int a, t_int b) { return b == 0 ? a : gcd(b, a % b); } } } /* sopt */ #endif sopt-2.0.0/cpp/sopt/padmm.h000066400000000000000000000220501277570055300155260ustar00rootroot00000000000000#ifndef SOPT_PROXIMAL_ADMM_H #define SOPT_PROXIMAL_ADMM_H #include "sopt/config.h" #include #include #include "sopt/exception.h" #include "sopt/linear_transform.h" #include "sopt/logging.h" #include "sopt/types.h" namespace sopt { namespace algorithm { //! \brief Proximal Alternate Direction method of mutltipliers //! \details \f$\min_{x, z} f(x) + h(z)\f$ subject to \f$Φx + z = y\f$. \f$y\f$ is a target vector. template class ProximalADMM { public: //! Scalar type typedef SCALAR value_type; //! Scalar type typedef value_type Scalar; //! Real type typedef typename real_type::type Real; //! Type of then underlying vectors typedef Vector t_Vector; //! Type of the Ψ and Ψ^H operations, as well as Φ and Φ^H typedef LinearTransform t_LinearTransform; //! Type of the convergence function typedef ConvergenceFunction t_IsConverged; //! Type of the convergence function typedef ProximalFunction t_Proximal; //! Values indicating how the algorithm ran struct Diagnostic { //! Number of iterations t_uint niters; //! Wether convergence was achieved bool good; //! the residual from the last iteration t_Vector residual; Diagnostic(t_uint niters = 0u, bool good = false) : niters(niters), good(good), residual(t_Vector::Zero(0)) {} Diagnostic(t_uint niters, bool good, t_Vector &&residual) : niters(niters), good(good), residual(std::move(residual)) {} }; //! Holds result vector as well struct DiagnosticAndResult : public Diagnostic { //! Output x t_Vector x; }; //! Setups ProximalADMM //! \param[in] f_proximal: proximal operator of the \f$f\f$ function. //! \param[in] g_proximal: proximal operator of the \f$g\f$ function template ProximalADMM(t_Proximal const &f_proximal, t_Proximal const &g_proximal, Eigen::MatrixBase const &target) : itermax_(std::numeric_limits::max()), gamma_(1e-8), nu_(1), lagrange_update_scale_(0.9), is_converged_(), Phi_(linear_transform_identity()), f_proximal_(f_proximal), g_proximal_(g_proximal), target_(target) {} virtual ~ProximalADMM() {} // Macro helps define properties that can be initialized as in // auto sdmm = ProximalADMM().prop0(value).prop1(value); #define SOPT_MACRO(NAME, TYPE) \ TYPE const &NAME() const { return NAME##_; } \ ProximalADMM &NAME(TYPE const &NAME) { \ NAME##_ = NAME; \ return *this; \ } \ \ protected: \ TYPE NAME##_; \ \ public: //! Maximum number of iterations SOPT_MACRO(itermax, t_uint); //! γ parameter SOPT_MACRO(gamma, Real); //! ν parameter SOPT_MACRO(nu, Real); //! Lagrange update scale β SOPT_MACRO(lagrange_update_scale, Real); //! A function verifying convergence SOPT_MACRO(is_converged, t_IsConverged); //! Measurement operator SOPT_MACRO(Phi, t_LinearTransform); //! First proximal SOPT_MACRO(f_proximal, t_Proximal); //! Second proximal SOPT_MACRO(g_proximal, t_Proximal); #undef SOPT_MACRO //! \brief Simplifies calling the proximal of f. void f_proximal(t_Vector &out, Real gamma, t_Vector const &x) const { f_proximal()(out, gamma, x); } //! \brief Simplifies calling the proximal of f. void g_proximal(t_Vector &out, Real gamma, t_Vector const &x) const { g_proximal()(out, gamma, x); } //! Vector of target measurements t_Vector const &target() const { return target_; } //! Sets the vector of target measurements template ProximalADMM &target(Eigen::MatrixBase const &target) { target_ = target; return *this; } //! Facilitates call to user-provided convergence function bool is_converged(t_Vector const &x) const { return static_cast(is_converged()) and is_converged()(x); } //! \brief Calls Proximal ADMM //! \param[out] out: Output vector x Diagnostic operator()(t_Vector &out) const { return operator()(out, initial_guess()); } //! \brief Calls Proximal ADMM //! \param[out] out: Output vector x //! \param[in] guess: initial guess Diagnostic operator()(t_Vector &out, std::tuple const &guess) const { return operator()(out, std::get<0>(guess), std::get<1>(guess)); } //! \brief Calls Proximal ADMM //! \param[in] guess: initial guess DiagnosticAndResult operator()(std::tuple const &guess) const { DiagnosticAndResult result; static_cast(result) = operator()(result.x, guess); return result; } //! \brief Calls Proximal ADMM //! \param[in] guess: initial guess DiagnosticAndResult operator()() const { DiagnosticAndResult result; static_cast(result) = operator()(result.x, initial_guess()); return result; } //! Makes it simple to chain different calls to PADMM DiagnosticAndResult operator()(DiagnosticAndResult const &warmstart) const { DiagnosticAndResult result = warmstart; static_cast(result) = operator()(result.x, warmstart.x, warmstart.residual); return result; } //! Set Φ and Φ^† using arguments that sopt::linear_transform understands template typename std::enable_if= 1, ProximalADMM &>::type Phi(ARGS &&... args) { Phi_ = linear_transform(std::forward(args)...); return *this; } //! \brief Computes initial guess for x and the residual using the targets //! \details with y the vector of measurements //! - x = Φ^T y / ν //! - residuals = Φ x - y std::tuple initial_guess() const { std::tuple guess; std::get<0>(guess) = Phi().adjoint() * target() / nu(); std::get<1>(guess) = Phi() * std::get<0>(guess) - target(); return guess; } protected: void iteration_step(t_Vector &out, t_Vector &residual, t_Vector &lambda, t_Vector &z) const; //! Checks input makes sense void sanity_check(t_Vector const &x_guess, t_Vector const &res_guess) const { if((Phi().adjoint() * target()).size() != x_guess.size()) SOPT_THROW("target, adjoint measurement operator and input vector have inconsistent sizes"); if(target().size() != res_guess.size()) SOPT_THROW("target and residual vector have inconsistent sizes"); if((Phi() * x_guess).size() != target().size()) SOPT_THROW("target, measurement operator and input vector have inconsistent sizes"); if(not static_cast(is_converged())) SOPT_WARN("No convergence function was provided: algorithm will run for {} steps", itermax()); } //! \brief Calls Proximal ADMM //! \param[out] out: Output vector x //! \param[in] guess: initial guess //! \param[in] residuals: initial residuals Diagnostic operator()(t_Vector &out, t_Vector const &guess, t_Vector const &res) const; //! Vector of measurements t_Vector target_; }; template void ProximalADMM::iteration_step(t_Vector &out, t_Vector &residual, t_Vector &lambda, t_Vector &z) const { g_proximal(z, gamma(), -lambda - residual); f_proximal(out, gamma() / nu(), out - Phi().adjoint() * (residual + lambda + z) / nu()); residual = Phi() * out - target(); lambda += lagrange_update_scale() * (residual + z); } template typename ProximalADMM::Diagnostic ProximalADMM:: operator()(t_Vector &out, t_Vector const &x_guess, t_Vector const &res_guess) const { SOPT_HIGH_LOG("Performing Proximal ADMM"); sanity_check(x_guess, res_guess); t_Vector lambda = t_Vector::Zero(target().size()); t_Vector z = t_Vector::Zero(target().size()); t_Vector residual = res_guess; out = x_guess; for(t_uint niters(0); niters < itermax(); ++niters) { SOPT_LOW_LOG(" - Iteration {}/{}", niters, itermax()); iteration_step(out, residual, lambda, z); SOPT_LOW_LOG(" - Sum of residuals: {}", residual.array().abs().sum()); if(is_converged(out)) { SOPT_MEDIUM_LOG(" - converged in {} of {} iterations", niters, itermax()); return {niters, true}; } } // check function exists, otherwise, don't know if convergence is meaningful if(static_cast(is_converged())) SOPT_ERROR(" - did not converge within {} iterations", itermax()); return {itermax(), false, std::move(residual)}; } } } /* sopt::algorithm */ #endif sopt-2.0.0/cpp/sopt/positive_quadrant.h000066400000000000000000000040121277570055300201670ustar00rootroot00000000000000#ifndef SOPT_PROJECTED_ALGORITHM_H #define SOPT_PROJECTED_ALGORITHM_H #include "sopt/linear_transform.h" #include "sopt/types.h" namespace sopt { namespace algorithm { //! \brief Computes according to given algorithm and then projects it to the positive quadrant //! \details C implementation of the reweighted algorithms uses this, even-though the solutions are //! already constrained to the positive quadrant. template class PositiveQuadrant { public: //! Underlying algorithm typedef ALGORITHM Algorithm; //! Underlying scalar typedef typename Algorithm::Scalar Scalar; //! Underlying vector typedef typename Algorithm::t_Vector t_Vector; //! Underlying convergence functions typedef typename Algorithm::t_IsConverged t_IsConverged; //! Underlying result type typedef typename ALGORITHM::Diagnostic Diagnostic; //! Underlying result type typedef typename ALGORITHM::DiagnosticAndResult DiagnosticAndResult; PositiveQuadrant(Algorithm const &algo) : algorithm_(algo) {} PositiveQuadrant(Algorithm &&algo) : algorithm_(std::move(algo)) {} Algorithm &algorithm() { return algorithm_; } Algorithm const &algorithm() const { return algorithm_; } //! Performs algorithm and project results onto positive quadrant template Diagnostic operator()(t_Vector &out, T const &... args) const { auto const diagnostic = algorithm()(out, std::forward(args)...); out = positive_quadrant(out); return diagnostic; }; //! Performs algorithm and project results onto positive quadrant template DiagnosticAndResult operator()(T const &... args) const { auto result = algorithm()(std::forward(args)...); result.x = positive_quadrant(result.x); return result; }; protected: //! Underlying algorithm Algorithm algorithm_; }; //! Extended algorithm where the solution is projected on the positive quadrant template PositiveQuadrant positive_quadrant(ALGORITHM const &algo) { return {algo}; } } } #endif sopt-2.0.0/cpp/sopt/power_method.h000066400000000000000000000117701277570055300171330ustar00rootroot00000000000000#ifndef SOPT_POWER_METHO_H #define SOPT_POWER_METHO_H #include "sopt/config.h" #include #include #include "sopt/exception.h" #include "sopt/linear_transform.h" #include "sopt/logging.h" #include "sopt/types.h" namespace sopt { namespace algorithm { //! \brief Eigenvalue and eigenvector for eigenvalue with largest magnitude template class PowerMethod { public: //! Scalar type typedef SCALAR value_type; //! Scalar type typedef value_type Scalar; //! Real type typedef typename real_type::type Real; //! Type of then underlying vectors typedef Vector t_Vector; //! Type of the Ψ and Ψ^H operations, as well as Φ and Φ^H typedef LinearTransform t_LinearTransform; //! Holds result vector as well struct DiagnosticAndResult { //! Number of iterations t_uint niters; //! Wether convergence was achieved bool good; //! Magnitude of the eigenvalue Scalar magnitude; //! Corresponding eigenvector if converged Vector eigenvector; }; //! Setups ProximalADMM PowerMethod() : itermax_(std::numeric_limits::max()), tolerance_(1e-8) {} virtual ~PowerMethod() {} // Macro helps define properties that can be initialized as in // auto sdmm = ProximalADMM().prop0(value).prop1(value); #define SOPT_MACRO(NAME, TYPE) \ TYPE const &NAME() const { return NAME##_; } \ PowerMethod &NAME(TYPE const &NAME) { \ NAME##_ = NAME; \ return *this; \ } \ \ protected: \ TYPE NAME##_; \ \ public: //! Maximum number of iterations SOPT_MACRO(itermax, t_uint); //! Convergence criteria SOPT_MACRO(tolerance, Real); #undef SOPT_MACRO //! \brief Calls the power method for A.adjoint() * A DiagnosticAndResult AtA(t_LinearTransform const &A, t_Vector const &input) const; //! \brief Calls the power method for A, with A a matrix template DiagnosticAndResult operator()(Eigen::DenseBase const &A, t_Vector const &input) const; //! \brief Calls the power method for a given matrix-vector multiplication function DiagnosticAndResult operator()(OperatorFunction const &op, t_Vector const &input) const; protected: }; template typename PowerMethod::DiagnosticAndResult PowerMethod::AtA(t_LinearTransform const &A, t_Vector const &input) const { auto const op = [&A](t_Vector &out, t_Vector const &input) -> void { out = A.adjoint() * (A * input).eval(); }; return operator()(op, input); } template template typename PowerMethod::DiagnosticAndResult PowerMethod:: operator()(Eigen::DenseBase const &A, t_Vector const &input) const { Matrix const Ad = A.derived(); auto const op = [&Ad](t_Vector &out, t_Vector const &input) -> void { out = Ad * input; }; return operator()(op, input); } template typename PowerMethod::DiagnosticAndResult PowerMethod:: operator()(OperatorFunction const &op, t_Vector const &input) const { SOPT_INFO("Computing the upper bound of a given operator"); SOPT_INFO(" - input vector {}", input.transpose()); t_Vector eigenvector = input.normalized(); SOPT_INFO(" - eigenvector norm {}", eigenvector.stableNorm()); typename t_Vector::Scalar previous_magnitude = 1; bool converged = false; t_uint niters = 0; for(; niters < itermax() and converged == false; ++niters) { op(eigenvector, eigenvector); typename t_Vector::Scalar const magnitude = eigenvector.stableNorm() / static_cast(eigenvector.size()); auto const rel_val = std::abs((magnitude - previous_magnitude) / previous_magnitude); converged = rel_val < tolerance(); SOPT_INFO(" - Iteration {}/{} -- norm: {}", niters, itermax(), magnitude); eigenvector /= magnitude; previous_magnitude = magnitude; } // check function exists, otherwise, don't know if convergence is meaningful if(not converged) { SOPT_WARN(" - did not converge within {} iterations", itermax()); } else { SOPT_INFO(" - converged in {} of {} iterations", niters, itermax()); } return DiagnosticAndResult{itermax(), converged, previous_magnitude, eigenvector.normalized()}; } } } /* sopt::algorithm */ #endif sopt-2.0.0/cpp/sopt/proximal.h000066400000000000000000000154201277570055300162660ustar00rootroot00000000000000#ifndef SOPT_PROXIMAL_H #define SOPT_PROXIMAL_H #include "sopt/config.h" #include #include #include "sopt/proximal_expression.h" #include "sopt/maths.h" namespace sopt { //! Holds some standard proximals namespace proximal { //! Proximal of euclidian norm struct EuclidianNorm { template void operator()(Vector &out, typename real_type::type const &t, Eigen::MatrixBase const &x) const { typedef typename T0::Scalar Scalar; auto const norm = x.stableNorm(); if(norm > t) out = (Scalar(1) - t / norm) * x; else out.fill(0); } //! Lazy version template ProximalExpression operator()(typename T0::Scalar const &t, Eigen::MatrixBase const &x) const { return {*this, t, x}; } }; //! Proximal of the euclidian norm template auto euclidian_norm(typename real_type::type const &t, Eigen::MatrixBase const &x) -> decltype(EuclidianNorm(), t, x) { return EuclidianNorm()(t, x); } //! Proximal of the l1 norm template void l1_norm(Eigen::DenseBase &out, typename real_type::type gamma, Eigen::DenseBase const &x) { out = sopt::soft_threshhold(x, gamma); } //! \brief Proximal of the l1 norm //! \detail This specialization makes it easier to use in algorithms, e.g. within `SDMM::append`. template void l1_norm(Vector &out, typename real_type::type gamma, Vector const &x) { l1_norm, Vector>(out, gamma, x); } //! \brief Proximal of l1 norm //! \details For more complex version involving linear transforms and weights, see L1TightFrame and //! L1 classes. In practice, this is an alias for soft_threshhold. template auto l1_norm(typename real_type::type gamma, Eigen::DenseBase const &x) -> decltype(sopt::soft_threshhold(x, gamma)) { return sopt::soft_threshhold(x, gamma); } //! Proximal for projection on the positive quadrant template void positive_quadrant(Vector &out, typename real_type::type, Vector const &x) { out = sopt::positive_quadrant(x); }; //! Proximal for indicator function of L2 ball template class L2Ball { public: typedef typename real_type::type Real; //! Constructs an L2 ball proximal of size epsilon L2Ball(Real epsilon) : epsilon_(epsilon) {} //! Calls proximal function void operator()(Vector &out, typename real_type::type, Vector const &x) const { return operator()(out, x); } //! Calls proximal function void operator()(Vector &out, Vector const &x) const { auto const norm = x.stableNorm(); if(norm > epsilon()) out = x * (epsilon() / norm); else out = x; } //! Lazy version template EnveloppeExpression operator()(Real const &, Eigen::MatrixBase const &x) const { return {*this, x}; } //! Lazy version template EnveloppeExpression operator()(Eigen::MatrixBase const &x) const { return {*this, x}; } //! Size of the ball Real epsilon() const { return epsilon_; } //! Size of the ball L2Ball &epsilon(Real eps) { epsilon_ = eps; return *this; } protected: //! Size of the ball Real epsilon_; }; template class WeightedL2Ball : public L2Ball { public: typedef typename L2Ball::Real Real; typedef Vector t_Vector; //! Constructs an L2 ball proximal of size epsilon with given weights template WeightedL2Ball(Real epsilon, Eigen::DenseBase const &w) : L2Ball(epsilon), weights_(w) {} //! Constructs an L2 ball proximal of size epsilon WeightedL2Ball(Real epsilon) : WeightedL2Ball(epsilon, t_Vector::Ones(1)) {} //! Calls proximal function void operator()(Vector &out, typename real_type::type, Vector const &x) const { return operator()(out, x); } //! Calls proximal function void operator()(Vector &out, Vector const &x) const { auto const norm = weights().size() == 1 ? x.stableNorm() * std::abs(weights()(0)) : (x.array() * weights().array()).matrix().stableNorm(); if(norm > epsilon()) out = x * (epsilon() / norm); else out = x; } //! Lazy version template EnveloppeExpression operator()(Real const &, Eigen::MatrixBase const &x) const { return {*this, x}; } //! Lazy version template EnveloppeExpression operator()(Eigen::MatrixBase const &x) const { return {*this, x}; } //! Weights associated with each dimension t_Vector const &weights() const { return weights_; } //! Weights associated with each dimension template WeightedL2Ball &weights(Eigen::MatrixBase const &w) { if((w.array() < 0e0).any()) SOPT_THROW("Weights cannot be negative"); if(w.stableNorm() < 1e-12) SOPT_THROW("Weights cannot be null"); weights_ = w; return *this; } //! Size of the ball Real epsilon() const { return L2Ball::epsilon(); } //! Size of the ball WeightedL2Ball &epsilon(Real const &eps) { L2Ball::epsilon(eps); return *this; } protected: t_Vector weights_; }; //! Translation over proximal function template class Translation { public: //! Creates proximal of translated function template Translation(FUNCTION const &func, T_VECTOR const &trans) : func(func), trans(trans) {} //! Computes proximal of translated function template typename std::enable_if::value, void>::type operator()(OUTPUT out, typename real_type::type const &t, Eigen::MatrixBase const &x) const { func(out, t, x + trans); out -= trans; } //! Computes proximal of translated function template void operator()(Vector &out, typename real_type::type const &t, Eigen::MatrixBase const &x) const { func(out, t, x + trans); out -= trans; } //! Lazy version template ProximalExpression const &, T0> operator()(typename T0::Scalar const &t, Eigen::MatrixBase const &x) const { return {*this, t, x}; } private: //! Function to translate FUNCTION const func; //! Translation VECTOR const trans; }; //! Translates given proximal by given vector template Translation translate(FUNCTION const &func, VECTOR const &translation) { return Translation(func, translation); } } } /* sopt::proximal */ #endif sopt-2.0.0/cpp/sopt/proximal_expression.h000066400000000000000000000066531277570055300205550ustar00rootroot00000000000000#ifndef SOPT_PROXIMAL_EXPRESSION_H #define SOPT_PROXIMAL_EXPRESSION_H #include "sopt/config.h" #include #include #include "sopt/maths.h" namespace sopt { //! Holds some standard proximals namespace proximal { namespace details { //! \brief Expression referencing a lazy proximal function call //! \details It helps transform the call ``proximal(out, gamma, input)`` //! to ``out = proximal(gamma, input)`` without incurring copy or allocation overhead if ``out`` //! already exists. template class DelayedProximalFunction : public Eigen::ReturnByValue> { public: typedef typename DERIVED::PlainObject PlainObject; typedef typename DERIVED::Index Index; typedef typename real_type::type Real; DelayedProximalFunction(FUNCTION const &func, Real const &gamma, DERIVED const &x) : func(func), gamma(gamma), x(x) {} DelayedProximalFunction(DelayedProximalFunction const &c) : func(c.func), gamma(c.gamma), x(c.x) {} DelayedProximalFunction(DelayedProximalFunction &&c) : func(std::move(c.func)), gamma(c.gamma), x(c.x) {} template void evalTo(DESTINATION &destination) const { destination.resizeLike(x); func(destination, gamma, x); } Index rows() const { return x.rows(); } Index cols() const { return x.cols(); } private: FUNCTION const func; Real const gamma; DERIVED const &x; }; //! \brief Expression referencing a lazy function call to envelope proximal //! \details It helps transform the call ``proximal(out, input)`` //! to ``out = proximal(input)`` without incurring copy or allocation overhead if ``out`` //! already exists. template class DelayedProximalEnveloppeFunction : public Eigen::ReturnByValue> { public: typedef typename DERIVED::PlainObject PlainObject; typedef typename DERIVED::Index Index; typedef typename real_type::type Real; DelayedProximalEnveloppeFunction(FUNCTION const &func, DERIVED const &x) : func(func), x(x) {} DelayedProximalEnveloppeFunction(DelayedProximalEnveloppeFunction const &c) : func(c.func), x(c.x) {} DelayedProximalEnveloppeFunction(DelayedProximalEnveloppeFunction &&c) : func(std::move(c.func)), x(c.x) {} template void evalTo(DESTINATION &destination) const { destination.resizeLike(x); func(destination, x); } Index rows() const { return x.rows(); } Index cols() const { return x.cols(); } private: FUNCTION const func; DERIVED const &x; }; } /* details */ //! Eigen expression from proximal functions template using ProximalExpression = details::DelayedProximalFunction>; //! Eigen expression from proximal enveloppe functions template using EnveloppeExpression = details::DelayedProximalEnveloppeFunction>; } } /* sopt::proximal */ namespace Eigen { namespace internal { template struct traits> { typedef typename VECTOR::PlainObject ReturnType; }; template struct traits> { typedef typename VECTOR::PlainObject ReturnType; }; } } #endif sopt-2.0.0/cpp/sopt/real_type.h000066400000000000000000000034701277570055300164210ustar00rootroot00000000000000#ifndef SOPT_REAL_TYPE_H #define SOPT_REAL_TYPE_H #include "sopt/config.h" #include #include namespace sopt { namespace details { // Checks wether a type has contains a type "value_type" template struct HasValueType { using Have = char[1]; using HaveNot = char[2]; struct Fallback { struct value_type {}; }; struct Derived : T, Fallback {}; template static Have &test(typename U::value_type *); template static HaveNot &test(U *); public: static constexpr bool value = sizeof(test(nullptr)) == sizeof(HaveNot); }; // Specialization for fundamental type that cannot be derived from template struct HasValueType::value>::type> : std::false_type {}; //! Detects whether a class contains a value_type type template ::value> class has_value_type; template class has_value_type : public std::true_type {}; template class has_value_type : public std::false_type {}; //! Computes inner-most element type template ::value> class underlying_value_type; template class underlying_value_type { public: typedef T type; }; template class underlying_value_type { public: typedef typename underlying_value_type::type type; }; } //! Gets to the underlying real type template using real_type = details::underlying_value_type; //! True if underlying type is complex template struct is_complex : public std::false_type {}; //! True if underlying type is complex template struct is_complex, void> : public std::true_type {}; } /* sopt */ #endif sopt-2.0.0/cpp/sopt/relative_variation.h000066400000000000000000000027411277570055300203240ustar00rootroot00000000000000#ifndef SOPT_RELATIVE_VARIATION_H #define SOPT_RELATIVE_VARIATION_H #include "sopt/config.h" #include #include "sopt/logging.h" #include "sopt/maths.h" namespace sopt { template class RelativeVariation { public: //! Underlying scalar type typedef TYPE Scalar; //! Underlying scalar type typedef typename real_type::type Real; //! Maximum variation from one step to the next RelativeVariation(Real epsilon) : epsilon_(epsilon), previous(typename Array::Index(0)){}; //! Copy constructor RelativeVariation(RelativeVariation const &c) : epsilon_(c.epsilon_), previous(c.previous){}; //! True if object has changed by less than epsilon template bool operator()(Eigen::MatrixBase const &input) { return operator()(input.array()); } //! True if object has changed by less than epsilon template bool operator()(Eigen::ArrayBase const &input) { if(previous.size() != input.size()) { previous = input; return false; } auto const norm = (input - previous).matrix().squaredNorm(); previous = input; SOPT_LOW_LOG(" - relative variation: {} previous; }; } /* sopt */ #endif sopt-2.0.0/cpp/sopt/reweighted.h000066400000000000000000000240621277570055300165640ustar00rootroot00000000000000#ifndef SOPT_REWEIGHTED_H #define SOPT_REWEIGHTED_H #include "sopt/linear_transform.h" #include "sopt/types.h" namespace sopt { namespace algorithm { template class Reweighted; //! Factory function to create an l0-approximation by reweighting an l1 norm template Reweighted reweighted(ALGORITHM const &algo, typename Reweighted::t_SetWeights const &set_weights, typename Reweighted::t_Reweightee const &reweightee); //! \brief L0-approximation algorithm, through reweighting //! \details This algorithm approximates \f$min_x ||Ψ^Tx||_0 + f(x)\f$ by solving the set of //! problems \f$j\f$, \f$min_x ||W_jΨ^Tx||_1 + f(x)\f$ where the *diagonal* matrix \f$W_j\f$ is set //! using the results from \f$j-1\f$: \f$ δ_j W_j^{-1} = δ_j + ||W_{j-1}Ψ^T||_1\f$. \f$δ_j\f$ //! prevents division by zero. It is a series which converges to zero. By default, //! \f$δ_{j+1}=0.1δ_j\f$. //! //! The algorithm proceeds needs three forms of input: //! - the inner algorithm, e.g. ImagingProximalADMM //! - a function returning Ψ^Tx given x //! - a function to modify the inner algorithm with new weights template class Reweighted { public: //! Inner-loop algorithm typedef ALGORITHM Algorithm; //! Scalar type typedef typename Algorithm::Scalar Scalar; //! Real type typedef typename real_type::type Real; //! Weight vector type typedef Vector WeightVector; //! Type of then underlying vectors typedef typename Algorithm::t_Vector XVector; //! Type of the convergence function typedef typename Algorithm::t_IsConverged t_IsConverged; //! \brief Type of the function that is subject to reweighting //! \details E.g. \f$Ψ^Tx\f$. Note that l1-norm is not applied here. typedef std::function t_Reweightee; //! Type of the function to set weights typedef std::function t_SetWeights; //! Function to update delta at each turn typedef std::function t_DeltaUpdate; //! output from running reweighting scheme struct ReweightedResult { //! Number of iterations (outer loop) t_uint niters; //! Wether convergence was achieved bool good; //! Weights at last iteration WeightVector weights; //! Result from last inner loop typename Algorithm::DiagnosticAndResult algo; //! Default construction ReweightedResult() : niters(0), good(false), weights(WeightVector::Ones(1)), algo() {} }; Reweighted(Algorithm const &algo, t_SetWeights const &setweights, t_Reweightee const &reweightee) : algo_(algo), setweights_(setweights), reweightee_(reweightee), itermax_(std::numeric_limits::max()), min_delta_(0e0), is_converged_(), update_delta_([](Real delta) { return 1e-1 * delta; }) {} //! Underlying "inner-loop" algorithm Algorithm &algorithm() { return algo_; } //! Underlying "inner-loop" algorithm Algorithm const &algorithm() const { return algo_; } //! Sets the underlying "inner-loop" algorithm Reweighted &algorithm(Algorithm const &algo) { algo_ = algo; return *this; } //! Sets the underlying "inner-loop" algorithm Reweighted &algorithm(Algorithm &&algo) { algo_ = std::move(algo); return *this; } //! Function to reset the weights in the algorithm t_SetWeights const &set_weights() const { return setweights_; } //! Function to reset the weights in the algorithm Reweighted &set_weights(t_SetWeights const &setweights) const { setweights_ = setweights; return *this; } //! Sets the weights on the underlying algorithm void set_weights(Algorithm &algo, WeightVector const &weights) const { return set_weights()(algo, weights); } //! Function that needs to be reweighted //! \details E.g. \f$Ψ^Tx\f$. Note that l1-norm is not applied here. Reweighted &reweightee(t_Reweightee const &rw) { reweightee_ = rw; return *this; } //! Function that needs to be reweighted t_Reweightee const &reweightee() const { return reweightee_; } //! Forwards to the reweightee function XVector reweightee(XVector const &x) const { return reweightee()(algorithm(), x); } //! Maximum number of reweighted iterations t_uint itermax() const { return itermax_; } Reweighted &itermax(t_uint i) { itermax_ = i; return *this; } //! Lower limit for delta Real min_delta() const { return min_delta_; } Reweighted &min_delta(Real min_delta) { min_delta_ = min_delta; return *this; } //! Checks convergence of the reweighting scheme t_IsConverged const &is_converged() const { return is_converged_; } Reweighted &is_converged(t_IsConverged const &convergence) { is_converged_ = convergence; return *this; } bool is_converged(XVector const &x) const { return is_converged() ? is_converged()(x) : false; } //! \brief Performs reweighting //! \details This overload will compute an initial result without initial weights set to one. template typename std::enable_if::value or std::is_same::value), ReweightedResult>::type operator()(INPUT const &input) const; //! \brief Performs reweighting //! \details This overload will compute an initial result without initial weights set to one. ReweightedResult operator()() const; //! Reweighted algorithm, from prior call to inner-algorithm ReweightedResult operator()(typename Algorithm::DiagnosticAndResult const &warm) const; //! Reweighted algorithm, from prior call to reweighting algorithm ReweightedResult operator()(ReweightedResult const &warm) const; //! Updates delta Real update_delta(Real delta) const { return update_delta()(delta); } //! Updates delta t_DeltaUpdate const &update_delta() const { return update_delta_; } //! Updates delta Reweighted update_delta(t_DeltaUpdate const &ud) const { return update_delta_ = ud; } protected: //! Inner loop algorithm Algorithm algo_; //! Function to set weights t_SetWeights setweights_; //! \brief Function that is subject to reweighting //! \details E.g. \f$Ψ^Tx\f$. Note that l1-norm is not applied here. t_Reweightee reweightee_; //! Maximum number of reweighted iterations t_uint itermax_; //! \brief Lower limit for delta Real min_delta_; //! Checks convergence t_IsConverged is_converged_; //! Updates delta at each turn t_DeltaUpdate update_delta_; }; template template typename std:: enable_if::value or std::is_same::ReweightedResult>::value), typename Reweighted::ReweightedResult>::type Reweighted::operator()(INPUT const &input) const { Algorithm algo = algorithm(); set_weights(algo, WeightVector::Ones(1)); return operator()(algo(input)); } template typename Reweighted::ReweightedResult Reweighted::operator()() const { Algorithm algo = algorithm(); set_weights(algo, WeightVector::Ones(1)); return operator()(algo()); } template typename Reweighted::ReweightedResult Reweighted:: operator()(typename Algorithm::DiagnosticAndResult const &warm) const { ReweightedResult result; result.algo = warm; result.weights = WeightVector::Ones(1); return operator()(result); } template typename Reweighted::ReweightedResult Reweighted:: operator()(ReweightedResult const &warm) const { SOPT_HIGH_LOG("Starting reweighted scheme"); // Copies inner algorithm, so that operator() can be constant Algorithm algo(algorithm()); ReweightedResult result(warm); auto delta = std::max(standard_deviation(reweightee(warm.algo.x)), min_delta()); SOPT_LOW_LOG("- Initial delta: {}", delta); for(result.niters = 0; result.niters < itermax(); ++result.niters) { SOPT_LOW_LOG("Reweigting iteration {}/{} ", result.niters, itermax()); SOPT_LOW_LOG(" - delta: {}", delta); result.weights = delta / (delta + reweightee(result.algo.x).array().abs()); set_weights(algo, result.weights); result.algo = algo(result.algo); if(is_converged(result.algo.x)) { SOPT_MEDIUM_LOG("Reweighting scheme did converge in {} iterations", result.niters); result.good = true; break; } delta = std::max(min_delta(), update_delta(delta)); } // result is always good if no convergence function is defined if(not is_converged()) result.good = true; else if(not result.good) SOPT_ERROR("Reweighting scheme did *not* converge in {} iterations", itermax()); return result; } //! Factory function to create an l0-approximation by reweighting an l1 norm template Reweighted reweighted(ALGORITHM const &algo, typename Reweighted::t_SetWeights const &set_weights, typename Reweighted::t_Reweightee const &reweightee) { return {algo, set_weights, reweightee}; } template class ImagingProximalADMM; template class PositiveQuadrant; template Eigen::CwiseUnaryOp, const T> positive_quadrant(Eigen::DenseBase const &input); template Reweighted>> reweighted(ImagingProximalADMM const &algo) { auto const posq = positive_quadrant(algo); typedef typename std::remove_const::type Algorithm; typedef Reweighted RW; auto const reweightee = [](Algorithm const &posq, typename RW::XVector const &x) -> typename RW::XVector { return posq.algorithm().Psi().adjoint() * x; }; auto const set_weights = [](Algorithm &posq, typename RW::WeightVector const &weights) -> void { posq.algorithm().l1_proximal_weights(weights); }; return {posq, set_weights, reweightee}; } } // namespace algorithm } // namespace sopt #endif sopt-2.0.0/cpp/sopt/sampling.cc000066400000000000000000000004541277570055300164040ustar00rootroot00000000000000#include "sopt/sampling.h" namespace sopt { Sampling::Sampling(t_uint size, t_uint samples) : indices(size), size(size) { std::iota(indices.begin(), indices.end(), 0); std::shuffle(indices.begin(), indices.end(), std::mt19937(std::random_device()())); indices.resize(samples); } } /* sopt */ sopt-2.0.0/cpp/sopt/sampling.h000066400000000000000000000061211277570055300162430ustar00rootroot00000000000000#ifndef SOPT_SAMPLING_H #define SOPT_SAMPLING_H #include "sopt/config.h" #include "sopt/config.h" #include #include #include #include #include "sopt/linear_transform.h" #include "sopt/types.h" namespace sopt { //! \brief An operator that samples a set of measurements. //! \details Picks some elements from a vector class Sampling { public: //! Constructs from a vector Sampling(t_uint size, std::vector const &indices) : indices(indices), size(size) {} //! Constructs from the size and the number of samples to pick template Sampling(t_uint size, t_uint samples, RNG &&rng); //! Constructs from the size and the number of samples to pick Sampling(t_uint size, t_uint samples) : Sampling(size, samples, std::mt19937_64(std::random_device()())) {} // Performs sampling template void operator()(Eigen::DenseBase &out, Eigen::DenseBase const &x) const; // Performs sampling template void operator()(Eigen::DenseBase &&out, Eigen::DenseBase const &x) const { operator()(out, x); } // Performs adjunct of sampling template void adjoint(Eigen::DenseBase &out, Eigen::DenseBase const &x) const; // Performs adjunct sampling template void adjoint(Eigen::DenseBase &&out, Eigen::DenseBase const &x) const { adjoint(out, x); } //! Size of the vector returned by the adjoint operation t_uint cols() const { return size; } //! Number of measurements t_uint rows() const { return indices.size(); } protected: //! Set of indices to pick std::vector indices; //! Original vector size t_uint size; }; template void Sampling::operator()(Eigen::DenseBase &out, Eigen::DenseBase const &x) const { out.resize(indices.size()); for(decltype(indices.size()) i(0); i < indices.size(); ++i) { assert(indices[i] < static_cast(x.size())); out[i] = x[indices[i]]; } } template void Sampling::adjoint(Eigen::DenseBase &out, Eigen::DenseBase const &x) const { assert(static_cast(x.size()) == indices.size()); out.resize(out.size()); out.fill(0); for(decltype(indices.size()) i(0); i < indices.size(); ++i) { assert(indices[i] < static_cast(out.size())); out[indices[i]] = x[i]; } } //! Returns linear transform version of this object. template LinearTransform> linear_transform(Sampling const &sampling) { return linear_transform>( [sampling](Vector &out, Vector const &x) { sampling(out, x); }, {{0, 1, static_cast(sampling.rows())}}, [sampling](Vector &out, Vector const &x) { sampling.adjoint(out, x); }, {{0, 1, static_cast(sampling.cols())}}); } template Sampling::Sampling(t_uint size, t_uint samples, RNG &&rng) : indices(size), size(size) { std::iota(indices.begin(), indices.end(), 0); std::shuffle(indices.begin(), indices.end(), rng); indices.resize(samples); } } /* sopt */ #endif sopt-2.0.0/cpp/sopt/sdmm.h000066400000000000000000000277121277570055300154020ustar00rootroot00000000000000#ifndef SOPT_SDMM_H #define SOPT_SDMM_H #include "sopt/config.h" #include #include #include #include "sopt/conjugate_gradient.h" #include "sopt/exception.h" #include "sopt/linear_transform.h" #include "sopt/logging.h" #include "sopt/proximal.h" #include "sopt/proximal_expression.h" #include "sopt/types.h" #include "sopt/wrapper.h" namespace sopt { namespace algorithm { //! \brief Simultaneous-direction method of the multipliers //! \details The algorithm is detailed in (doi) 10.1093/mnras/stu202. template class SDMM { public: //! Values indicating how the algorithm ran struct Diagnostic { //! Number of iterations t_uint niters; //! Wether convergence was achieved bool good; //! Conjugate gradient result ConjugateGradient::Diagnostic cg_diagnostic; }; struct DiagnosticAndResult : public Diagnostic { //! Vector which minimizes the sum of functions. Vector x; }; //! Scalar type typedef SCALAR value_type; //! Scalar type typedef value_type Scalar; //! Real type typedef typename real_type::type Real; //! Type of then underlying vectors typedef Vector t_Vector; //! Type of the A and A^H operations typedef LinearTransform t_LinearTransform; //! Type of the proximal functions typedef ProximalFunction t_Proximal; //! Type of the convergence function typedef ConvergenceFunction t_IsConverged; SDMM() : itermax_(std::numeric_limits::max()), gamma_(1e-8), conjugate_gradient_(std::numeric_limits::max(), 1e-6), is_converged_([](t_Vector const &) { return false; }) {} virtual ~SDMM() {} // Macro helps define properties that can be initialized as in // auto sdmm = SDMM().prop0(value).prop1(value); #define SOPT_MACRO(NAME, TYPE) \ TYPE const &NAME() const { return NAME##_; } \ SDMM &NAME(TYPE const &NAME) { \ NAME##_ = NAME; \ return *this; \ } \ \ protected: \ TYPE NAME##_; \ \ public: //! Maximum number of iterations SOPT_MACRO(itermax, t_uint); //! Gamma SOPT_MACRO(gamma, Real); //! Conjugate gradient SOPT_MACRO(conjugate_gradient, ConjugateGradient); //! A function verifying convergence SOPT_MACRO(is_converged, t_IsConverged); #undef SOPT_MACRO //! Helps setup conjugate gradient SDMM &conjugate_gradient(t_uint itermax, t_real tolerance) { conjugate_gradient_.itermax(itermax); conjugate_gradient_.tolerance(tolerance); return *this; } //! \brief Appends a proximal and linear transform template SDMM &append(PROXIMAL proximal, T args) { proximals().emplace_back(proximal); transforms().emplace_back(linear_transform(args)); return *this; } //! \brief Appends a proximal with identity as the linear transform template SDMM &append(PROXIMAL proximal) { return append(proximal, linear_transform_identity()); } //! \brief Appends a proximal with the linear transform as pair of functions template SDMM &append(PROXIMAL proximal, L l, LADJOINT ladjoint) { return append(proximal, linear_transform(l, ladjoint)); } //! \brief Appends a proximal with the linear transform as pair of functions template SDMM &append(PROXIMAL proximal, L l, LADJOINT ladjoint, std::array sizes) { return append(proximal, linear_transform(l, ladjoint, sizes)); } //! \brief Appends a proximal with the linear transform as pair of functions template SDMM &append(PROXIMAL proximal, L l, std::array dsizes, LADJOINT ladjoint, std::array isizes) { return append(proximal, linear_transform(l, dsizes, ladjoint, isizes)); } //! \brief Implements SDMM //! \details Follows Combettes and Pesquet "Proximal Splitting Methods in Signal Processing", //! arXiv:0912.3522v4 [math.OC] (2010), equation 65. //! See therein for notation Diagnostic operator()(t_Vector &out, t_Vector const &input) const; DiagnosticAndResult operator()(t_Vector const &input) const { DiagnosticAndResult result; static_cast(result) = operator()(result.x, input); return result; } //! Makes it simple to chain different calls to SDMM DiagnosticAndResult operator()(DiagnosticAndResult const &warmstart) const { DiagnosticAndResult result; static_cast(result) = operator()(result.x, warmstart.x); return result; } //! Linear transforms associated with each objective function std::vector const &transforms() const { return transforms_; } //! Linear transforms associated with each objective function std::vector &transforms() { return transforms_; } //! Linear transform associated with a given objective function t_LinearTransform const &transforms(t_uint i) const { return transforms_[i]; } //! Linear transform associated with a given objective function t_LinearTransform &transforms(t_uint i) { return transforms_[i]; } //! Proximal of each objective function std::vector const &proximals() const { return proximals_; } //! Linear transforms associated with each objective function std::vector &proximals() { return proximals_; } //! Proximal associated with a given objective function t_Proximal const &proximals(t_uint i) const { return proximals_[i]; } //! Proximal associated with a given objective function t_Proximal &proximals(t_uint i) { return proximals_[i]; } //! Lazy call to specific proximal function template proximal::ProximalExpression proximals(t_uint i, Eigen::MatrixBase const &x) const { return {proximals()[i], gamma(), x}; } //! Number of terms t_uint size() const { return proximals().size(); } // We must declare the first argument explicitly so that the function never // match the getter with the same name. //! \brief Forwards to internal conjugage gradient object //! \details Removes the need for ugly extra brackets. template auto conjugate_gradient(T0 &&t0, T &&... args) const -> decltype(this->conjugate_gradient()(std::forward(t0), std::forward(args)...)) { return conjugate_gradient()(std::forward(t0), std::forward(args)...); } //! Forwards to convergence function parameter bool is_converged(t_Vector const &x) const { return is_converged()(x); } protected: //! Linear transforms associated with each objective function std::vector transforms_; //! Proximal of each objective function std::vector proximals_; //! Type of the list of vectors typedef std::vector t_Vectors; //! Conjugate gradient step virtual ConjugateGradient::Diagnostic solve_for_xn(t_Vector &out, t_Vectors const &y, t_Vectors const &z) const; //! Direction step virtual void update_directions(t_Vectors &y, t_Vectors &z, t_Vector const &x) const; //! Initializes intermediate values virtual void initialization(t_Vectors &y, t_Vectors &z, t_Vector const &x) const; //! Checks that the input make sense virtual void sanity_check(t_Vector const &input) const; }; template typename SDMM::Diagnostic SDMM:: operator()(t_Vector &out, t_Vector const &input) const { sanity_check(input); bool convergence = false; t_uint niters(0); // Figures out where itermax or convergence reached auto const has_finished = [&convergence, &niters, this](t_Vector const &out) { convergence = is_converged(out); return niters >= itermax() or convergence; }; SOPT_HIGH_LOG("Performing SDMM "); out = input; t_Vectors y(transforms().size()), z(transforms().size()); // Initial step replaces iteration update with initialization initialization(y, z, input); auto cg_diagnostic = solve_for_xn(out, y, z); while(not has_finished(out)) { SOPT_LOW_LOG("Iteration {}/{}. ", niters, itermax()); // computes y and z from out and transforms update_directions(y, z, out); SOPT_LOW_LOG(" - sum z_ij = {}", std::accumulate(z.begin(), z.end(), Scalar(0e0), [](Scalar const &a, t_Vector const &z) { return a + z.sum(); })); // computes x = L^-1 y cg_diagnostic = solve_for_xn(out, y, z); SOPT_LOW_LOG(" - CG Residual = {} in {}/{} iterations", cg_diagnostic.residual, cg_diagnostic.niters, conjugate_gradient().itermax()); ++niters; } return {niters, convergence, cg_diagnostic}; } template ConjugateGradient::Diagnostic SDMM::solve_for_xn(t_Vector &out, t_Vectors const &y, t_Vectors const &z) const { assert(z.size() == transforms().size()); assert(y.size() == transforms().size()); SOPT_TRACE("Solving for x_n"); // Initialize b of A x = b = sum_i L_i^H(z_i - y_i) t_Vector b = out.Zero(out.size()); for(t_uint i(0); i < transforms().size(); ++i) b += transforms(i).adjoint() * (y[i] - z[i]); if(b.stableNorm() < 1e-12) { out.fill(0e0); return {0, 0, true}; } // Then create operator A auto A = [this](t_Vector &out, t_Vector const &input) { out = out.Zero(input.size()); for(auto const &transform : this->transforms()) out += transform.adjoint() * (transform * input).eval(); }; // Call conjugate gradient auto const diagnostic = this->conjugate_gradient(out, A, b); if(not diagnostic.good) { SOPT_ERROR("CG error - iterations: {}/{} - residuals {}\n", diagnostic.niters, conjugate_gradient().itermax(), diagnostic.residual); SOPT_THROW("Conjugate gradient failed to converge"); } return diagnostic; } template void SDMM::update_directions(t_Vectors &y, t_Vectors &z, t_Vector const &x) const { SOPT_TRACE("Updating directions"); for(t_uint i(0); i < transforms().size(); ++i) { z[i] += transforms(i) * x; y[i] = proximals(i, z[i]); z[i] -= y[i]; } } template void SDMM::initialization(t_Vectors &y, t_Vectors &z, t_Vector const &x) const { SOPT_TRACE("Initializing SDMM"); for(t_uint i(0); i < transforms().size(); i++) { y[i] = transforms(i) * x; z[i].resize(y[i].size()); z[i].fill(0); assert(z[i].size() == y[i].size()); SOPT_TRACE(" - transform {}: {}", i, y[i].transpose()); } } template void SDMM::sanity_check(t_Vector const &x) const { bool doexit = false; if(proximals().size() != transforms().size()) { SOPT_ERROR("Internal error: number of proximals and transforms do not match"); doexit = true; } if(x.size() == 0) SOPT_WARN("Input vector has zero size"); if(size() == 0) SOPT_WARN("No operators - SDMM is empty"); for(t_uint i(0); i < size(); ++i) { auto const xdual = t_Vector::Zero((transforms(i) * x).size()); auto const r = (transforms(i).adjoint() * xdual).size(); if(r != x.size()) { SOPT_ERROR("Output size of transform {} and input do not match: {} vs {}", i, r, x.size()); doexit = true; } } if(doexit) SOPT_THROW("Input to SDMM is inconsistent"); } } } /* sopt::algorithm */ #endif sopt-2.0.0/cpp/sopt/types.h000066400000000000000000000034051277570055300155770ustar00rootroot00000000000000#ifndef BICO_TRAITS_H #define BICO_TRAITS_H #include "sopt/config.h" #include #include #include #include "sopt/real_type.h" namespace sopt { //! Root of the type hierarchy for signed integers typedef int t_int; //! Root of the type hierarchy for unsigned integers typedef size_t t_uint; //! Root of the type hierarchy for real numbers typedef double t_real; //! Root of the type hierarchy for (real) complex numbers typedef std::complex t_complex; //! \brief A vector of a given type //! \details Operates as mathematical vector. template using Vector = Eigen::Matrix; //! \brief A matrix of a given type //! \details Operates as mathematical matrix. template using Matrix = Eigen::Matrix; //! \brief A 1-dimensional list of elements of given type //! \details Operates coefficient-wise, not matrix-vector-wise template using Array = Eigen::Array; //! \brief A 2-dimensional list of elements of given type //! \details Operates coefficient-wise, not matrix-vector-wise template using Image = Eigen::Array; //! Typical function out = A*x template > using OperatorFunction = std::function; //! Typical function signature for calls to proximal template using ProximalFunction = std::function &, typename real_type::type, Vector const &)>; //! Typical function signature for convergence template using ConvergenceFunction = std::function const &)>; } #endif sopt-2.0.0/cpp/sopt/utilities.cc000066400000000000000000000065401277570055300166070ustar00rootroot00000000000000#include "sopt/utilities.h" #include "sopt/config.h" #include #include #include #include "sopt/exception.h" #include "sopt/logging.h" #include "sopt/types.h" namespace { //! A single pixel //! Converts ABGR to greyscale double value double convert_to_greyscale(uint32_t &pixel) { uint8_t const blue = TIFFGetB(pixel); uint8_t const green = TIFFGetG(pixel); uint8_t const red = TIFFGetR(pixel); return static_cast(blue + green + red) / (3e0 * 255e0); } //! Converts greyscale double value to RGBA uint32_t convert_from_greyscale(double pixel) { uint32_t result = 0; uint8_t *ptr = (uint8_t *)&result; auto const g = [](double p) -> uint8_t { auto const scaled = 255e0 * p; if(scaled < 0) return 0; return scaled > 255 ? 255 : uint8_t(scaled); }; ptr[0] = g(pixel); ptr[1] = g(pixel); ptr[2] = g(pixel); ptr[3] = 255; return result; } } namespace sopt { namespace utilities { Image<> read_tiff(std::string const &filename) { SOPT_MEDIUM_LOG("Reading image file {} ", filename); TIFF *tif = TIFFOpen(filename.c_str(), "r"); if(not tif) SOPT_THROW("Could not open file ") << filename; uint32 width, height, t; TIFFGetField(tif, TIFFTAG_IMAGEWIDTH, &width); TIFFGetField(tif, TIFFTAG_IMAGELENGTH, &height); TIFFGetField(tif, TIFFTAG_PHOTOMETRIC, &t); SOPT_LOW_LOG("- image size {}, {} ", width, height); Image<> result = Image<>::Zero(height, width); uint32 *raster = (uint32 *)_TIFFmalloc(width * height * sizeof(uint32)); if(not raster) SOPT_THROW("Could not allocate memory to read file ") << filename; if(not TIFFReadRGBAImage(tif, width, height, raster, 0)) SOPT_THROW("Could not read file ") << filename; uint32_t *pixel = (uint32_t *)raster; for(uint32 i(0); i < height; ++i) for(uint32 j(0); j < width; ++j, ++pixel) result(i, j) = convert_to_greyscale(*pixel); _TIFFfree(raster); TIFFClose(tif); return result; } void write_tiff(Image<> const &image, std::string const &filename) { SOPT_MEDIUM_LOG("Writing image file {} ", filename); SOPT_LOW_LOG("- image size {}, {} ", image.rows(), image.cols()); TIFF *tif = TIFFOpen(filename.c_str(), "w"); if(not tif) SOPT_THROW("Could not open file ") << filename; uint32 const width = image.cols(); uint32 const height = image.rows(); SOPT_TRACE("Allocating buffer"); std::vector raster(width * height); TIFFSetField(tif, TIFFTAG_IMAGEWIDTH, width); TIFFSetField(tif, TIFFTAG_IMAGELENGTH, height); TIFFSetField(tif, TIFFTAG_SAMPLESPERPIXEL, sizeof(decltype(raster)::value_type)); TIFFSetField(tif, TIFFTAG_BITSPERSAMPLE, 8); TIFFSetField(tif, TIFFTAG_ROWSPERSTRIP, height); TIFFSetField(tif, TIFFTAG_PLANARCONFIG, PLANARCONFIG_CONTIG); TIFFSetField(tif, TIFFTAG_COMPRESSION, COMPRESSION_LZW); TIFFSetField(tif, TIFFTAG_PHOTOMETRIC, PHOTOMETRIC_RGB); TIFFSetField(tif, TIFFTAG_ORIENTATION, ORIENTATION_BOTLEFT); SOPT_TRACE("Initializing buffer"); auto pixel = raster.begin(); for(uint32 i(0); i < height; ++i) for(uint32 j(0); j < width; ++j, ++pixel) *pixel = convert_from_greyscale(image(i, j)); SOPT_TRACE("Writing strip"); TIFFWriteEncodedStrip(tif, 0, &raster[0], width * height * sizeof(decltype(raster)::value_type)); TIFFWriteDirectory(tif); SOPT_TRACE("Closing tif"); TIFFClose(tif); SOPT_TRACE("Freeing raster"); } } // utilities } // sopt sopt-2.0.0/cpp/sopt/utilities.h000066400000000000000000000005601277570055300164450ustar00rootroot00000000000000#ifndef SOPT_UTILITIES_H #define SOPT_UTILITIES_H #include "sopt/config.h" #include #include "sopt/types.h" namespace sopt { namespace utilities { //! Reads tiff image sopt::Image<> read_tiff(std::string const &name); //! Writes a tiff greyscale file void write_tiff(Image<> const &image, std::string const &filename); } } /* sopt::utilities */ #endif sopt-2.0.0/cpp/sopt/wavelets.h000066400000000000000000000073401277570055300162670ustar00rootroot00000000000000#ifndef SOPT_WAVELETS_H #define SOPT_WAVELETS_H // Convenience header to include wavelets headers and additional utilities #include "sopt/config.h" #include "sopt/linear_transform.h" #include "sopt/wavelets/sara.h" #include "sopt/wavelets/wavelets.h" namespace sopt { namespace details { namespace { //! Thin linear-transform wrapper around some operator accepting direct and indirect template LinearTransform> linear_transform(OP const &op) { return LinearTransform>( [&op](Vector &out, Vector const &x) { op.indirect(x.array(), out.array()); }, [&op](Vector &out, Vector const &x) { op.direct(out.array(), x.array()); }); } //! \brief Thin linear-transform wrapper around 2d wavelets //! \details Goes back and forth between vector representations and image representations, where //! images can be 2d. //! \param[in] op: Wavelet operator //! \param[in] rows: Number of rows in the image //! \param[in] cols: Number of columns in the image //! \param[in] factor: Allows for SARA transforms, i.e. more than one wavelet basis template LinearTransform> linear_transform(OP const &op, t_uint rows, t_uint cols, t_uint factor = 1) { if(rows == 1 or cols == 1) return linear_transform(op); return LinearTransform>( [&op, rows, cols, factor](Vector &out, Vector const &x) { assert(static_cast(x.size()) == rows * cols * factor); out.resize(rows * cols); auto signal = Image::Map(out.data(), rows, cols); auto const coeffs = Image::Map(x.data(), rows, cols * factor); op.indirect(coeffs, signal); }, {{0, 1, static_cast(rows * cols)}}, [&op, rows, cols, factor](Vector &out, Vector const &x) { assert(static_cast(x.size()) == rows * cols); out.resize(rows * cols * factor); auto const signal = Image::Map(x.data(), rows, cols); auto coeffs = Image::Map(out.data(), rows, cols * factor); op.direct(coeffs, signal); }, {{0, 1, static_cast(factor * rows * cols)}}); } } // anonymous } // details //! \brief Thin linear-transform wrapper around 1d wavelets //! \warning Because of the way Purify defines things, Ψ^T is actually the transform from signal to //! coefficients. template LinearTransform> linear_transform(wavelets::Wavelet const &wavelet) { return details::linear_transform(wavelet); } //! \brief Thin linear-transform wrapper around 1d sara operator //! \note Because of the way Purify defines things, Ψ^T is actually the transform from signal to //! coefficients. template LinearTransform> linear_transform(wavelets::SARA const &sara) { return details::linear_transform(sara); } //! \brief Thin linear-transform wrapper around 2d wavelets //! \note Because of the way Purify defines things, Ψ^T is actually the transform from signal to //! coefficients. template LinearTransform> linear_transform(wavelets::Wavelet const &wavelet, t_uint rows, t_uint cols = 1) { return details::linear_transform(wavelet, rows, cols); } //! \brief Thin linear-transform wrapper around 2d wavelets //! \param[in] sara: SARA wavelet dictionary //! \param[in] rows: Number of rows in the image //! \param[in] cols: Number of columns in the image //! \note Because of the way Purify defines things, Ψ^T is actually the transform from signal to //! coefficients. template LinearTransform> linear_transform(wavelets::SARA const &sara, t_uint rows, t_uint cols = 1) { return details::linear_transform(sara, rows, cols, sara.size()); } } /* sopt */ #endif sopt-2.0.0/cpp/sopt/wavelets/000077500000000000000000000000001277570055300161125ustar00rootroot00000000000000sopt-2.0.0/cpp/sopt/wavelets/direct.h000066400000000000000000000114171277570055300175410ustar00rootroot00000000000000#ifndef SOPT_WAVELETS_DIRECT_H #define SOPT_WAVELETS_DIRECT_H #include "sopt/config.h" #include #include "sopt/types.h" #include "sopt/wavelets/wavelet_data.h" // Function inside anonymouns namespace won't appear in library namespace sopt { namespace wavelets { namespace { //! \brief Single-level 1d direct transform //! \param[out] coeffs_: output of the function (despite the const) //! \param[in] signal: input signal for which to compute wavelet transform //! \param[in] wavelet: contains wavelet coefficients template typename std::enable_if::type direct_transform_impl(Eigen::ArrayBase const &coeffs_, Eigen::ArrayBase const &signal, WaveletData const &wavelet) { Eigen::ArrayBase &coeffs = const_cast &>(coeffs_); assert(coeffs.size() == signal.size()); assert(wavelet.direct_filter.low.size() == wavelet.direct_filter.high.size()); auto const N = signal.size() / 2; down_convolve(coeffs.head(N), signal, wavelet.direct_filter.low); down_convolve(coeffs.tail(coeffs.size() - N), signal, wavelet.direct_filter.high); } //! Single-level 2d direct transform //! \param[out] coeffs_: output of the function (despite the const) //! \param[inout] signal: input signal for which to compute wavelet transform. Input is modified. //! \param[in] wavelet: contains wavelet coefficients template typename std::enable_if::type direct_transform_impl(Eigen::ArrayBase const &coeffs_, Eigen::ArrayBase const &signal_, WaveletData const &wavelet) { Eigen::ArrayBase &coeffs = const_cast &>(coeffs_); Eigen::ArrayBase &signal = const_cast &>(signal_); assert(coeffs.rows() == signal.rows()); assert(coeffs.cols() == signal.cols()); assert(wavelet.direct_filter.low.size() == wavelet.direct_filter.high.size()); for(t_uint i(0); i < static_cast(coeffs.rows()); ++i) direct_transform_impl(coeffs.row(i).transpose(), signal.row(i).transpose(), wavelet); for(t_uint i(0); i < static_cast(coeffs.cols()); ++i) { signal.col(i) = coeffs.col(i); direct_transform_impl(coeffs.col(i), signal.col(i), wavelet); } } } //! \brief N-levels 1d direct transform //! \param[out] coeffs_: output of the function (despite the const) //! \param[in] signal: input signal for which to compute wavelet transfor //! \param[in] wavelet: contains wavelet coefficients //! \note The size of the coefficients should a multiple of $2^l$ where $l$ is the number of //! levels. template typename std::enable_if::type direct_transform(Eigen::ArrayBase &coeffs, Eigen::ArrayBase const &signal, t_uint levels, WaveletData const &wavelet) { assert(coeffs.rows() == signal.rows()); assert(coeffs.cols() == signal.cols()); auto input = copy(signal); if(levels > 0) direct_transform_impl(coeffs, input, wavelet); for(t_uint level(1); level < levels; ++level) { auto const N = static_cast(signal.size()) >> level; input.head(N) = coeffs.head(N); direct_transform_impl(coeffs.head(N), input.head(N), wavelet); } } //! N-levels 2d direct transform //! \param[in] signal: input signal for which to compute wavelet transfor //! \param[in] wavelet: contains wavelet coefficients //! \note The size of the coefficients should a multiple of $2^l$ where $l$ is the number of //! levels. template typename std::enable_if::type direct_transform(Eigen::ArrayBase const &coeffs_, Eigen::ArrayBase const &signal, t_uint levels, WaveletData const &wavelet) { assert(coeffs_.rows() == signal.rows()); assert(coeffs_.cols() == signal.cols()); Eigen::ArrayBase &coeffs = const_cast &>(coeffs_); if(levels == 0) { coeffs = signal; return; } auto input = copy(signal); direct_transform_impl(coeffs, input, wavelet); for(t_uint level(1); level < levels; ++level) { auto const Nx = static_cast(signal.rows()) >> level; auto const Ny = static_cast(signal.cols()) >> level; input.topLeftCorner(Nx, Ny) = coeffs.topLeftCorner(Nx, Ny); direct_transform_impl(coeffs.topLeftCorner(Nx, Ny), input.topLeftCorner(Nx, Ny), wavelet); } } //! \brief Direct 1d and 2d transform //! \note The size of the coefficients should a multiple of $2^l$ where $l$ is the number of //! levels. template auto direct_transform(Eigen::ArrayBase const &signal, t_uint levels, WaveletData const &wavelet) -> decltype(copy(signal)) { auto result = copy(signal); direct_transform(result, signal, levels, wavelet); return result; } } } #endif sopt-2.0.0/cpp/sopt/wavelets/indirect.h000066400000000000000000000117111277570055300200650ustar00rootroot00000000000000#ifndef SOPT_WAVELET_INDIRECT_H #define SOPT_WAVELET_INDIRECT_H #include "sopt/config.h" #include "sopt/types.h" #include "sopt/wavelets/innards.impl.h" #include "sopt/wavelets/wavelet_data.h" // Function inside anonymouns namespace won't appear in library namespace sopt { namespace wavelets { namespace { //! Single-level 1d indirect transform //! \param[in] coeffs_: input coefficients //! \param[out] signal: output with the reconstituted signal //! \param[in] wavelet: contains wavelet coefficients template typename std::enable_if::type indirect_transform_impl(Eigen::ArrayBase const &coeffs, Eigen::ArrayBase const &signal_, WaveletData const &wavelet) { Eigen::ArrayBase &signal = const_cast &>(signal_); assert(coeffs.size() == signal.size()); assert(coeffs.size() % 2 == 0); up_convolve_sum(signal, coeffs, wavelet.indirect_filter.low_even, wavelet.indirect_filter.low_odd, wavelet.indirect_filter.high_even, wavelet.indirect_filter.high_odd); } //! Single-level 2d indirect transform //! \param[in] coeffs_: input coefficients //! \param[out] signal: output with the reconstituted signal //! \param[in] wavelet: contains wavelet coefficients template typename std::enable_if::type indirect_transform_impl(Eigen::ArrayBase const &coeffs_, Eigen::ArrayBase const &signal_, WaveletData const &wavelet) { Eigen::ArrayBase &coeffs = const_cast &>(coeffs_); Eigen::ArrayBase &signal = const_cast &>(signal_); assert(coeffs.rows() == signal.rows() and coeffs.cols() == signal.cols()); assert(coeffs.rows() % 2 == 0 and coeffs.cols() % 2 == 0); for(typename T0::Index i(0); i < signal.rows(); ++i) indirect_transform_impl(coeffs.row(i).transpose(), signal.row(i).transpose(), wavelet); coeffs = signal; for(typename T0::Index j(0); j < signal.cols(); ++j) indirect_transform_impl(coeffs.col(j), signal.col(j), wavelet); } } //! \brief N-levels 1d indirect transform //! \param[in] coeffs_: input coefficients //! \param[out] signal: output with the reconstituted signal //! \param[in] wavelet: contains wavelet coefficients //! \note The size of the coefficients should a multiple of $2^l$ where $l$ is the number of //! levels. template typename std::enable_if::type indirect_transform(Eigen::ArrayBase const &coeffs, Eigen::ArrayBase &signal, t_uint levels, WaveletData const &wavelet) { if(levels == 0) return; assert(coeffs.rows() == signal.rows()); assert(coeffs.cols() == signal.cols()); assert(coeffs.size() % (1u << levels) == 0); auto input = copy(coeffs); for(t_uint level(levels - 1); level > 0; --level) { auto const N = static_cast(signal.size()) >> level; indirect_transform_impl(input.head(N), signal.head(N), wavelet); input.head(N) = signal.head(N); } indirect_transform_impl(input, signal, wavelet); } //! \brief N-levels 2d indirect transform //! \param[in] coeffs_: input coefficients //! \param[out] signal: output with the reconstituted signal //! \param[in] wavelet: contains wavelet coefficients //! \note The size of the signal and coefficients should a multiple of $2^l$ where $l$ is the //! number of levels. template typename std::enable_if::type indirect_transform(Eigen::ArrayBase const &coeffs_, Eigen::ArrayBase const &signal_, t_uint levels, WaveletData const &wavelet) { Eigen::ArrayBase &coeffs = const_cast &>(coeffs_); Eigen::ArrayBase &signal = const_cast &>(signal_); assert(coeffs.rows() == signal.rows()); assert(coeffs.cols() == signal.cols()); assert(coeffs.size() % (1u << levels) == 0); if(levels == 0) { signal = coeffs_; return; } auto input = copy(coeffs); for(t_uint level(levels - 1); level > 0; --level) { auto const Nx = static_cast(signal.rows()) >> level; auto const Ny = static_cast(signal.cols()) >> level; indirect_transform_impl(input.topLeftCorner(Nx, Ny), signal.topLeftCorner(Nx, Ny), wavelet); input.topLeftCorner(Nx, Ny) = signal.topLeftCorner(Nx, Ny); } indirect_transform_impl(input, signal, wavelet); } //! Indirect 1d and 2d transform //! \param[in] coeffs_: input coefficients //! \param[in] wavelet: contains wavelet coefficients //! \returns the reconstituted signal //! \note The size of the coefficients should a multiple of $2^l$ where $l$ is the number of //! levels. template auto indirect_transform(Eigen::ArrayBase const &coeffs, t_uint levels, WaveletData const &wavelet) -> decltype(copy(coeffs)) { auto result = copy(coeffs); indirect_transform(coeffs, result, levels, wavelet); return result; } } } #endif sopt-2.0.0/cpp/sopt/wavelets/innards.impl.h000066400000000000000000000137401277570055300206660ustar00rootroot00000000000000#ifndef SOPT_WAVELETS_INNARDS_H #define SOPT_WAVELETS_INNARDS_H #include "sopt/config.h" #include // Function inside anonymouns namespace won't appear in library namespace sopt { namespace wavelets { namespace { //! \brief Returns evaluated expression or copy of input //! \details Gets C++ to figure out what the exact type is. Eigen tries and avoids copies. But //! sometimes we actually want a copy, to make sure the arguments of a function are not modified template auto copy(Eigen::ArrayBase const &a) -> typename std::remove_const::type>::type { return a.eval(); } //! \brief Applies scalar product of the size b //! \details In practice, the operation is equivalent to the following //! - `a` can be seen as a periodic vector: `a[i] == a[i % a.size()]` //! - `a'` is the segment `a` = a[offset:offset+b.size()]` //! - the result is the scalar product of `a'` by `b`. template typename T0::Scalar periodic_scalar_product(Eigen::ArrayBase const &a, Eigen::ArrayBase const &b, typename T0::Index offset) { auto const Na = static_cast(a.size()); auto const Nb = static_cast(b.size()); // offset in [0, a.size()[ offset %= Na; if(offset < 0) offset += Na; // Simple case, just do it if(Na > Nb + offset) { return (a.segment(offset, Nb) * b).sum(); } // Wrap around, do it, but carefully typename T0::Scalar result(0); for(typename T0::Index i(0), j(offset); i < Nb; ++i, ++j) result += a(j % Na) * b(i); return result; } //! \brief Convolves the signal by the filter //! \details The signal is seen as a periodic vector during convolution template void convolve(Eigen::ArrayBase &result, Eigen::ArrayBase const &signal, Eigen::ArrayBase const &filter) { assert(result.size() == signal.size()); for(typename T0::Index i(0); i < t_int(signal.size()); ++i) result(i) = periodic_scalar_product(signal, filter, i); } //! \brief Convolve variation for vector blocks template void convolve(Eigen::VectorBlock &&result, Eigen::ArrayBase const &signal, Eigen::ArrayBase const &filter) { return convolve(result, signal, filter); } //! \brief Convolves and down-samples the signal by the filter //! \details Just like convolve, but does every other point. template void down_convolve(Eigen::ArrayBase &result, Eigen::ArrayBase const &signal, Eigen::ArrayBase const &filter) { assert(result.size() * 2 <= signal.size()); if(signal.rows() == 1) for(typename T0::Index i(0); i < result.size(); ++i) result(i) = periodic_scalar_product(signal.transpose(), filter, 2 * i); else for(typename T0::Index i(0); i < result.size(); ++i) result(i) = periodic_scalar_product(signal, filter, 2 * i); } //! \brief Dowsampling + convolve variation for vector blocks template void down_convolve(Eigen::VectorBlock &&result, Eigen::ArrayBase const &signal, Eigen::ArrayBase const &filter) { down_convolve(result, signal, filter); } //! Convolve and sims low and high pass of a signal template void convolve_sum(Eigen::ArrayBase &result, Eigen::ArrayBase const &low_pass_signal, Eigen::ArrayBase const &low_pass, Eigen::ArrayBase const &high_pass_signal, Eigen::ArrayBase const &high_pass) { assert(result.size() == low_pass_signal.size()); assert(result.size() == high_pass_signal.size()); assert(low_pass.size() == high_pass.size()); static_assert(std::is_signed::value, "loffset, hoffset expect signed values"); auto const loffset = 1 - static_cast(low_pass.size()); auto const hoffset = 1 - static_cast(high_pass.size()); for(typename T0::Index i(0); i < result.size(); ++i) { result(i) = periodic_scalar_product(low_pass_signal, low_pass, i + loffset) + periodic_scalar_product(high_pass_signal, high_pass, i + hoffset); } } //! \brief Convolves and up-samples at the same time //! \details If Cn shuffles the (periodic) vector right, U does the upsampling, //! and O is the convolution operation, then the normal upsample + convolution //! is O(Cn[U [a]], b). What we are doing here is U[O(Cn'[a'], b')]. This avoids //! some unnecessary operations (multiplying and summing zeros) and removes the //! need for temporary copies. Testing shows the operations are equivalent. But //! I certainly cannot show how on paper. template void up_convolve_sum(Eigen::ArrayBase &result, Eigen::ArrayBase const &coeffs, Eigen::ArrayBase const &low_even, Eigen::ArrayBase const &low_odd, Eigen::ArrayBase const &high_even, Eigen::ArrayBase const &high_odd) { assert(result.size() <= coeffs.size()); assert(result.size() % 2 == 0); assert(low_even.size() == high_even.size()); assert(low_odd.size() == high_odd.size()); auto const Nlow = (coeffs.size() + 1) / 2; auto const Nhigh = coeffs.size() / 2; auto const size = low_even.size() + low_odd.size(); auto const is_even = size % 2 == 0; auto const even_offset = (1 - size) / 2; auto const odd_offset = (1 - size) / 2 + (is_even ? 0 : 1); for(typename T0::Index i(0); i + 1 < result.size(); i += 2) { result(i + (is_even ? 1 : 0)) = periodic_scalar_product(coeffs.head(Nlow), low_even, i / 2 + even_offset) + periodic_scalar_product(coeffs.tail(Nhigh), high_even, i / 2 + even_offset); result(i + (is_even ? 0 : 1)) = periodic_scalar_product(coeffs.head(Nlow), low_odd, i / 2 + odd_offset) + periodic_scalar_product(coeffs.tail(Nhigh), high_odd, i / 2 + odd_offset); } } } } } #endif sopt-2.0.0/cpp/sopt/wavelets/sara.h000066400000000000000000000217711277570055300172210ustar00rootroot00000000000000#ifndef SOPT_WAVELETS_SARA_H #define SOPT_WAVELETS_SARA_H #include "sopt/config.h" #include #include #include #include #include "sopt/logging.h" #include "sopt/wavelets/wavelets.h" namespace sopt { namespace wavelets { //! Sparsity Averaging Reweighted Analysis class SARA : public std::vector { public: #ifndef SOPT_HAS_NOT_USING // Constructors using std::vector::vector; #else //! Default constructor SARA() : std::vector(){}; #endif //! Easy constructor SARA(std::initializer_list> const &init) : SARA(init.begin(), init.end()) {} //! Construct from any iterator over a (std:string, t_uint) tuple template (*std::declval())), std::string>::value and std::is_convertible(*std::declval())), t_uint>::value>::type> SARA(ITERATOR first, ITERATOR last) { for(; first != last; ++first) emplace_back(std::get<0>(*first), std::get<1>(*first)); } //! Destructor virtual ~SARA() {} //! \brief Direct transform //! \param[in] signal: computes wavelet coefficients for this signal. Its size must be a //! multiple of $2^l$ where $l$ is the maximum number of levels. Can be a matrix (2d-transform) //! or a column vector (1-d transform). //! \return wavelets coefficients arranged by columns: if the input is n by m, then the output //! is n by m * d, with d the number of wavelets. //! \details Supports 1 and 2 dimensional tranforms for real and complex data. template typename T0::PlainObject direct(Eigen::ArrayBase const &signal) const; //! \brief Direct transform //! \param[inout] coefficients: Output wavelet coefficients. Must be of the type as the input. //! If the input is n by m, and d is the number of wavelets, then the output should be n by (m * //! d). //! \param[in] signal: computes wavelet coefficients for this signal. Its size must be a //! multiple of $2^l$ where $l$ is the maximum number of levels. Can be a matrix (2d-transform) //! or a column vector (1-d transform). //! \details Supports 1 and 2 dimensional tranforms for real and complex data. template void direct(Eigen::ArrayBase &coefficients, Eigen::ArrayBase const &signal) const; //! \brief Direct transform //! \param[inout] coefficients: Output wavelet coefficients. Must be of the type as the input. //! If the input is n by m, and l is the number of wavelets, then the output should be n by (m * //! l). //! \param[in] signal: computes wavelet coefficients for this signal. Its size must be a //! multiple of $2^l$ where $l$ is the number of levels. Can be a matrix (2d-transform) or a //! column vector (1-d transform). //! \details Supports 1 and 2 dimensional tranforms for real and complex data. This version //! allows non-constant Eigen expressions to be passe on without the ugly `const_cast` of the //! cannonical approach. template void direct(Eigen::ArrayBase &&coefficients, Eigen::ArrayBase const &signal) const { direct(coefficients, signal); } //! \brief Indirect transform //! \param[in] coefficients: Input wavelet coefficients. Its size must be a multiple of $2^l$ //! where $l$ is the number of levels. Can be a matrix (2d-transform) or a column vector (1-d //! transform). //! \details Supports 1 and 2 dimensional tranforms for real and complex data. template typename T0::PlainObject indirect(Eigen::ArrayBase const &coeffs) const; //! \brief Indirect transform //! \param[in] coefficients: Input wavelet coefficients. Its size must be a multiple of $2^l$ //! where $l$ is the number of levels. Can be a matrix (2d-transform) or a column vector (1-d //! \param[inout] signal: Reconstructed signal. Must be of the same size and type as the input. //! \details Supports 1 and 2 dimensional tranforms for real and complex data. template void indirect(Eigen::ArrayBase const &coefficients, Eigen::ArrayBase &signal) const; //! \brief Indirect transform //! \param[in] coefficients: Input wavelet coefficients. Its size must be a multiple of $2^l$ //! where $l$ is the number of levels. Can be a matrix (2d-transform) or a column vector (1-d //! \param[inout] signal: Reconstructed signal. Must be of the same size and type as the input. //! \details Supports 1 and 2 dimensional tranforms for real and complex data. This version //! allows non-constant Eigen expressions to be passe on without the ugly `const_cast` of the //! cannonical approach. template void indirect(Eigen::ArrayBase const &coeffs, Eigen::ArrayBase &&signal) const { indirect(coeffs, signal); } //! Number of levels over which to do transform t_uint max_levels() const { auto cmp = [](Wavelet const &a, Wavelet const &b) { return a.levels() < b.levels(); }; return std::max_element(begin(), end(), cmp)->levels(); } //! Adds a wavelet of specific type void emplace_back(std::string const &name, t_uint nlevels) { std::vector::emplace_back(factory(name, nlevels)); } }; #define SOPT_WAVELET_ERROR_MACRO(INPUT) \ if(INPUT.rows() % (1u << max_levels()) != 0) \ throw std::length_error("Inconsistent number of columns and wavelet levels"); \ else if(INPUT.cols() != 1 and INPUT.cols() % (1u << max_levels())) \ throw std::length_error("Inconsistent number of rows and wavelet levels"); template void SARA::direct(Eigen::ArrayBase &coeffs, Eigen::ArrayBase const &signal) const { SOPT_WAVELET_ERROR_MACRO(signal); if(coeffs.rows() != signal.rows() or coeffs.cols() != signal.cols() * static_cast(size())) coeffs.derived().resize(signal.rows(), signal.cols() * size()); if(coeffs.rows() != signal.rows() or coeffs.cols() != signal.cols() * static_cast(size())) throw std::length_error("Incorrect size for output matrix(or could not resize)"); auto const Ncols = signal.cols(); #ifndef SOPT_OPENMP SOPT_TRACE("Calling direct sara without threads"); for(size_type i(0); i < size(); ++i) at(i).direct(coeffs.leftCols((i + 1) * Ncols).rightCols(Ncols), signal); #else #pragma omp parallel { if(omp_get_thread_num() == 0) { SOPT_TRACE("Calling direct sara with {} threads of {}", omp_get_num_threads(), omp_get_max_threads()); } #pragma omp for for(size_type i = 0; i < size(); ++i) at(i).direct(coeffs.leftCols((i + 1) * Ncols).rightCols(Ncols), signal); } #endif coeffs /= std::sqrt(size()); } template void SARA::indirect(Eigen::ArrayBase const &coeffs, Eigen::ArrayBase &signal) const { SOPT_WAVELET_ERROR_MACRO(coeffs); if(coeffs.cols() % size() != 0) throw std::length_error( "Columns of coefficient matrix and number of wavelets are inconsistent"); if(coeffs.rows() != signal.rows() or coeffs.cols() != signal.cols() * static_cast(size())) signal.derived().resize(coeffs.rows(), coeffs.cols() / size()); if(coeffs.rows() != signal.rows() or coeffs.cols() != signal.cols() * static_cast(size())) throw std::length_error("Incorrect size for output matrix(or could not resize)"); auto const Ncols = signal.cols(); #ifndef SOPT_OPENMP SOPT_TRACE("Calling indirect sara without threads"); signal = front().indirect(coeffs.leftCols(Ncols).rightCols(Ncols)); for(size_type i(1); i < size(); ++i) signal += at(i).indirect(coeffs.leftCols((i + 1) * Ncols).rightCols(Ncols)); #else signal.fill(0); #pragma omp parallel { if(omp_get_thread_num() == 0) { SOPT_TRACE("Calling indirect sara with {} threads of {}", omp_get_num_threads(), omp_get_max_threads()); } Image reductor = Image::Zero(signal.rows(), Ncols); #pragma omp for for(size_type i = 0; i < size(); ++i) reductor += at(i).indirect(coeffs.leftCols((i + 1) * Ncols).rightCols(Ncols)); #pragma omp critical signal += reductor; } #endif signal /= std::sqrt(size()); } #undef SOPT_WAVELET_ERROR_MACRO template typename T0::PlainObject SARA::indirect(Eigen::ArrayBase const &coeffs) const { typedef decltype(this->front().indirect(coeffs)) t_Output; t_Output signal = t_Output::Zero(coeffs.rows(), coeffs.cols() / size()); (*this).indirect(coeffs, signal); return signal; } template typename T0::PlainObject SARA::direct(Eigen::ArrayBase const &signal) const { typedef decltype(this->front().direct(signal)) t_Output; t_Output result = t_Output::Zero(signal.rows(), signal.cols() * size()); (*this).direct(result, signal); return result; } } } #endif sopt-2.0.0/cpp/sopt/wavelets/wavelet_data.cc000066400000000000000000004037301277570055300210700ustar00rootroot00000000000000#include "sopt/wavelets/wavelet_data.h" #include "sopt/config.h" #include #include #include "sopt/types.h" namespace sopt { namespace wavelets { namespace { //! Vector setup from initializer list, because easier WaveletData::t_vector init_db(std::initializer_list const &input) { sopt::Vector<> result(input.size()); std::copy(input.begin(), input.end(), result.data()); return result; } //! Every other element is negative WaveletData::t_vector negate_even(WaveletData::t_vector const &coeffs) { WaveletData::t_vector result(coeffs); for(t_int i(0); i < static_cast(coeffs.size()); i += 2) result(i) = -result(i); return result; } //! Odd elements only WaveletData::t_vector odd(WaveletData::t_vector const &coeffs) { WaveletData::t_vector result(coeffs.size() / 2); for(t_int i(1); i < static_cast(coeffs.size()); i += 2) result(i / 2) = coeffs(i); return result; } //! Even elements only WaveletData::t_vector even(WaveletData::t_vector const &coeffs) { WaveletData::t_vector result((coeffs.size() + 1) / 2); for(t_int i(0), j(0); i < static_cast(coeffs.size()); i += 2, ++j) result(j) = coeffs(i); return result; } } WaveletData::WaveletData(std::initializer_list const &coeffs) : WaveletData(init_db(coeffs)) {} WaveletData::WaveletData(t_vector const &coeffs) : coefficients(coeffs), direct_filter{coeffs, negate_even(coeffs.reverse())}, indirect_filter{even(coeffs.reverse()), odd(coeffs.reverse()), even(coeffs), -odd(coeffs)} {} WaveletData const Daubechies1({7.071067811865475244008443621048490392848359376884740365883398e-01, 7.071067811865475244008443621048490392848359376884740365883398e-01}); WaveletData const Daubechies2({4.829629131445341433748715998644486838169524195042022752011715e-01, 8.365163037378079055752937809168732034593703883484392934953414e-01, 2.241438680420133810259727622404003554678835181842717613871683e-01, -1.294095225512603811744494188120241641745344506599652569070016e-01}); WaveletData const Daubechies3({3.326705529500826159985115891390056300129233992450683597084705e-01, 8.068915093110925764944936040887134905192973949948236181650920e-01, 4.598775021184915700951519421476167208081101774314923066433867e-01, -1.350110200102545886963899066993744805622198452237811919756862e-01, -8.544127388202666169281916918177331153619763898808662976351748e-02, 3.522629188570953660274066471551002932775838791743161039893406e-02}); WaveletData const Daubechies4({2.303778133088965008632911830440708500016152482483092977910968e-01, 7.148465705529156470899219552739926037076084010993081758450110e-01, 6.308807679298589078817163383006152202032229226771951174057473e-01, -2.798376941685985421141374718007538541198732022449175284003358e-02, -1.870348117190930840795706727890814195845441743745800912057770e-01, 3.084138183556076362721936253495905017031482172003403341821219e-02, 3.288301166688519973540751354924438866454194113754971259727278e-02, -1.059740178506903210488320852402722918109996490637641983484974e-02}); WaveletData const Daubechies5({1.601023979741929144807237480204207336505441246250578327725699e-01, 6.038292697971896705401193065250621075074221631016986987969283e-01, 7.243085284377729277280712441022186407687562182320073725767335e-01, 1.384281459013207315053971463390246973141057911739561022694652e-01, -2.422948870663820318625713794746163619914908080626185983913726e-01, -3.224486958463837464847975506213492831356498416379847225434268e-02, 7.757149384004571352313048938860181980623099452012527983210146e-02, -6.241490212798274274190519112920192970763557165687607323417435e-03, -1.258075199908199946850973993177579294920459162609785020169232e-02, 3.335725285473771277998183415817355747636524742305315099706428e-03}); WaveletData const Daubechies6({1.115407433501094636213239172409234390425395919844216759082360e-01, 4.946238903984530856772041768778555886377863828962743623531834e-01, 7.511339080210953506789344984397316855802547833382612009730420e-01, 3.152503517091976290859896548109263966495199235172945244404163e-01, -2.262646939654398200763145006609034656705401539728969940143487e-01, -1.297668675672619355622896058765854608452337492235814701599310e-01, 9.750160558732304910234355253812534233983074749525514279893193e-02, 2.752286553030572862554083950419321365738758783043454321494202e-02, -3.158203931748602956507908069984866905747953237314842337511464e-02, 5.538422011614961392519183980465012206110262773864964295476524e-04, 4.777257510945510639635975246820707050230501216581434297593254e-03, -1.077301085308479564852621609587200035235233609334419689818580e-03}); WaveletData const Daubechies7({7.785205408500917901996352195789374837918305292795568438702937e-02, 3.965393194819173065390003909368428563587151149333287401110499e-01, 7.291320908462351199169430703392820517179660611901363782697715e-01, 4.697822874051931224715911609744517386817913056787359532392529e-01, -1.439060039285649754050683622130460017952735705499084834401753e-01, -2.240361849938749826381404202332509644757830896773246552665095e-01, 7.130921926683026475087657050112904822711327451412314659575113e-02, 8.061260915108307191292248035938190585823820965629489058139218e-02, -3.802993693501441357959206160185803585446196938467869898283122e-02, -1.657454163066688065410767489170265479204504394820713705239272e-02, 1.255099855609984061298988603418777957289474046048710038411818e-02, 4.295779729213665211321291228197322228235350396942409742946366e-04, -1.801640704047490915268262912739550962585651469641090625323864e-03, 3.537137999745202484462958363064254310959060059520040012524275e-04}); WaveletData const Daubechies8({5.441584224310400995500940520299935503599554294733050397729280e-02, 3.128715909142999706591623755057177219497319740370229185698712e-01, 6.756307362972898068078007670471831499869115906336364227766759e-01, 5.853546836542067127712655200450981944303266678053369055707175e-01, -1.582910525634930566738054787646630415774471154502826559735335e-02, -2.840155429615469265162031323741647324684350124871451793599204e-01, 4.724845739132827703605900098258949861948011288770074644084096e-04, 1.287474266204784588570292875097083843022601575556488795577000e-01, -1.736930100180754616961614886809598311413086529488394316977315e-02, -4.408825393079475150676372323896350189751839190110996472750391e-02, 1.398102791739828164872293057263345144239559532934347169146368e-02, 8.746094047405776716382743246475640180402147081140676742686747e-03, -4.870352993451574310422181557109824016634978512157003764736208e-03, -3.917403733769470462980803573237762675229350073890493724492694e-04, 6.754494064505693663695475738792991218489630013558432103617077e-04, -1.174767841247695337306282316988909444086693950311503927620013e-04}); WaveletData const Daubechies9({3.807794736387834658869765887955118448771714496278417476647192e-02, 2.438346746125903537320415816492844155263611085609231361429088e-01, 6.048231236901111119030768674342361708959562711896117565333713e-01, 6.572880780513005380782126390451732140305858669245918854436034e-01, 1.331973858250075761909549458997955536921780768433661136154346e-01, -2.932737832791749088064031952421987310438961628589906825725112e-01, -9.684078322297646051350813353769660224825458104599099679471267e-02, 1.485407493381063801350727175060423024791258577280603060771649e-01, 3.072568147933337921231740072037882714105805024670744781503060e-02, -6.763282906132997367564227482971901592578790871353739900748331e-02, 2.509471148314519575871897499885543315176271993709633321834164e-04, 2.236166212367909720537378270269095241855646688308853754721816e-02, -4.723204757751397277925707848242465405729514912627938018758526e-03, -4.281503682463429834496795002314531876481181811463288374860455e-03, 1.847646883056226476619129491125677051121081359600318160732515e-03, 2.303857635231959672052163928245421692940662052463711972260006e-04, -2.519631889427101369749886842878606607282181543478028214134265e-04, 3.934732031627159948068988306589150707782477055517013507359938e-05}); WaveletData const Daubechies10({2.667005790055555358661744877130858277192498290851289932779975e-02, 1.881768000776914890208929736790939942702546758640393484348595e-01, 5.272011889317255864817448279595081924981402680840223445318549e-01, 6.884590394536035657418717825492358539771364042407339537279681e-01, 2.811723436605774607487269984455892876243888859026150413831543e-01, -2.498464243273153794161018979207791000564669737132073715013121e-01, -1.959462743773770435042992543190981318766776476382778474396781e-01, 1.273693403357932600826772332014009770786177480422245995563097e-01, 9.305736460357235116035228983545273226942917998946925868063974e-02, -7.139414716639708714533609307605064767292611983702150917523756e-02, -2.945753682187581285828323760141839199388200516064948779769654e-02, 3.321267405934100173976365318215912897978337413267096043323351e-02, 3.606553566956169655423291417133403299517350518618994762730612e-03, -1.073317548333057504431811410651364448111548781143923213370333e-02, 1.395351747052901165789318447957707567660542855688552426721117e-03, 1.992405295185056117158742242640643211762555365514105280067936e-03, -6.858566949597116265613709819265714196625043336786920516211903e-04, -1.164668551292854509514809710258991891527461854347597362819235e-04, 9.358867032006959133405013034222854399688456215297276443521873e-05, -1.326420289452124481243667531226683305749240960605829756400674e-05}); WaveletData const Daubechies11({1.869429776147108402543572939561975728967774455921958543286692e-02, 1.440670211506245127951915849361001143023718967556239604318852e-01, 4.498997643560453347688940373853603677806895378648933474599655e-01, 6.856867749162005111209386316963097935940204964567703495051589e-01, 4.119643689479074629259396485710667307430400410187845315697242e-01, -1.622752450274903622405827269985511540744264324212130209649667e-01, -2.742308468179469612021009452835266628648089521775178221905778e-01, 6.604358819668319190061457888126302656753142168940791541113457e-02, 1.498120124663784964066562617044193298588272420267484653796909e-01, -4.647995511668418727161722589023744577223260966848260747450320e-02, -6.643878569502520527899215536971203191819566896079739622858574e-02, 3.133509021904607603094798408303144536358105680880031964936445e-02, 2.084090436018106302294811255656491015157761832734715691126692e-02, -1.536482090620159942619811609958822744014326495773000120205848e-02, -3.340858873014445606090808617982406101930658359499190845656731e-03, 4.928417656059041123170739741708273690285547729915802418397458e-03, -3.085928588151431651754590726278953307180216605078488581921562e-04, -8.930232506662646133900824622648653989879519878620728793133358e-04, 2.491525235528234988712216872666801088221199302855425381971392e-04, 5.443907469936847167357856879576832191936678525600793978043688e-05, -3.463498418698499554128085159974043214506488048233458035943601e-05, 4.494274277236510095415648282310130916410497987383753460571741e-06}); WaveletData const Daubechies12({1.311225795722951750674609088893328065665510641931325007748280e-02, 1.095662728211851546057045050248905426075680503066774046383657e-01, 3.773551352142126570928212604879206149010941706057526334705839e-01, 6.571987225793070893027611286641169834250203289988412141394281e-01, 5.158864784278156087560326480543032700677693087036090056127647e-01, -4.476388565377462666762747311540166529284543631505924139071704e-02, -3.161784537527855368648029353478031098508839032547364389574203e-01, -2.377925725606972768399754609133225784553366558331741152482612e-02, 1.824786059275796798540436116189241710294771448096302698329011e-01, 5.359569674352150328276276729768332288862665184192705821636342e-03, -9.643212009650708202650320534322484127430880143045220514346402e-02, 1.084913025582218438089010237748152188661630567603334659322512e-02, 4.154627749508444073927094681906574864513532221388374861287078e-02, -1.221864906974828071998798266471567712982466093116558175344811e-02, -1.284082519830068329466034471894728496206109832314097633275225e-02, 6.711499008795509177767027068215672450648112185856456740379455e-03, 2.248607240995237599950865211267234018343199786146177099262010e-03, -2.179503618627760471598903379584171187840075291860571264980942e-03, 6.545128212509595566500430399327110729111770568897356630714552e-06, 3.886530628209314435897288837795981791917488573420177523436096e-04, -8.850410920820432420821645961553726598738322151471932808015443e-05, -2.424154575703078402978915320531719580423778362664282239377532e-05, 1.277695221937976658714046362616620887375960941439428756055353e-05, -1.529071758068510902712239164522901223197615439660340672602696e-06}); WaveletData const Daubechies13({9.202133538962367972970163475644184667534171916416562386009703e-03, 8.286124387290277964432027131230466405208113332890135072514277e-02, 3.119963221604380633960784112214049693946683528967180317160390e-01, 6.110558511587876528211995136744180562073612676018239438526582e-01, 5.888895704312189080710395347395333927665986382812836042235573e-01, 8.698572617964723731023739838087494399231884076619701250882016e-02, -3.149729077113886329981698255932282582876888450678789025950306e-01, -1.245767307508152589413808336021260180792739295173634719572069e-01, 1.794760794293398432348450072339369013581966256244133393042881e-01, 7.294893365677716380902830610477661983325929026879873553627963e-02, -1.058076181879343264509667304196464849478860754801236658232360e-01, -2.648840647534369463963912248034785726419604844297697016264224e-02, 5.613947710028342886214501998387331119988378792543100244737056e-02, 2.379972254059078811465170958554208358094394612051934868475139e-03, -2.383142071032364903206403067757739134252922717636226274077298e-02, 3.923941448797416243316370220815526558824746623451404043918407e-03, 7.255589401617566194518393300502698898973529679646683695269828e-03, -2.761911234656862178014576266098445995350093330501818024966316e-03, -1.315673911892298936613835370593643376060412592653652307238124e-03, 9.323261308672633862226517802548514100918088299801952307991569e-04, 4.925152512628946192140957387866596210103778299388823500840094e-05, -1.651289885565054894616687709238000755898548214659776703347801e-04, 3.067853757932549346649483228575476236600428217237900563128230e-05, 1.044193057140813708170714991080596951670706436217328169641474e-05, -4.700416479360868325650195165061771321650383582970958556568059e-06, 5.220035098454864691736424354843176976747052155243557001531901e-07}); WaveletData const Daubechies14({6.461153460087947818166397448622814272327159419201199218101404e-03, 6.236475884939889832798566758434877428305333693407667164602518e-02, 2.548502677926213536659077886778286686187042416367137443780084e-01, 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2.375106683660860777161950832380341362257503761490580896617678e-05, -2.731390824654337912922346414722045404779935825834384250023192e-05, -1.183471059985615942783182762352360917304348034947412986608322e-06, 8.372218198160788432628056043217491552198857358432112275253310e-06, -1.586145782434577495502614631566211839722879492827911790709498e-06, -1.870811602859180713762972281154953528056257451900381097476968e-06, 8.311421279707778528163597405935375886855029592150424544500718e-07, 2.548423522556577831218519052844387478819866531902854523544709e-07, -2.455377658434232699135878286794578515387138194247693201846263e-07, 2.753249073339512254085076456700241929492720457889076058451072e-09, 4.799043465450992009934526867650497683545716858606119786327559e-08, -1.156093688817008406756913949175208452083765368825442482226093e-08, -5.612784343327791397474114357094368557982413895802980814813369e-09, 3.138841695782424018351567952158415003571380699236147752239001e-09, 1.090815553713751810964713058800448676068475673611349566405716e-10, -4.512545778563249634425200856088490195004077806062978067796020e-10, 8.962418203859611987065968320295929679774693465791367610044773e-11, 3.037429098112535221800013609576297196061786927734556635696416e-11, -1.599716689261357143200396922409448515398648489795044468046420e-11, 8.876846287217374213524399682895564055949886050748321818411161e-13, 1.070969357114017002424433471621197579059927261727846375968378e-12, -3.029285026974877268896134589769473854669758797446795757329862e-13, 5.542263182639804235231685861028995158694397223907295269180336e-15, 1.338071386299105896025578761458472955294763310766371178363783e-14, -3.204628543401749860439316638848579711789176444320134355253750e-15, 3.339971984818693213132578777712503670014459411167839211495237e-16, -1.403274175373190617489823209168013922564353495443487431242610e-17}); WaveletData const Daubechies37({2.022060862498392121815038335333633351464174415618614893795880e-06, 4.942343750628132004714286117434454499485737947791397867195910e-05, 5.662418377066724013768394373249439163518654840493603575144737e-04, 4.024140368257286770702140124893772447952256842478891548092703e-03, 1.976228615387959153244055502205017461538589475705618414896893e-02, 7.058482597718160832030361890793007659963483925312132741868671e-02, 1.873263318620649448028843491747601576761901656888288838192023e-01, 3.684409724003061409445838616964941132670287724754729425204047e-01, 5.181670408556228873104519667534437205387109579265718071174178e-01, 4.622075536616057145505448401528172070050768534504278694229363e-01, 1.308789632330201726057701201017649601034381070893275586898075e-01, -2.461804297610834132869018581145720710365433914584680691693717e-01, -2.943759152626617722808219575932673733674290772235644691367427e-01, 1.967150045235938977077768648740052380288156507222647187301894e-02, 2.515232543602686933435224095078166291442923992611593827552710e-01, 8.180602838721862339029076982652411696000045533716726027662147e-02, -1.819622917786080007408824256525225216444443143868752611284260e-01, -1.084517138233017845554078812341876568514835176341639783558543e-01, 1.299296469598537527842528895259188653120602318620944502979726e-01, 1.017802968388141797470948228505865617480048287983176581607964e-01, -9.660754061668439030915405045955772715988585374771282291315496e-02, -8.233021190655740867404073660920379414988302492018783774702028e-02, 7.504761994836017933579005072594245435071674452882148228583865e-02, 5.956741087152995245435589042520108066877114768216272503684398e-02, -5.925681563265897095153806724965924334077555174281436189512239e-02, -3.825382947938424882011108885090442116802994193611884738133373e-02, 4.580794415126833246633256156110381805848138158784734496981778e-02, 2.097280059259754883313769469036393294461497749083921162354229e-02, -3.352358406410096994358662875913243067234786296009238949920582e-02, -8.833493890410232394064187990625563257107429109130726291528648e-03, 2.261865154459947356571431658958802912061105608212828675323452e-02, 1.690472383484423743663952859090705636512807161536954018400081e-03, -1.376398196289478433857985486097070339786225136728067000591187e-02, 1.519305778833399218481261844599507408563295102235964076544334e-03, 7.387757452855583640107787619408806919082115520707105052944171e-03, -2.248053187003824706127276829147166466869908326245810952521710e-03, -3.394523276408398601988475786247462646314228994098320665709345e-03, 1.816871343801423525477184531347879515909226877688306010517914e-03, 1.263934258117477182626760951047019242187910977671449470318766e-03, -1.111484865318630197259018233162929628309920117691177260742614e-03, -3.280788470880198419407186455190899535706232295554613820907245e-04, 5.490532773373631230219769273898345809368332716288071475378651e-04, 1.534439023195503211083338679106161291342621676983096723309776e-05, -2.208944032455493852493630802748509781675182699536797043565515e-04, 4.336726125945695214852398433524024058216834313839357806404424e-05, 7.055138782065465075838703109997365141906130284669094131032488e-05, -3.098662927619930052417611453170793938796310141219293329658062e-05, -1.639162496160583099236044020495877311072716199713679670940295e-05, 1.354327718416781810683349121150634031343717637827354228989989e-05, 1.849945003115590390789683032647334516600314304175482456338006e-06, -4.309941556597092389020622638271988877959028012481278949268461e-06, 4.854731396996411681769911684430785681028852413859386141424939e-07, 1.002121399297177629772998172241869405763288457224082581829033e-06, -3.494948603445727645895194867933547164628229076947330682199174e-07, -1.509885388671583553484927666148474078148724554849968758642331e-07, 1.109031232216439389999036327867142640916239658806376290861690e-07, 5.350657515461434290618742656970344024396382191417247602674540e-09, -2.252193836724805775389816424695618411834716065179297102428180e-08, 4.224485706362419268050011630338101126995607958955688879525896e-09, 2.793974465953982659829387370821677112004867350709951380622807e-09, -1.297205001469435139867686007585972538983682739297235604327668e-09, -1.031411129096974965677950646498153071722880698222864687038596e-10, 1.946164894082315021308714557636277980079559327508927751052218e-10, -3.203398244123241367987902201268363088933939831689591684670080e-11, -1.398415715537641487959551682557483348661602836709278513081908e-11, 6.334955440973913249611879065201632922100533284261000819747915e-12, -2.096363194234800541614775742755555713279549381264881030843258e-13, -4.421612409872105367333572734854401373201808896976552663098518e-13, 1.138052830921439682522395208295427884729893377395129205716662e-13, -4.518889607463726394454509623712773172513778367070839294449849e-16, -5.243025691884205832260354503748325334301994904062750850180233e-15, 1.189012387508252879928637969242590755033933791160383262132698e-15, -1.199280335852879554967035114674445327319437557227036460257649e-16, 4.906615064935203694857690087429901193139905690549533773201453e-18}); WaveletData const Daubechies38({1.425776641674131672055420247567865803211784397464191115245081e-06, 3.576251994264023012742569014888876217958307227940126418281357e-05, 4.211702664727116432247014444906469155300573201130549739553848e-04, 3.083088119253751774288740090262741910177322520624582862578292e-03, 1.563724934757215617277490102724080070486270026632620664785632e-02, 5.788994361285925649727664279317241952513246287766481213301801e-02, 1.600719935641106973482800861166599685169395465055048951307626e-01, 3.307757814110146511493637534404611754800768677041577030757306e-01, 4.965911753117180976599171147718708939352414838951726087564419e-01, 4.933560785171007975728485346997317064969513623594359091115804e-01, 2.130505713555785138286743353458562451255624665951160445122307e-01, -1.828676677083358907975548507946239135218223185041410632924815e-01, -3.216756378089978628483471725406916361929841940528189059002548e-01, -6.226650604782432226643360160478765847565862101045597180310490e-02, 2.321259638353531085028708104285994998671615563662858079262996e-01, 1.499851196187170199586403453788927307298226028262603028635758e-01, -1.417956859730596216710053144522330276392591055375830654519080e-01, -1.599125651582443618288533214523534937804208844386102639177693e-01, 8.563812155615105741612217814369165313487129645536001850276987e-02, 1.414147340733826800884683119379170594092606174915755283496153e-01, -5.658645863072738145681787657843320646815509410635114234947902e-02, -1.147311707107443752394144019458942779715665489230169950201022e-01, 4.309589543304764288137871223616030624246568683595408792078602e-02, 8.720439826203975011910714164154456762073786124233088471855868e-02, -3.660510340287429567372071039506772372567938710943432838908247e-02, -6.176620870841315993604736705613246241897497782373337911398117e-02, 3.198987753153780630818381136366859026137035450576631134176875e-02, 4.005498110511594820952087086241114309038577379366732959648548e-02, -2.689149388089451438550851767715967313417890393287236700072071e-02, -2.311413402054931680856913553585621248925303865540203357180768e-02, 2.090464525565524340215982365351342094670261491526831672682244e-02, 1.129049727868596484270081487761544232851115891449843967151657e-02, -1.470188206539868213708986402816605045648481224662435114088245e-02, -4.131306656031089274123231103326745723188134548520938157995702e-03, 9.214785032197180512031534870181734003522861645903894504302286e-03, 5.625715748403532005741565594881148757066703437214522101740941e-04, -5.071314509218348093935061417505663002006821323958752649640329e-03, 7.169821821064019257784165364894915621888541496773370435889585e-04, 2.400697781890973183892306914082592143984140550210130139535193e-03, -8.448626665537775009068937851465856973251363010924003314643612e-04, -9.424614077227377964015942271780098283910230639908018778588910e-04, 5.810759750532863662020321063678196633409555706981476723988312e-04, 2.817639250380670746018048967535608190123523180612961062603672e-04, -3.031020460726611993600629020329784682496477106470427787747855e-04, -4.555682696668420274688683005987764360677217149927938344795290e-05, 1.262043350166170705382346537131817701361522387904917335958705e-04, -1.155409103833717192628479047983460953381959342642374175822863e-05, -4.175141648540397797296325065775711309197411926289412468280801e-05, 1.334176149921350382547503457286060922218070031330137601427324e-05, 1.037359184045599795632258335010065103524959844966094870217687e-05, -6.456730428469619160379910439617575420986972394137121953806236e-06, -1.550844350118602575853380148525912999401292473185534395740371e-06, 2.149960269939665207789548199790770596890252405076394885606038e-06, -8.487087586072593071869805266089426629606479876982221840833098e-08, -5.187733738874144426008474683378542368066310000602823096009187e-07, 1.396377545508355481227961581059961184519872502493462010264633e-07, 8.400351046895965526933587176781279507953080669259318722910523e-08, -4.884757937459286762082185411608763964041010392101914854918157e-08, -5.424274800287298511126684174854414928447521710664476410973981e-09, 1.034704539274858480924046490952803937328239537222908159451039e-08, -1.436329487795135706854539856979275911183628476521636251660849e-09, -1.349197753983448821850381770889786301246741304307934955997111e-09, 5.261132557357598494535766638772624572100332209198979659077082e-10, 6.732336490189308685740626964182623159759767536724844030164551e-11, -8.278256522538134727330692938158991115335384611795874767521731e-11, 1.101692934599454551150832622160224231280195362919498540913658e-11, 6.291537317039508581580913620859140835852886308989584198166174e-12, -2.484789237563642857043361214502760723611468591833262675852242e-12, 2.626496504065252070488282876470525379851429538389481576454618e-14, 1.808661236274530582267084846343959377085922019067808145635263e-13, -4.249817819571463006966616371554206572863122562744916796556474e-14, -4.563397162127373109101691643047923747796563449194075621854491e-16, 2.045099676788988907802272564402310095398641092819367167252952e-15, -4.405307042483461342449027139838301611006835285455050155842865e-16, 4.304596839558790016251867477122791508849697688058169053134463e-17, -1.716152451088744188732404281737964277713026087224248235541071e-18}); WaveletData const &daubechies_data(t_uint n) { WaveletData const *result = nullptr; switch(n) { case 1: result = &Daubechies1; break; case 2: result = &Daubechies2; break; case 3: result = &Daubechies3; break; case 4: result = &Daubechies4; break; case 5: result = &Daubechies5; break; case 6: result = &Daubechies6; break; case 7: result = &Daubechies7; break; case 8: result = &Daubechies8; break; case 9: result = &Daubechies9; break; case 10: result = &Daubechies10; break; case 11: result = &Daubechies11; break; case 12: result = &Daubechies12; break; case 13: result = &Daubechies13; break; case 14: result = &Daubechies14; break; case 15: result = &Daubechies15; break; case 16: result = &Daubechies16; break; case 17: result = &Daubechies17; break; case 18: result = &Daubechies18; break; case 19: result = &Daubechies19; break; case 20: result = &Daubechies20; break; case 21: result = &Daubechies21; break; case 22: result = &Daubechies22; break; case 23: result = &Daubechies23; break; case 24: result = &Daubechies24; break; case 25: result = &Daubechies25; break; case 26: result = &Daubechies26; break; case 27: result = &Daubechies27; break; case 28: result = &Daubechies28; break; case 29: result = &Daubechies29; break; case 30: result = &Daubechies30; break; case 31: result = &Daubechies31; break; case 32: result = &Daubechies32; break; case 33: result = &Daubechies33; break; case 34: result = &Daubechies34; break; case 35: result = &Daubechies35; break; case 36: result = &Daubechies36; break; case 37: result = &Daubechies37; break; case 38: result = &Daubechies38; break; default: throw std::out_of_range("Daubechies coefficient not implemented for given index"); break; } return *result; } } } sopt-2.0.0/cpp/sopt/wavelets/wavelet_data.h000066400000000000000000000022311277570055300207210ustar00rootroot00000000000000#ifndef SOPT_WAVELET_WAVELET_DATA_H #define SOPT_WAVELET_WAVELET_DATA_H #include "sopt/config.h" #include "sopt/types.h" #include "sopt/wavelets/innards.impl.h" namespace sopt { namespace wavelets { //! Holds wavelets coefficients struct WaveletData { //! Type of the underlying scalar typedef t_real t_scalar; //! Type of the underlying vector typedef Array t_vector; //! Wavelet coefficient per-se t_vector const coefficients; //! Holds filters for direct transform struct DirectFilter { //! Low-pass filter for direct transform t_vector low; //! High-pass filter for direct transform t_vector high; } const direct_filter; //! Holds filters for indirect transform struct { //! High-pass filter for direct transform t_vector low_even; t_vector low_odd; t_vector high_even; t_vector high_odd; } const indirect_filter; //! Constructs from initializers WaveletData(std::initializer_list const &coefs); //! Constructs from vector WaveletData(t_vector const &coefs); }; //! Factory function returning specific daubechie wavelet data WaveletData const &daubechies_data(t_uint); } } #endif sopt-2.0.0/cpp/sopt/wavelets/wavelets.cc000066400000000000000000000012541277570055300202550ustar00rootroot00000000000000#include "sopt/wavelets/wavelets.h" #include "sopt/config.h" #include #include "sopt/logging.h" namespace sopt { namespace wavelets { Wavelet factory(std::string name, t_uint nlevels) { if(name == "dirac" or name == "Dirac") { SOPT_MEDIUM_LOG("Creating Dirac Wavelet"); return Wavelet(daubechies_data(1), 0); } if(name.substr(0, 2) == "DB" or name.substr(0, 2) == "db") { std::istringstream sstr(name.substr(2, name.size() - 2)); t_uint l(0); sstr >> l; SOPT_MEDIUM_LOG("Creating Daubechies Wavelet {}, level {}", l, nlevels); return Wavelet(daubechies_data(l), nlevels); } // Unknown input wavelet throw std::exception(); } } } sopt-2.0.0/cpp/sopt/wavelets/wavelets.h000066400000000000000000000157211277570055300201230ustar00rootroot00000000000000#ifndef SOPT_WAVELETS_WAVELETS_H #define SOPT_WAVELETS_WAVELETS_H #include "sopt/config.h" #include "sopt/types.h" #include "sopt/wavelets/direct.h" #include "sopt/wavelets/indirect.h" #include "sopt/wavelets/wavelet_data.h" namespace sopt { namespace wavelets { // Advance declaration so we can define the subsequent friend function class Wavelet; //! \brief Creates a wavelet transform object Wavelet factory(std::string name = "DB1", t_uint nlevels = 1); //! Performs direct and indirect wavelet transforms class Wavelet : public WaveletData { friend Wavelet factory(std::string name, t_uint nlevels); protected: //! Should be called through factory function Wavelet(WaveletData const &c, t_uint nlevels) : WaveletData(c), levels_(nlevels) {} public: //! Destructor virtual ~Wavelet() {} // Temporary macros that checks constraints on input #define SOPT_WAVELET_MACRO_MULTIPLE(NAME) \ if((NAME.rows() == 1 or NAME.cols() == 1)) { \ if(NAME.size() % (1 << levels()) != 0) \ throw std::length_error("Size of " #NAME " must number a multiple of 2^levels or 1"); \ } else if(NAME.rows() != 1 and NAME.rows() % (1 << levels()) != 0) \ throw std::length_error("Rows of " #NAME " must number a multiple of 2^levels or 1"); \ else if(NAME.cols() % (1 << levels()) != 0) \ throw std::length_error("Columns of " #NAME " must number a multiple of 2^levels"); #define SOPT_WAVELET_MACRO_EQUAL_SIZE(A, B) \ if(A.rows() != B.rows() or A.cols() != B.cols()) \ A.derived().resize(B.rows(), B.cols()); \ if(A.rows() != B.rows() or A.cols() != B.cols()) \ throw std::length_error("Incorrect size for output matrix(or could not resize)") //! \brief Direct transform //! \param[in] signal: computes wavelet coefficients for this signal. Its size must be a //! multiple of $2^l$ where $l$ is the number of levels. Can be a matrix (2d-transform) or a //! column vector (1-d transform). //! \return wavelet coefficients //! \details Supports 1 and 2 dimensional tranforms for real and complex data. template auto direct(Eigen::ArrayBase const &signal) const -> decltype(direct_transform(signal, 1, *this)) { SOPT_WAVELET_MACRO_MULTIPLE(signal); return direct_transform(signal, levels(), *this); } //! \brief Direct transform //! \param[inout] coefficients: Output wavelet coefficients. Must be of the same size and type //! as the input. //! \param[in] signal: computes wavelet coefficients for this signal. Its size must be a //! multiple of $2^l$ where $l$ is the number of levels. Can be a matrix (2d-transform) or a //! column vector //! (1-d transform). //! \details Supports 1 and 2 dimensional tranforms for real and complex data. template auto direct(Eigen::ArrayBase &coefficients, Eigen::ArrayBase const &signal) const -> decltype(direct_transform(coefficients, signal, 1, *this)) { SOPT_WAVELET_MACRO_MULTIPLE(signal); SOPT_WAVELET_MACRO_EQUAL_SIZE(coefficients, signal); return direct_transform(coefficients, signal, levels(), *this); } //! \brief Direct transform //! \param[inout] coefficients: Output wavelet coefficients. Must be of the same size and type //! as the input. //! \param[in] signal: computes wavelet coefficients for this signal. Its size must be a //! multiple of $2^l$ where $l$ is the number of levels. Can be a matrix (2d-transform) or a //! column vector //! (1-d transform). //! \details Supports 1 and 2 dimensional tranforms for real and complex data. This version //! allows non-constant Eigen expressions to be passe on without the ugly `const_cast` of the //! cannonical approach. template auto direct(Eigen::ArrayBase &&coefficients, Eigen::ArrayBase const &signal) const -> decltype(direct_transform(coefficients, signal, 1, *this)) { SOPT_WAVELET_MACRO_MULTIPLE(signal); SOPT_WAVELET_MACRO_EQUAL_SIZE(coefficients, signal); return direct_transform(coefficients, signal, levels(), *this); } //! \brief Indirect transform //! \param[in] coefficients: Input wavelet coefficients. Its size must be a multiple of $2^l$ //! where $l$ is the number of levels. Can be a matrix (2d-transform) or a column vector (1-d //! transform). //! \details Supports 1 and 2 dimensional tranforms for real and complex data. template auto indirect(Eigen::ArrayBase const &coefficients) const -> decltype(indirect_transform(coefficients, 1, *this)) { SOPT_WAVELET_MACRO_MULTIPLE(coefficients); return indirect_transform(coefficients, levels(), *this); } //! \brief Indirect transform //! \param[in] coefficients: Input wavelet coefficients. Its size must be a multiple of $2^l$ //! where $l$ is the number of levels. Can be a matrix (2d-transform) or a column vector (1-d //! \param[inout] signal: Reconstructed signal. Must be of the same size and type as the input. //! \details Supports 1 and 2 dimensional tranforms for real and complex data. template auto indirect(Eigen::ArrayBase const &coefficients, Eigen::ArrayBase &signal) const -> decltype(indirect_transform(coefficients, signal, 1, *this)) { SOPT_WAVELET_MACRO_MULTIPLE(coefficients); SOPT_WAVELET_MACRO_EQUAL_SIZE(signal, coefficients); return indirect_transform(coefficients, signal, levels(), *this); } //! \brief Indirect transform //! \param[in] coefficients: Input wavelet coefficients. Its size must be a multiple of $2^l$ //! where $l$ is the number of levels. Can be a matrix (2d-transform) or a column vector (1-d //! \param[inout] signal: Reconstructed signal. Must be of the same size and type as the input. //! \details Supports 1 and 2 dimensional tranforms for real and complex data. This version //! allows non-constant Eigen expressions to be passe on without the ugly `const_cast` of the //! cannonical approach. template auto indirect(Eigen::ArrayBase const &coeffs, Eigen::ArrayBase &&signal) const -> decltype(indirect_transform(coeffs, signal, 1, *this)) { SOPT_WAVELET_MACRO_MULTIPLE(coeffs); SOPT_WAVELET_MACRO_EQUAL_SIZE(signal, coeffs); return indirect_transform(coeffs, signal, levels(), *this); } #undef SOPT_WAVELET_MACRO_MULTIPLE #undef SOPT_WAVELET_MACRO_EQUAL_SIZE //! Number of levels over which to do transform t_uint levels() const { return levels_; } //! Sets number of levels over which to do transform void levels(t_uint l) { levels_ = l; } protected: //! Number of levels in the wavelet t_uint levels_; }; } } #endif sopt-2.0.0/cpp/sopt/wrapper.h000066400000000000000000000114441277570055300161150ustar00rootroot00000000000000#ifndef SOPT_WRAP #define SOPT_WRAP #include #include #include "sopt/config.h" #include "sopt/exception.h" #include "sopt/types.h" #include "sopt/maths.h" namespace sopt { namespace details { //! Expression referencing the result of a function call template class AppliedFunction : public Eigen::ReturnByValue> { public: typedef typename DERIVED::PlainObject PlainObject; typedef typename DERIVED::Index Index; AppliedFunction(FUNCTION const &func, DERIVED const &x, Index rows) : func(func), x(x), rows_(rows) {} AppliedFunction(FUNCTION const &func, DERIVED const &x) : func(func), x(x), rows_(x.rows()) {} AppliedFunction(AppliedFunction const &c) : func(c.func), x(c.x), rows_(c.rows_) {} AppliedFunction(AppliedFunction &&c) : func(std::move(c.func)), x(c.x), rows_(c.rows_) {} template void evalTo(DESTINATION &destination) const { func(destination, x); } Index rows() const { return rows_; } Index cols() const { return x.cols(); } private: FUNCTION const func; DERIVED const &x; Index const rows_; }; //! \brief Wraps an std::function to return an expression //! \details This makes writing the application of a function more beautiful on the eye. //! A function call `func(output, input)` can be made to look like `output = func(input)` or `output //! = func * input`. template class WrapFunction { public: //! Type of function wrapped here typedef OperatorFunction t_Function; //! Initializes the wrapper //! \param[in] func: function to wrap //! \param[in] sizes: three integer vector (a, b, c) //! if N is the size of the input, then (N * a) / b + c is the output //! b cannot be zero. WrapFunction(t_Function const &func, std::array sizes = {{1, 1, 0}}) : func(func), sizes_(sizes) { // cannot devide by zero assert(sizes_[1] != 0); } WrapFunction(WrapFunction const &c) : func(c.func), sizes_(c.sizes_) {} WrapFunction(WrapFunction const &&c) : func(std::move(c.func)), sizes_(std::move(c.sizes_)) {} void operator=(WrapFunction const &c) { func = c.func; sizes_ = c.sizes_; } void operator=(WrapFunction &&c) { func = std::move(c.func); sizes_ = std::move(c.sizes_); } //! Function application form template AppliedFunction> operator()(Eigen::ArrayBase const &x) const { return AppliedFunction>(func, x, rows(x)); } //! Multiplication application form template AppliedFunction> operator*(Eigen::ArrayBase const &x) const { return AppliedFunction>(func, x, rows(x)); } //! Function application form template AppliedFunction> operator()(Eigen::MatrixBase const &x) const { return AppliedFunction>(func, x, rows(x)); } //! Multiplication application form template AppliedFunction> operator*(Eigen::MatrixBase const &x) const { return AppliedFunction>(func, x, rows(x)); } //! \brief Defines relation-ship between input and output sizes //! \details An integer tuple (a, b, c) where, if N is the size of the input, then //! \f$(N * a) / b + c\f$ is the output. \f$b\f$ cannot be zero. //! In the simplest case where this objects wraps a square matrix, then the sizes are (1, 1, 0). //! If this objects wraps a rectangular matrix which halves the number of elements, then the //! sizes would be (1, 2, 0). std::array const &sizes() const { return sizes_; } //! Output vector size for a input with `xsize` elements template typename std::enable_if::value, T>::type rows(T xsize) const { auto const result = (static_cast(xsize) * sizes_[0]) / sizes_[1] + sizes_[2]; assert(result > 0); return static_cast(result); } protected: template t_uint rows(Eigen::DenseBase const &x) const { return rows(x.size()); } private: //! Reference function t_Function func; //! Ratio between input and output size std::array sizes_; }; //! Helper function to wrap functor into expression-able object template WrapFunction wrap(OperatorFunction const &func, std::array sizes = {{1, 1, 0}}) { return WrapFunction(func, sizes); } } } namespace Eigen { namespace internal { template struct traits> { typedef typename VECTOR::PlainObject ReturnType; }; } } #endif sopt-2.0.0/cpp/tests/000077500000000000000000000000001277570055300144355ustar00rootroot00000000000000sopt-2.0.0/cpp/tests/CMakeLists.txt000066400000000000000000000022101277570055300171700ustar00rootroot00000000000000if(logging) add_library(common_catch_main_object OBJECT "common_catch_main.cc") if(SPDLOG_INCLUDE_DIR) target_include_directories(common_catch_main_object SYSTEM PUBLIC ${SPDLOG_INCLUDE_DIR}) endif() if(CATCH_INCLUDE_DIR) target_include_directories(common_catch_main_object SYSTEM PUBLIC ${CATCH_INCLUDE_DIR}) endif() target_include_directories(common_catch_main_object PUBLIC "${PROJECT_BINARY_DIR}/include/" "${CMAKE_CURRENT_SOURCE_DIR}/.." ) add_dependencies(common_catch_main_object lookup_dependencies) endif() add_catch_test(wavelets LIBRARIES sopt) add_catch_test(sara LIBRARIES sopt) add_catch_test(maths LIBRARIES sopt) add_catch_test(wrapper LIBRARIES sopt) add_catch_test(conjugate_gradient LIBRARIES sopt) add_catch_test(linear_transform LIBRARIES sopt) add_catch_test(sdmm LIBRARIES sopt) add_catch_test(sdmm_warm_start LIBRARIES sopt) add_catch_test(proximal LIBRARIES sopt) add_catch_test_with_seed(seeded_proximal test_proximal 1449580491) add_catch_test(padmm LIBRARIES sopt) add_catch_test(padmm_warm_start LIBRARIES sopt) add_catch_test(reweighted LIBRARIES sopt) add_catch_test(power_method LIBRARIES sopt) sopt-2.0.0/cpp/tests/common_catch_main.cc000066400000000000000000000011011277570055300203730ustar00rootroot00000000000000#define CATCH_CONFIG_RUNNER #include "sopt/config.h" #include #include #include #include "sopt/logging.h" std::unique_ptr mersenne(new std::mt19937_64(0)); int main(int argc, char const *argv[]) { Catch::Session session; // There must be exactly once instance int returnCode = session.applyCommandLine(argc, argv); if(returnCode != 0) // Indicates a command line error return returnCode; mersenne.reset(new std::mt19937_64(session.configData().rngSeed)); sopt::logging::initialize(); return session.run(); } sopt-2.0.0/cpp/tests/conjugate_gradient.cc000066400000000000000000000025601277570055300206030ustar00rootroot00000000000000#include #include "catch.hpp" #include "sopt/conjugate_gradient.h" TEST_CASE("Conjugate gradient", "[cg]") { using namespace sopt; ConjugateGradient const cg(std::numeric_limits::max(), 1e-12); SECTION("Real valued") { auto const A = Image<>::Random(10, 10).eval(); auto const AtA = (A.transpose().matrix() * A.matrix()).eval(); auto const expected = Array<>::Random(A.rows()).eval(); auto const actual = cg(AtA, (A.transpose().matrix() * expected.matrix()).eval()); CHECK(actual.niters > 0); CHECK(actual.residual == Approx(0)); CAPTURE(actual.residual); CAPTURE((A.matrix() * actual.result).transpose()); CAPTURE(expected.transpose()); CHECK((A.matrix() * actual.result).isApprox(expected.matrix(), 1e-6)); } SECTION("Complex valued") { auto const A = Image::Random(10, 10).eval(); auto const AhA = (A.conjugate().transpose().matrix() * A.matrix()).eval(); auto const expected = Array::Random(A.rows()).eval(); auto const actual = cg(AhA, (A.conjugate().transpose().matrix() * expected.matrix()).eval()); CHECK(actual.niters > 0); CHECK(actual.residual == Approx(0)); CAPTURE(actual.residual); CAPTURE((A.matrix() * actual.result).transpose()); CAPTURE(expected.transpose()); CHECK((A.matrix() * actual.result).isApprox(expected.matrix(), 1e-6)); } } sopt-2.0.0/cpp/tests/linear_transform.cc000066400000000000000000000050251277570055300203130ustar00rootroot00000000000000#include #include #include #include "sopt/linear_transform.h" TEST_CASE("Linear Transforms", "[ops]") { using namespace sopt; typedef int SCALAR; typedef Array t_Vector; typedef Image t_Matrix; auto const N = 10; SECTION("Functions") { auto direct = [](t_Vector &out, t_Vector const &input) { out = input * 2 - 1; }; auto indirect = [](t_Vector &out, t_Vector const &input) { out = input * 4 - 1; }; t_Vector const x = t_Vector::Random(2 * N) * 5; auto op = sopt::linear_transform(direct, indirect); CHECK((op * x).eval().cols() == x.cols()); CHECK((op * x).eval().rows() == x.rows()); CHECK((op * x).cols() == x.cols()); CHECK((op * x).rows() == x.rows()); CHECK((op * x).matrix() == (2 * x - 1).matrix()); CHECK((op.adjoint() * x).matrix() == (4 * x - 1).matrix()); } SECTION("Matrix") { t_Matrix const L = t_Matrix::Random(N, N); t_Vector const x = t_Vector::Random(N) * 5; auto op = sopt::linear_transform(L.matrix()); CHECK((op * x.matrix()).eval().cols() == x.cols()); CHECK((op * x.matrix()).eval().rows() == x.rows()); CHECK((op * x.matrix()).cols() == x.cols()); CHECK((op * x.matrix()).rows() == x.rows()); CHECK(op * x.matrix() == L.matrix() * x.matrix()); CHECK(op.adjoint() * x.matrix() == L.conjugate().transpose().matrix() * x.matrix()); } SECTION("Rectangular matrix") { t_Matrix const L = t_Matrix::Random(N, 2 * N); t_Vector const x = t_Vector::Random(2 * N) * 5; auto op = sopt::linear_transform(L.matrix()); CHECK((op * x.matrix()).eval().cols() == 1); CHECK((op * x.matrix()).eval().rows() == N); CHECK((op * x.matrix()).cols() == 1); CHECK((op * x.matrix()).rows() == N); CHECK(op * x.matrix() == L.matrix() * x.matrix()); CHECK(op.adjoint() * x.head(N).matrix() == L.conjugate().transpose().matrix() * x.head(N).matrix()); } } TEST_CASE("Array of Linear transforms", "[ops]") { using namespace sopt; typedef int SCALAR; typedef Vector t_Vector; typedef Matrix t_Matrix; auto const N = 10; t_Vector const x = t_Vector::Random(N) * 5; std::vector Ls{t_Matrix::Random(N, N), t_Matrix::Random(N, N)}; std::vector> ops; for(auto const &matrix : Ls) ops.emplace_back(sopt::linear_transform(matrix)); for(decltype(Ls)::size_type i(0); i < ops.size(); ++i) { CHECK(ops[i] * x == Ls[i] * x); CHECK(ops[i].adjoint() * x == Ls[i].conjugate().transpose() * x); } } sopt-2.0.0/cpp/tests/maths.cc000066400000000000000000000136441277570055300160700ustar00rootroot00000000000000#include #include #include #include #include "sopt/maths.h" #include "sopt/relative_variation.h" #include "sopt/sampling.h" #include "sopt/types.h" TEST_CASE("Projector on positive quadrant", "[utility][project]") { using namespace sopt; SECTION("Real matrix") { Image<> input = Image<>::Ones(5, 5) + Image<>::Random(5, 5) * 0.55; input(1, 1) *= -1; input(3, 2) *= -1; auto const expr = positive_quadrant(input); CAPTURE(input); CAPTURE(expr); CHECK(expr(1, 1) == Approx(0)); CHECK(expr(3, 2) == Approx(0)); auto value = expr.eval(); CHECK(value(1, 1) == Approx(0)); CHECK(value(3, 2) == Approx(0)); value(1, 1) = input(1, 1); value(3, 2) = input(3, 2); CHECK(value.isApprox(input)); } SECTION("Complex matrix") { Image input = Image::Ones(5, 5) + Image::Random(5, 5) * 0.55; input.real()(1, 1) *= -1; input.real()(3, 2) *= -1; auto const expr = positive_quadrant(input); CAPTURE(input); CAPTURE(expr); CHECK(expr.imag().isApprox(Image<>::Zero(5, 5))); auto value = expr.eval(); CHECK(value.real()(1, 1) == Approx(0)); CHECK(value.real()(3, 2) == Approx(0)); value(1, 1) = input.real()(1, 1); value(3, 2) = input.real()(3, 2); CHECK(value.real().isApprox(input.real())); CHECK(value.imag().isApprox(0e0 * input.real())); } } TEST_CASE("Weighted l1 norm", "[utility][l1]") { sopt::Array<> weight(4); weight << 1, 2, 3, 4; SECTION("Real valued") { sopt::Array<> input(4); input << 5, -6, 7, -8; CHECK(sopt::l1_norm(input, weight) == Approx(5 + 12 + 21 + 32)); } SECTION("Complex valued") { sopt::t_complex const i(0, 1); sopt::Array input(4); input << 5. + 5. * i, 6. + 6. * i, 7. + 7. * i, 8. + 8. * i; CHECK(sopt::l1_norm(input, weight) == Approx(std::sqrt(2) * (5 + 12 + 21 + 32))); } } TEST_CASE("Soft threshhold", "[utility][threshhold]") { sopt::Array<> input(6); input << 1e1, 2e1, 3e1, 4e1, 1e4, 2e4; SECTION("Single-valued threshhold") { // check thresshold CHECK(sopt::soft_threshhold(input, 1.1e1)(0) == Approx(0)); CHECK(not(sopt::soft_threshhold(input, 1.1e1)(1) == Approx(0))); // check linearity auto a = sopt::soft_threshhold(input, 9e0)(0); auto b = sopt::soft_threshhold(input, 4.5e0)(0); auto c = sopt::soft_threshhold(input, 2.25e0)(0); CAPTURE(b - a); CAPTURE(c - b); CHECK((b - a) == Approx(2 * (c - b))); } SECTION("Multi-values threshhold") { using namespace sopt; Array<> threshhold(6); input[2] *= -1; threshhold << 1.1e1, 1.1e1, 1e0, 4.5, 2.25, 2.26; SECTION("Real input") { Array<> const actual = soft_threshhold(input, threshhold); CHECK(actual(0) == 0e0); CHECK(actual(1) == input(1) - threshhold(1)); CHECK(actual(2) == input(2) + threshhold(2)); CHECK(actual(3) == input(3) - threshhold(3)); CHECK_THROWS_AS(soft_threshhold(input, threshhold.head(2)), sopt::Exception); } SECTION("Complex input") { Array const actual = soft_threshhold(input.cast(), threshhold); CHECK(actual(0) == 0e0); CHECK(actual(1) == input(1) - threshhold(1)); CHECK(actual(2) == input(2) + threshhold(2)); CHECK(actual(3) == input(3) - threshhold(3)); CHECK_THROWS_AS(soft_threshhold(input, threshhold.head(2)), sopt::Exception); } } } TEST_CASE("Sampling", "[utility][sampling]") { typedef sopt::Vector t_Vector; t_Vector const input = t_Vector::Random(12); sopt::Sampling const sampling(12, {1, 3, 6, 5}); t_Vector down = t_Vector::Zero(4); sampling(down, input); CHECK(down.size() == 4); CHECK(down(0) == input(1)); CHECK(down(1) == input(3)); CHECK(down(2) == input(6)); CHECK(down(3) == input(5)); t_Vector up = t_Vector::Zero(input.size()); sampling.adjoint(up, down); CHECK(up(1) == input(1)); CHECK(up(3) == input(3)); CHECK(up(6) == input(6)); CHECK(up(5) == input(5)); up(1) = 0; up(3) = 0; up(6) = 0; up(5) = 0; CHECK(up == t_Vector::Zero(up.size())); } TEST_CASE("Relative variation", "[utility][convergence]") { sopt::RelativeVariation relvar(1e-8); sopt::Array<> input = sopt::Array<>::Random(6); CHECK(not relvar(input)); CHECK(relvar(input)); CHECK(relvar(input + relvar.epsilon() * 0.5 / 6. * sopt::Array<>::Random(6))); CHECK(not relvar(input + relvar.epsilon() * 1.1 * sopt::Array<>::Ones(6))); } TEST_CASE("Standard deviation", "[utility]") { sopt::Array input = sopt::Array::Random(6) + 1e0; sopt::t_complex mean = input.mean(); sopt::t_real stddev = 0e0; for(sopt::Vector<>::Index i(0); i < input.size(); ++i) stddev += std::real(std::conj(input(i) - mean) * (input(i) - mean)); stddev = std::sqrt(stddev) / std::sqrt(sopt::t_real(input.size())); CHECK(std::abs(sopt::standard_deviation(input) - stddev) < 1e-8); CHECK(std::abs(sopt::standard_deviation(input.matrix()) - stddev) < 1e-8); } // Checks type traits work static_assert(not sopt::details::HasValueType::value, ""); static_assert(not sopt::details::HasValueType>::value, ""); static_assert(sopt::details::HasValueType>::value, ""); static_assert(sopt::details::HasValueType::Scalar>::value, ""); static_assert(std::is_same::type, sopt::t_real>::value, ""); static_assert(std::is_same::type, sopt::t_real>::value, ""); static_assert(sopt::is_complex>::value, "Testing is_complex"); static_assert(sopt::is_complex>::value, "Testing is_complex"); static_assert(not sopt::is_complex::value, "Testing is_complex"); static_assert(not sopt::is_complex>::value, "Testing is_complex"); static_assert(not sopt::is_complex>>::value, "Testing is_complex"); sopt-2.0.0/cpp/tests/padmm.cc000066400000000000000000000106551277570055300160510ustar00rootroot00000000000000#include #include #include #include "sopt/imaging_padmm.h" #include "sopt/padmm.h" #include "sopt/proximal.h" #include "sopt/types.h" sopt::t_int random_integer(sopt::t_int min, sopt::t_int max) { extern std::unique_ptr mersenne; std::uniform_int_distribution uniform_dist(min, max); return uniform_dist(*mersenne); }; typedef sopt::t_real Scalar; typedef sopt::Vector t_Vector; typedef sopt::Matrix t_Matrix; auto const N = 5; TEST_CASE("Proximal ADMM with ||x - x0||_2 functions", "[padmm][integration]") { using namespace sopt; t_Vector const target0 = t_Vector::Random(N); t_Vector const target1 = t_Vector::Random(N) * 4; auto const g0 = proximal::translate(proximal::EuclidianNorm(), -target0); auto const g1 = proximal::translate(proximal::EuclidianNorm(), -target1); t_Matrix const mId = -t_Matrix::Identity(N, N); t_Vector const translation = t_Vector::Ones(N) * 5; auto const padmm = algorithm::ProximalADMM(g0, g1, t_Vector::Zero(N)) .Phi(mId) .itermax(3000) .gamma(0.01); auto const result = padmm(); t_Vector const segment = (target1 - target0).normalized(); t_real const alpha = (result.x - target0).transpose() * segment; CHECK((target1 - target0).transpose() * segment >= alpha); CHECK(alpha >= 0e0); CAPTURE(segment.transpose()); CAPTURE((result.x - target0).transpose()); CAPTURE((result.x - target1).transpose()); CHECK((result.x - target0 - alpha * segment).stableNorm() < 1e-8); } template struct is_imaging_proximal_ref : public std::is_same &, T> {}; TEST_CASE("Check type returned on setting variables") { // Yeah, could be static asserts using namespace sopt; using namespace sopt::algorithm; ImagingProximalADMM admm(Vector::Zero(0)); CHECK(is_imaging_proximal_ref::value); CHECK(is_imaging_proximal_ref::value); CHECK(is_imaging_proximal_ref::value); CHECK(is_imaging_proximal_ref::value); CHECK(is_imaging_proximal_ref::value); CHECK(is_imaging_proximal_ref::value); CHECK(is_imaging_proximal_ref::value); CHECK(is_imaging_proximal_ref::value); CHECK(is_imaging_proximal_ref::value); CHECK(is_imaging_proximal_ref::value); CHECK(is_imaging_proximal_ref::value); CHECK(is_imaging_proximal_ref::value); CHECK(is_imaging_proximal_ref::value); CHECK(is_imaging_proximal_ref::Zero(0)))>::value); typedef ConvergenceFunction ConvFunc; CHECK(is_imaging_proximal_ref()))>::value); CHECK(is_imaging_proximal_ref()))>::value); CHECK(is_imaging_proximal_ref()))>::value); CHECK(is_imaging_proximal_ref()))>::value); typedef LinearTransform> LinTrans; CHECK(is_imaging_proximal_ref()))>::value); CHECK(is_imaging_proximal_ref()))>::value); CHECK(is_imaging_proximal_ref()))>::value); CHECK(is_imaging_proximal_ref()))>::value); CHECK(is_imaging_proximal_ref()))>::value); CHECK(is_imaging_proximal_ref()))>::value); CHECK(is_imaging_proximal_ref()))>::value); CHECK(is_imaging_proximal_ref()))>::value); CHECK(is_imaging_proximal_ref()))>::value); CHECK(is_imaging_proximal_ref()))>::value); } sopt-2.0.0/cpp/tests/padmm_warm_start.cc000066400000000000000000000037421277570055300203130ustar00rootroot00000000000000#include #include #include #include "sopt/padmm.h" #include "sopt/proximal.h" #include "sopt/types.h" typedef sopt::t_real Scalar; typedef sopt::Vector t_Vector; typedef sopt::Matrix t_Matrix; auto const N = 30; SCENARIO("ProximalADMM with warm start", "[padmm][integration]") { using namespace sopt; GIVEN("A ProximalADMM instance with its input") { t_Vector const target0 = t_Vector::Random(N); t_Vector const target1 = t_Vector::Random(N) * 4; auto const g0 = proximal::translate(proximal::EuclidianNorm(), -target0); auto const g1 = proximal::translate(proximal::EuclidianNorm(), -target1); t_Matrix const mId = -t_Matrix::Identity(N, N); auto convergence = [&target1, &target0](t_Vector const &x) -> bool { t_Vector const segment = (target1 - target0).normalized(); t_real const alpha = (x - target0).transpose() * segment; SOPT_TRACE(" {} {}", alpha, (x - target0 - alpha * segment).stableNorm()); return alpha >= 0e0 and (target1 - target0).transpose() * segment >= alpha and (x - target0 - alpha * segment).stableNorm() < 1e-8; }; auto padmm = algorithm::ProximalADMM(g0, g1, t_Vector::Zero(N)) .Phi(mId) .itermax(3000) .gamma(0.5) .is_converged(convergence); WHEN("the algorithms runs") { auto const full = padmm(); THEN("it converges") { CHECK(full.niters > 20); CHECK(full.good); } WHEN("It is set to stop before convergence") { auto const first_half = padmm.itermax(full.niters - 5)(); THEN("It is not converged") { CHECK(not first_half.good); } WHEN("A warm restart is attempted") { auto const second_half = padmm.itermax(5000)(first_half); THEN("The warm restart is validated by the fast convergence") { CHECK(second_half.niters < 10); } } } } } } sopt-2.0.0/cpp/tests/power_method.cc000066400000000000000000000027151277570055300174450ustar00rootroot00000000000000#include #include #include #include "catch.hpp" #include "sopt/power_method.h" TEST_CASE("Power Method") { using namespace sopt; typedef t_real Scalar; auto const N = 10; Eigen::EigenSolver> es; Matrix A = Matrix::Random(N, N); es.compute(A.adjoint() * A, true); auto const eigenvalues = es.eigenvalues(); auto const eigenvectors = es.eigenvectors(); Eigen::DenseIndex index; (eigenvalues.transpose() * eigenvalues).real().maxCoeff(&index); auto const eigenvalue = eigenvalues(index); Vector const eigenvector = eigenvectors.col(index); // Create input vector close to solution Vector const input = eigenvector * 1e-4 + Vector::Random(N); auto const pm = algorithm::PowerMethod().tolerance(1e-12); SECTION("AtA") { auto const lt = linear_transform(A.cast()); auto const result = pm.AtA(lt, input); CHECK(result.good); CAPTURE(eigenvalue); CAPTURE(result.magnitude); CAPTURE(result.eigenvector.transpose() * eigenvector); CHECK(std::abs(result.magnitude - std::abs(eigenvalue)) < 1e-8); } SECTION("A") { auto const result = pm((A.adjoint() * A).cast(), input); CHECK(result.good); CAPTURE(eigenvalue); CAPTURE(result.magnitude); CAPTURE(result.eigenvector.transpose() * eigenvector); CHECK(std::abs(result.magnitude - std::abs(eigenvalue)) < 1e-8); } } sopt-2.0.0/cpp/tests/proximal.cc000066400000000000000000000243361277570055300166070ustar00rootroot00000000000000#include #include #include #include #include "sopt/l1_proximal.h" #include "sopt/proximal.h" #include "sopt/types.h" template sopt::Matrix concatenated_permutations(sopt::t_uint i, sopt::t_uint j) { extern std::unique_ptr mersenne; std::vector cols(j); std::iota(cols.begin(), cols.end(), 0); std::shuffle(cols.begin(), cols.end(), *mersenne); assert(j % i == 0); auto const N = j / i; auto const elem = 1e0 / std::sqrt(static_cast::type>(N)); sopt::Matrix result = sopt::Matrix::Zero(i, cols.size()); for(typename sopt::Matrix::Index k(0); k < result.cols(); ++k) result(cols[k] / N, k) = elem; return result; } TEST_CASE("L2Ball", "[proximal]") { using namespace sopt; proximal::L2Ball ball(0.5); Vector out; Vector x(5); x << 1, 2, 3, 4, 5; out = ball(0, x); CHECK(x.isApprox(out / 0.5 * x.stableNorm())); ball.epsilon(x.stableNorm() * 1.001); out = ball(0, x); CHECK(x.isApprox(out)); } TEST_CASE("WeightedL2Ball", "[proximal]") { using namespace sopt; Vector const weights = 0.01 * Vector::Random(5).array() + 1e0; Vector x(5); x << 1, 2, 3, 4, 5; proximal::WeightedL2Ball wball(0.5, weights); proximal::L2Ball ball(0.5); Vector const expected = ball((x.array() * weights.array()).matrix()).array() / weights.array(); Vector const actual = wball(x); CHECK(actual.isApprox(expected)); wball.epsilon((x.array() * weights.array()).matrix().stableNorm() * 1.001); CHECK(x.isApprox(wball(x))); } TEST_CASE("Euclidian norm", "[proximal]") { using namespace sopt; proximal::EuclidianNorm eucl; Vector out(5); Vector x(5); x << 1, 2, 3, 4, 5; eucl(out, x.stableNorm() * 1.001, x); CHECK(out.isApprox(Vector::Zero(x.size()))); out = eucl(0.1, x); CHECK(out.isApprox(x * (1e0 - 0.1 / x.stableNorm()))); } TEST_CASE("Translation", "[proximal]") { using namespace sopt; Vector out(5); Vector x(5); x << 1, 2, 3, 4, 5; proximal::L2Ball ball(5000); // Pass in a reference, so we can modify ball.epsilon later in the test. auto const translated = proximal::translate(std::ref(ball), -x * 0.5); translated(out, 0, x); CHECK(out.isApprox(x)); ball.epsilon(0.125); out = translated(0, x); Vector expected = ball(1, x * 0.5) + x * 0.5; CHECK(out.isApprox(expected)); } TEST_CASE("Tight-Frame L1 proximal", "[l1][proximal]") { using namespace sopt; auto l1 = proximal::L1TightFrame(); auto check_is_minimum = [&l1](Vector const &x, t_real gamma = 1e0) { typedef t_complex Scalar; Vector const p = l1(gamma, x); auto const mini = l1.objective(x, p, gamma); auto const eps = 1e-4; for(Vector::Index i(0); i < p.size(); ++i) { for(auto const dir : {Scalar(eps, 0), Scalar(0, eps), Scalar(-eps, 0), Scalar(0, -eps)}) { Vector p_plus = p; p_plus[i] += dir; CHECK(l1.objective(x, p_plus, gamma) >= mini); } } }; Vector const input = Vector::Random(8); // no weights SECTION("Scalar weights") { CHECK(l1(1, input).isApprox(proximal::l1_norm(1, input))); CHECK(l1(0.3, input).isApprox(proximal::l1_norm(0.3, input))); check_is_minimum(input, 0.664); } // with weights == 1 SECTION("vector weights") { l1.weights(Vector::Ones(input.size())); CHECK(l1(1, input).isApprox(proximal::l1_norm(1, input))); CHECK(l1(0.2, input).isApprox(proximal::l1_norm(0.2, input))); check_is_minimum(input, 0.664); } SECTION("vector weights with random values") { l1.weights(Vector::Random(input.size()).array().abs().matrix()); check_is_minimum(input, 0.235); } SECTION("Psi is a concatenation of permutations") { auto const psi = concatenated_permutations(input.size(), input.size() * 10); l1.Psi(psi).weights(1e0); check_is_minimum(input, 0.235); } SECTION("Weights cannot be negative") { CHECK_THROWS_AS(l1.weights(-1e0), Exception); Vector weights = Vector::Random(5).array().abs().matrix(); weights[2] = -1; CHECK_THROWS_AS(l1.weights(weights), Exception); } } TEST_CASE("L1 proximal utilities", "[l1][utilities]") { using namespace sopt; typedef t_complex Scalar; SECTION("Mixing") { auto const input = Vector::Random(10).eval(); Vector output; SECTION("No Mixing") { proximal::L1::NoMixing()(output, 2.1 * input, 0); CHECK(output.isApprox(2.1 * input)); proximal::L1::NoMixing()(output, 4.1 * input, 10); CHECK(output.isApprox(4.1 * input)); } SECTION("Fista Mixing") { proximal::L1::FistaMixing fista; // step zero: no mixing yet fista(output, 2.1 * input, 0); CHECK(output.isApprox(2.1 * input)); // step one: first mixing fista(output, 3.1 * input, 1); auto const alpha = (fista.next(1) - 1) / fista.next(fista.next(1)); Vector const first = (1e0 + alpha) * 3.1 * input - alpha * 2.1 * input; CHECK(output.isApprox(first)); // step two: second mixing fista(output, 4.1 * input, 1); auto const beta = (fista.next(fista.next(1)) - 1) / fista.next(fista.next(fista.next(1))); Vector const second = (1e0 + alpha) * 4.1 * input - alpha * first; CHECK(output.isApprox(second)); } } SECTION("Breaker") { proximal::L1::Breaker breaker(2e0); SECTION("Finds convergence") { std::vector objectives = {1.0, 0.9, 0.5, 0.6, 0.4, 0.4 + 0.41 * 1e-8, 0.3, 0.3 + 0.29 * 1e-8}; for(size_t i(0); i < objectives.size() - 1; ++i) { CHECK(not breaker(objectives[i])); CHECK(breaker.current() == Approx(objectives[i]).epsilon(1e-12)); } CHECK(breaker(objectives.back())); CHECK(not breaker.two_cycle()); CHECK(breaker.converged()); } SECTION("Find cycle") { std::vector objectives = {1.0, 0.9, 0.5, 0.6, 0.4, 0.3, 0.4, 0.3}; for(size_t i(0); i < objectives.size() - 1; ++i) { CHECK(not breaker(objectives[i])); CHECK(breaker.current() == Approx(objectives[i]).epsilon(1e-12)); } CHECK(breaker(objectives.back())); CHECK(breaker.two_cycle()); CHECK(not breaker.converged()); } } } TEST_CASE("L1 proximal", "[l1][proximal]") { using namespace sopt; typedef t_complex Scalar; auto l1 = proximal::L1().tolerance(1e-10); Vector const input = Vector::Random(4); SECTION("Check against tight-frame") { l1.fista_mixing(false); SECTION("Scalar weights") { auto const result = l1(1, input); CHECK(result.good); CHECK(result.niters > 0); CHECK(l1.itermax() == 0); CHECK(result.proximal.isApprox(proximal::L1TightFrame()(1, input))); } SECTION("Vector weights and more complex Psi") { auto const Psi = concatenated_permutations(input.size(), input.size() * 10); auto const weights = Vector::Random(Psi.cols()).array().abs().matrix().eval(); auto const gamma = 1e-1 / Psi.array().abs().sum(); l1.Psi(Psi).weights(weights).tolerance(1e-12); auto const result = l1(gamma, input); CHECK(result.good); CHECK(result.niters > 0); auto const expected = l1.tight_frame(gamma, input).eval(); CHECK(result.objective == Approx(l1.objective(input, expected, gamma))); CAPTURE((result.proximal - expected).array().abs().transpose()); CHECK(result.proximal.isApprox(expected)); } } SECTION("General case") { auto check_is_minimum = [&l1, &input](t_real gamma, Vector const &proximal) { // returns false if did not converge. // Looks like computing the proximal does not always work... auto const mini = l1.objective(input, proximal, gamma); auto const eps = 1e-4; // check alongst specific directions for(Vector::Index i(0); i < proximal.size(); ++i) { for(auto const dir : {Scalar(eps, 0), Scalar(0, eps), Scalar(-eps, 0), Scalar(0, -eps)}) { Vector p_plus = proximal; p_plus[i] += dir; if(l1.positivity_constraint()) p_plus = sopt::positive_quadrant(p_plus); else if(l1.real_constraint()) p_plus = p_plus.real().cast(); auto const rel_var = std::abs((l1.objective(input, p_plus, gamma) - mini) / mini); CHECK((l1.objective(input, p_plus, gamma) > mini or rel_var < l1.tolerance() * 10)); } } // check alongst non-specific directions for(size_t i(0); i < 10; ++i) { Vector p_plus = proximal + proximal.Random(proximal.size()) * eps; if(l1.positivity_constraint()) p_plus = sopt::positive_quadrant(p_plus); else if(l1.real_constraint()) p_plus = p_plus.real().cast(); auto const rel_var = std::abs((l1.objective(input, p_plus, gamma) - mini) / mini); CHECK((l1.objective(input, p_plus, gamma) > mini or rel_var < l1.tolerance() * 10)); } }; auto const Psi = Matrix::Random(input.size(), input.size() * 10).eval(); auto const weights = Vector::Random(Psi.cols()).array().abs().matrix().eval(); auto const gamma = 1e-1 / Psi.array().abs().sum(); l1.Psi(Psi).weights(weights).fista_mixing(true).tolerance(1e-10).itermax(5000); SECTION("No constraints") { CHECK(not l1.positivity_constraint()); CHECK(not l1.real_constraint()); auto const result = l1(gamma, input); CHECK(result.good); check_is_minimum(gamma, result.proximal); } SECTION("Positivity constraints") { l1.positivity_constraint(true); CHECK(l1.positivity_constraint()); CHECK(not l1.real_constraint()); auto const result = l1(gamma, input); CHECK(result.good); check_is_minimum(gamma, result.proximal); } SECTION("Real constraints") { l1.real_constraint(true); CHECK(l1.real_constraint()); CHECK(not l1.positivity_constraint()); auto const result = l1(gamma, input); CHECK(result.good); check_is_minimum(gamma, result.proximal); } } } sopt-2.0.0/cpp/tests/reweighted.cc000066400000000000000000000074601277570055300171020ustar00rootroot00000000000000#include #include #include #include "sopt/imaging_padmm.h" #include "sopt/reweighted.h" using namespace sopt; //! \brief Minimum set of functions and typedefs needed by reweighting //! \details The attributes are public and static so we can access them during the tests. struct DummyAlgorithm { typedef t_real Scalar; typedef Vector t_Vector; typedef ConvergenceFunction t_IsConverged; struct DiagnosticAndResult { //! Expected by reweighted algorithm static t_Vector x; }; DiagnosticAndResult operator()(t_Vector const &x) const { ++called_with_x; DiagnosticAndResult::x = x.array() + 0.1; return {}; } DiagnosticAndResult operator()(DiagnosticAndResult const &warm) const { ++called_with_warm; DiagnosticAndResult::x = warm.x.array() + 0.1; return {}; } //! Applies Ψ^T * x static t_Vector reweightee(DummyAlgorithm const &, t_Vector const &x) { ++DummyAlgorithm::called_reweightee; return x * 2; } //! sets the weights static void set_weights(DummyAlgorithm &, t_Vector const &weights) { ++DummyAlgorithm::called_weights; DummyAlgorithm::weights = weights; } static t_Vector weights; static int called_with_x; static int called_with_warm; static int called_reweightee; static int called_weights; }; int DummyAlgorithm::called_with_x = 0; int DummyAlgorithm::called_with_warm = 0; int DummyAlgorithm::called_reweightee = 0; int DummyAlgorithm::called_weights = 0; DummyAlgorithm::t_Vector DummyAlgorithm::DiagnosticAndResult::x; DummyAlgorithm::t_Vector DummyAlgorithm::weights; TEST_CASE("L0-Approximation") { auto const N = 6; DummyAlgorithm::t_Vector const input = DummyAlgorithm::t_Vector::Random(N); auto l0algo = algorithm::reweighted(DummyAlgorithm(), DummyAlgorithm::set_weights, DummyAlgorithm::reweightee); DummyAlgorithm::called_with_x = 0; DummyAlgorithm::called_with_warm = 0; DummyAlgorithm::called_reweightee = 0; DummyAlgorithm::called_weights = 0; DummyAlgorithm::DiagnosticAndResult::x = DummyAlgorithm::t_Vector::Zero(0); DummyAlgorithm::weights = DummyAlgorithm::t_Vector::Zero(0); GIVEN("The maximum number of iteration is zero") { l0algo.itermax(0); WHEN("The reweighting algorithm is called") { auto const result = l0algo(input); THEN("The algorithm exited at the first iteration") { CHECK(result.niters == 0); CHECK(result.good == true); } THEN("The weights is set to 1") { CHECK(result.weights.size() == 1); CHECK(std::abs(result.weights(0) - 1) < 1e-12); } THEN("The inner algorithm was called once") { CHECK(DummyAlgorithm::called_with_x == 1); CHECK(DummyAlgorithm::called_with_warm == 0); CHECK(result.algo.x.array().isApprox(input.array() + 0.1)); } } } GIVEN("The maximum number of iterations is one") { l0algo.itermax(1); WHEN("The reweighting algorithm is called") { auto const result = l0algo(input); THEN("The algorithm exited at the second iteration") { CHECK(result.niters == 1); CHECK(result.good == true); } THEN("The weights are not one") { CHECK(result.weights.size() == input.size()); // standard deviation of Ψ^T x, with x the output of the first call to the inner algorithm Vector<> const PsiT_x = DummyAlgorithm::reweightee({}, input.array() + 0.1); auto delta = standard_deviation(PsiT_x); CHECK(result.weights.array().isApprox(delta / (delta + PsiT_x.array().abs()))); } THEN("The inner algorithm was called twice") { CHECK(DummyAlgorithm::called_with_x == 1); CHECK(DummyAlgorithm::called_with_warm == 1); CHECK(result.algo.x.array().isApprox(input.array() + 0.2)); } } } } sopt-2.0.0/cpp/tests/sara.cc000066400000000000000000000070171277570055300156770ustar00rootroot00000000000000#include #include #include #include #include "sopt/wavelets.h" #include "sopt/wavelets/sara.h" sopt::t_int random_integer(sopt::t_int min, sopt::t_int max) { extern std::unique_ptr mersenne; std::uniform_int_distribution uniform_dist(min, max); return uniform_dist(*mersenne); }; TEST_CASE("Check SARA implementation mechanically", "[wavelet]") { using namespace sopt::wavelets; using namespace sopt; typedef std::tuple t_i; SARA const sara{t_i{std::string{"DB3"}, 1u}, t_i{std::string{"DB1"}, 2u}, t_i{std::string{"DB1"}, 3u}}; SECTION("Construction and vector functionality") { CHECK(sara.size() == 3); CHECK(sara[0].levels() == 1); CHECK(sara[1].levels() == 2); CHECK(sara[2].levels() == 3); CHECK(sara.max_levels() == 3); CHECK(sara[0].coefficients.isApprox(factory("DB3", 1).coefficients)); CHECK(sara[1].coefficients.isApprox(factory("DB1", 1).coefficients)); CHECK(sara[2].coefficients.isApprox(factory("DB1", 1).coefficients)); } Image<> input = Image<>::Random((1u << sara.max_levels()) * 3, (1u << sara.max_levels())); Image<> coeffs; sara.direct(coeffs, input); SECTION("Direct transform") { Image<> const first = sara[0].direct(input) / std::sqrt(sara.size()); Image<> const second = sara[1].direct(input) / std::sqrt(sara.size()); Image<> const third = sara[2].direct(input) / std::sqrt(sara.size()); auto const N = input.cols(); CAPTURE(coeffs.leftCols(N)); CAPTURE(first); CHECK(coeffs.leftCols(N).isApprox(first)); CHECK(coeffs.leftCols(2 * N).rightCols(N).isApprox(second)); CHECK(coeffs.rightCols(N).isApprox(third)); } SECTION("Indirect transform") { auto const output = sara.indirect(coeffs); CHECK(output.isApprox(input)); } } TEST_CASE("Linear-transform wrapper", "[wavelet]") { using namespace sopt::wavelets; using namespace sopt; SARA const sara{std::make_tuple(std::string{"DB3"}, 1u), std::make_tuple(std::string{"DB1"}, 2u), std::make_tuple(std::string{"DB1"}, 3u)}; auto const rows = 256, cols = 256; auto const Psi = linear_transform(sara, rows, cols); SECTION("Indirect transform") { Image<> const image = Image<>::Random(rows, cols); Image<> const expected = sara.direct(image); // The linear transform expects a column vector as input auto const as_vector = Vector<>::Map(image.data(), image.size()); // And it returns a column vector as well Vector<> const actual = Psi.adjoint() * as_vector; CHECK(actual.size() == expected.size()); auto const coeffs = Image<>::Map(actual.data(), image.rows(), image.cols() * sara.size()); CHECK(expected.rows() == coeffs.rows()); CHECK(expected.cols() == coeffs.cols()); CHECK(coeffs.isApprox(expected, 1e-8)); } SECTION("direct transform") { Image<> const coeffs = Image<>::Random(rows, cols * sara.size()); Image<> const expected = sara.indirect(coeffs); // The linear transform expects a column vector as input auto const as_vector = Vector<>::Map(coeffs.data(), coeffs.size()); // And it returns a column vector as well Vector<> const actual = Psi * as_vector; CHECK(actual.size() == expected.size()); CHECK(coeffs.cols() % sara.size() == 0); auto const image = Image<>::Map(actual.data(), coeffs.rows(), coeffs.cols() / sara.size()); CHECK(expected.rows() == image.rows()); CHECK(expected.cols() == image.cols()); CHECK(image.isApprox(expected, 1e-8)); } } sopt-2.0.0/cpp/tests/sdmm.cc000066400000000000000000000214551277570055300157130ustar00rootroot00000000000000#include #include #include #include "sopt/proximal.h" #include "sopt/sdmm.h" #include "sopt/types.h" sopt::t_int random_integer(sopt::t_int min, sopt::t_int max) { extern std::unique_ptr mersenne; std::uniform_int_distribution uniform_dist(min, max); return uniform_dist(*mersenne); }; typedef sopt::t_real Scalar; typedef sopt::Vector t_Vector; typedef sopt::Matrix t_Matrix; auto const N = 4; // Makes members public so we can test one at a time class IntrospectSDMM : public sopt::algorithm::SDMM { public: using sopt::algorithm::SDMM::initialization; using sopt::algorithm::SDMM::solve_for_xn; using sopt::algorithm::SDMM::update_directions; using sopt::algorithm::SDMM::t_Vectors; using sopt::algorithm::SDMM::t_Vector; }; TEST_CASE("Proximal translation", "[proximal]") { using namespace sopt; t_Vector const translation = t_Vector::Ones(N) * 5; auto const g = proximal::EuclidianNorm(); auto const gT = proximal::translate(g, -translation); t_Vector const input = t_Vector::Random(N).array() + 1e0; CHECK(g(0.1, input).isApprox((1e0 - 0.1 / input.stableNorm()) * input)); auto const gamma = input.stableNorm() * 0.5; CHECK(g(gamma, input).isApprox((1e0 - gamma / input.stableNorm()) * input)); CHECK(g(gamma * 2 + 1, input).isApprox(input.Zero(N))); CHECK(gT(0.1, input) .isApprox((1e0 - 0.1 / (input - translation).stableNorm()) * (input - translation) + translation)); } // Iterate through algorithm for special case where the L_i are identies and the objective functions // are simple euclidian norms TEST_CASE("Introspect SDMM with L_i = Identity and Euclidian objectives", "[sdmm]") { using namespace sopt; t_Matrix const Id = t_Matrix::Identity(N, N).eval(); t_Vector const target0 = t_Vector::Zero(N); t_Vector const target1 = t_Vector::Random(N); auto const g0 = proximal::translate(proximal::EuclidianNorm(), -target0); auto const g1 = proximal::translate(proximal::EuclidianNorm(), -target1); t_Vector const input = 10 * t_Vector::Random(N); IntrospectSDMM sdmm = IntrospectSDMM(); sdmm.itermax(10) .gamma(0.01) .conjugate_gradient(std::numeric_limits::max(), 1e-12) .append(g0, Id) .append(g1, Id); SECTION("Step by Step") { INFO("Initialization"); t_Vector out = input; IntrospectSDMM::t_Vectors y(sdmm.transforms().size(), t_Vector::Zero(out.size())); IntrospectSDMM::t_Vectors z(sdmm.transforms().size(), t_Vector::Zero(out.size())); sdmm.initialization(y, z, out); CHECK(y[0].isApprox(input)); CHECK(y[1].isApprox(input)); INFO("\nThen solve for conjugate gradient"); auto const diagnostic0 = sdmm.solve_for_xn(out, y, z); CHECK(diagnostic0.good); CAPTURE(out.transpose()); CAPTURE(input.transpose()); CAPTURE(0.5 * (y[0] + y[1]).transpose()); CHECK(out.isApprox(0.5 * (y[0] + y[1]), 1e-8)); CHECK(out.isApprox(input, 1e-8)); INFO("\nWe move on to first iteration!"); INFO("- updates y and z"); sdmm.update_directions(y, z, out); CHECK(y[0].isApprox(g0(sdmm.gamma(), input))); CHECK(y[1].isApprox(g1(sdmm.gamma(), input))); CHECK(z[0].isApprox(input - y[0])); CHECK(z[1].isApprox(input - y[1])); INFO("- solve for conjugate gradient"); auto const diagnostic1 = sdmm.solve_for_xn(out, y, z); CHECK(diagnostic1.good); CAPTURE(out.transpose()); CAPTURE((0.5 * (y[0] - z[0] + y[1] - z[1])).transpose()); CHECK(out.isApprox(0.5 * (y[0] - z[0] + y[1] - z[1]))); t_Vector const x1 = g0(sdmm.gamma(), input) + g1(sdmm.gamma(), input) - input; CHECK(out.isApprox(x1)); INFO("\nWe move on to second iteration!"); INFO("- updates y and z"); sdmm.update_directions(y, z, out); CHECK(y[0].isApprox(g0(sdmm.gamma(), g1(sdmm.gamma(), input)))); CHECK(y[1].isApprox(g1(sdmm.gamma(), g0(sdmm.gamma(), input)))); CHECK(z[0].isApprox(g1(sdmm.gamma(), input) - y[0])); CHECK(z[1].isApprox(g0(sdmm.gamma(), input) - y[1])); INFO("- solve for conjugate gradient"); auto const diagnostic2 = sdmm.solve_for_xn(out, y, z); CHECK(diagnostic2.good); CHECK(out.isApprox(0.5 * (y[0] - z[0] + y[1] - z[1]))); t_Vector const x2 = g0(sdmm.gamma(), g1(sdmm.gamma(), input)) + g1(sdmm.gamma(), g0(sdmm.gamma(), input)) - 0.5 * g1(sdmm.gamma(), input) - 0.5 * g0(sdmm.gamma(), input); CHECK(out.isApprox(x2)); } SECTION("Iteration by Iteration") { t_Vector out; SECTION("First Iteration") { sdmm.itermax(1); auto const diagnostic = sdmm(out, input); CHECK(not diagnostic.good); CHECK(diagnostic.niters == 1); CHECK(out.isApprox(g0(sdmm.gamma(), input) + g1(sdmm.gamma(), input) - input)); } SECTION("Second Iteration") { sdmm.itermax(2); auto const diagnostic = sdmm(out, input); CHECK(not diagnostic.good); CHECK(diagnostic.niters == 2); t_Vector const x2 = g0(sdmm.gamma(), g1(sdmm.gamma(), input)) + g1(sdmm.gamma(), g0(sdmm.gamma(), input)) - 0.5 * g1(sdmm.gamma(), input) - 0.5 * g0(sdmm.gamma(), input); CHECK(out.isApprox(x2)); } SECTION("Nth Iterations") { sdmm.gamma(1); for(t_uint itermax(0); itermax < 10; ++itermax) { t_Vector x = input; t_Vector y[2] = {x, x}; t_Vector z[2] = {t_Vector::Zero(N).eval(), t_Vector::Zero(N).eval()}; for(t_uint i(0); i < itermax; ++i) { y[0] = g0(sdmm.gamma(), x + z[0]); y[1] = g1(sdmm.gamma(), x + z[1]); z[0] += x - g0(sdmm.gamma(), x + z[0]); z[1] += x - g1(sdmm.gamma(), x + z[1]); x = 0.5 * (y[0] - z[0] + y[1] - z[1]); } sdmm.itermax(itermax); auto const diagnostic = sdmm(out, input); CHECK(out.isApprox(x, 1e-8)); CHECK(not diagnostic.good); CHECK(diagnostic.niters == itermax); } } } } TEST_CASE("SDMM with ||x - x0||_2 functions", "[sdmm][integration]") { using namespace sopt; t_Matrix const Id = t_Matrix::Identity(N, N).eval(); t_Vector const target0 = t_Vector::Random(N); t_Vector target1 = t_Vector::Random(N) * 4; // for(t_uint i(0); i < N; ++i) target1(i) = i + 1; auto sdmm = algorithm::SDMM() .itermax(5000) .gamma(1) .conjugate_gradient(std::numeric_limits::max(), 1e-12) .append(proximal::translate(proximal::EuclidianNorm(), -target0), Id) .append(proximal::translate(proximal::EuclidianNorm(), -target1), Id); t_Vector result; SECTION("Just two operators") { auto const diagnostic = sdmm(result, t_Vector::Random(N)); CHECK(not diagnostic.good); CHECK(diagnostic.niters == sdmm.itermax()); t_Vector const segment = (target1 - target0).normalized(); t_real const alpha = (result - target0).transpose() * segment; CAPTURE(target0.transpose()); CAPTURE(target1.transpose()); CHECK((target1 - target0).transpose() * segment >= alpha); CHECK(alpha >= 0e0); CHECK((result - target0 - alpha * segment).stableNorm() < 1e-8); } SECTION("Three operators") { t_Vector const target2 = t_Vector::Random(N) * 8; sdmm.append(proximal::translate(proximal::EuclidianNorm(), -target2), Id); auto const diagnostic = sdmm(result, t_Vector::Random(N)); CHECK(not diagnostic.good); CHECK(diagnostic.niters == sdmm.itermax()); CAPTURE(result.transpose()); auto const func = [&target0, &target1, &target2](t_Vector const &x) { return (x - target0).stableNorm() + (x - target1).stableNorm() + (x - target2).stableNorm(); }; for(int i(0); i < N; ++i) { t_Vector epsilon = t_Vector::Zero(N); epsilon(i) = 1e-6; CHECK(func(result) < func(result + epsilon)); CHECK(func(result) < func(result - epsilon)); } } SECTION("With different L") { t_Matrix const L0 = t_Matrix::Random(N, N) * 2; t_Matrix const L1 = t_Matrix::Random(N, N) * 4; REQUIRE(std::abs((L0.transpose() * L0 + L1.transpose() * L1).determinant()) > 1e-8); sdmm.itermax(300); sdmm.transforms(0) = linear_transform(L0); sdmm.transforms(1) = linear_transform(L1); auto const diagnostic = sdmm(result, t_Vector::Random(N)); CHECK(not diagnostic.good); CHECK(diagnostic.niters == sdmm.itermax()); CAPTURE(result.transpose()); auto const func = [&target0, &target1, &L0, &L1](t_Vector const &x) { return (L0 * x - target0).stableNorm() + (L1 * x - target1).stableNorm(); }; for(int i(0); i < N; ++i) { t_Vector epsilon = t_Vector::Zero(N); epsilon(i) = 1e-4; CAPTURE(epsilon.transpose()); CHECK(func(result) <= func(result + epsilon)); CHECK(func(result) <= func(result - epsilon)); } } } sopt-2.0.0/cpp/tests/sdmm_warm_start.cc000066400000000000000000000037251277570055300201560ustar00rootroot00000000000000#include #include #include #include "sopt/proximal.h" #include "sopt/sdmm.h" #include "sopt/types.h" typedef sopt::t_real Scalar; typedef sopt::Vector t_Vector; typedef sopt::Matrix t_Matrix; auto const N = 30; SCENARIO("SDMM with warm start", "[sdmm][integration]") { using namespace sopt; GIVEN("An SDMM instance with its input") { t_Matrix const Id = t_Matrix::Identity(N, N).eval(); t_Vector const target0 = t_Vector::Random(N); t_Vector target1 = t_Vector::Random(N) * 4; auto convergence = [&target1, &target0](t_Vector const &x) -> bool { t_Vector const segment = (target1 - target0).normalized(); t_real const alpha = (x - target0).transpose() * segment; return alpha >= 0e0 and (target1 - target0).transpose() * segment >= alpha and (x - target0 - alpha * segment).stableNorm() < 1e-8; }; auto sdmm = algorithm::SDMM() .is_converged(convergence) .itermax(5000) .gamma(1) .conjugate_gradient(std::numeric_limits::max(), 1e-12) .append(proximal::translate(proximal::EuclidianNorm(), -target0), Id) .append(proximal::translate(proximal::EuclidianNorm(), -target1), Id); t_Vector input = t_Vector::Random(N); WHEN("the algorithms runs") { auto const full = sdmm(input); THEN("it converges") { CHECK(full.niters > 20); CHECK(full.good); } WHEN("It is set to stop before convergence") { auto const first_half = sdmm.itermax(full.niters - 5)(input); THEN("It is not converged") { CHECK(not first_half.good); } WHEN("A warm restart is attempted") { auto const second_half = sdmm.itermax(5000)(first_half); THEN("The warm restart is validated by the fast convergence") { CHECK(second_half.niters < 10); } } } } } } sopt-2.0.0/cpp/tests/wavelets.cc000066400000000000000000000271451277570055300166070ustar00rootroot00000000000000#include #include #include "sopt/types.h" #include "sopt/wavelets/direct.h" #include "sopt/wavelets/indirect.h" #include "sopt/wavelets/wavelet_data.h" #include "sopt/wavelets/wavelets.h" typedef sopt::Array t_iVector; t_iVector even(t_iVector const &x) { t_iVector result((x.size() + 1) / 2); for(t_iVector::Index i(0); i < x.size(); i += 2) result(i / 2) = x(i); return result; }; t_iVector odd(t_iVector const &x) { t_iVector result(x.size() / 2); for(t_iVector::Index i(1); i < x.size(); i += 2) result(i / 2) = x(i); return result; }; template Eigen::Array upsample(Eigen::ArrayBase const &input) { typedef Eigen::Array Matrix; Matrix result(input.size() * 2); for(t_iVector::Index i(0); i < input.size(); ++i) { result(2 * i) = input(i); result(2 * i + 1) = 0; } return result; }; sopt::t_int random_integer(sopt::t_int min, sopt::t_int max) { extern std::unique_ptr mersenne; std::uniform_int_distribution uniform_dist(min, max); return uniform_dist(*mersenne); }; t_iVector random_ivector(sopt::t_int size, sopt::t_int min, sopt::t_int max) { extern std::unique_ptr mersenne; t_iVector result(size); std::uniform_int_distribution uniform_dist(min, max); for(t_iVector::Index i(0); i < result.size(); ++i) result(i) = uniform_dist(*mersenne); return result; }; // Checks round trip operation template void check_round_trip(Eigen::ArrayBase const &input_, sopt::t_uint db, sopt::t_uint nlevels = 1) { auto const input = input_.eval(); auto const &dbwave = sopt::wavelets::daubechies_data(db); auto const transform = sopt::wavelets::direct_transform(input, nlevels, dbwave); auto const actual = sopt::wavelets::indirect_transform(transform, nlevels, dbwave); CAPTURE(actual); CAPTURE(input); CAPTURE(transform); CHECK(input.isApprox(actual, 1e-14)); CHECK(not transform.isApprox(sopt::wavelets::direct_transform(input, nlevels - 1, dbwave), 1e-4)); } TEST_CASE("Wavelet transform innards with integer data", "[wavelet]") { using namespace sopt::wavelets; t_iVector small(3); small << 1, 2, 3; t_iVector large(6); large << 4, 5, 6, 7, 8, 9; SECTION("Periodic scalar product") { // no wrapping CHECK(periodic_scalar_product(large, small, 0) == 1 * 4 + 2 * 5 + 3 * 6); CHECK(periodic_scalar_product(large, small, 1) == 1 * 5 + 2 * 6 + 3 * 7); CHECK(periodic_scalar_product(large, small, 3) == 1 * 7 + 2 * 8 + 3 * 9); // with wrapping CHECK(periodic_scalar_product(large, small, 4) == 1 * 8 + 2 * 9 + 3 * 4); // with wrapping and expression CHECK(periodic_scalar_product(large, small.reverse(), 4) == 3 * 8 + 2 * 9 + 1 * 4); // wrapping works with offset as well CHECK(periodic_scalar_product(large, small, 4 + large.size()) == 1 * 8 + 2 * 9 + 3 * 4); CHECK(periodic_scalar_product(large, small, 4 - 3 * large.size()) == 1 * 8 + 2 * 9 + 3 * 4); // signal smaller than filter CHECK(periodic_scalar_product(small, large.head(4), 1) == 4 * 2 + 5 * 3 + 6 * 1 + 7 * 2); } SECTION("Convolve") { t_iVector result(large.size()); convolve(result, large, small); CHECK(result(0) == 1 * 4 + 2 * 5 + 3 * 6); CHECK(result(1) == 1 * 5 + 2 * 6 + 3 * 7); CHECK(result(3) == 1 * 7 + 2 * 8 + 3 * 9); CHECK(result(4) == 1 * 8 + 2 * 9 + 3 * 4); } SECTION("Convolve and sum") { t_iVector result(large.size()); t_iVector noOffset(large.size()); // Check that if high pass is zero, then this is an offseted convolution convolve_sum(result, large, small, large, 0 * small); convolve(noOffset, large, small); CHECK(result(small.size() - 1) == noOffset(0)); CHECK(result(0) == noOffset(result.size() - small.size() + 1)); // Check same for low pass convolve_sum(result, large, 0 * small, large, small); CHECK(result(small.size() - 1) == noOffset(0)); CHECK(result(0) == noOffset(result.size() - small.size() + 1)); // Check symmetry relationships auto const trial = [&small, &large](int a, int b, int c, int d) { t_iVector result(large.size()); convolve_sum(result, a * large, b * small, c * large, d * small); return result; }; // should all be ok as long as arguments sum: (a * b) + (c * d) == (a' * b') + (c' * d') CHECK((trial(0, 1, 3, 1) == trial(0, 1, 1, 3)).all()); CHECK((trial(5, 1, 3, 1) == trial(3, 1, 5, 1)).all()); CHECK((trial(1, 5, 3, 1) == trial(3, 1, 5, 1)).all()); CHECK((trial(1, 3, 5, 1) == trial(3, 1, 5, 1)).all()); CHECK((trial(1, 3, 1, 5) == trial(3, 1, 5, 1)).all()); CHECK((trial(1, 0, 4, 2) == trial(3, 1, 5, 1)).all()); CHECK((trial(1, -1, 1, 1) == trial(0, 1, 0, 1)).all()); CHECK((trial(4, -3, 2, 6) == trial(0, 1, 0, 1)).all()); } SECTION("Convolve and Down-sample simultaneously") { t_iVector expected(large.size()); convolve(expected, large, small); t_iVector actual(large.size() / 2); down_convolve(actual, large, small); for(size_t i(0); i < static_cast(actual.size()); ++i) CHECK(expected(i * 2) == actual(i)); } SECTION("Convolve output to expression") { t_iVector actual(large.size() * 2); t_iVector expected(large.size()); convolve(actual.head(large.size()), large, small); convolve(expected, large, small); CHECK((actual.head(large.size()) == expected).all()); } SECTION("Copy does copy") { auto result = copy(large); CHECK(large.data() != result.data()); auto actual = copy(large.head(3)); CHECK(large.data() != actual.data()); CHECK(large.data() == large.head(3).data()); } SECTION("Convolve, Sum and Up-sample simultaneously") { for(sopt::t_int i(0); i < 100; ++i) { auto const Ncoeffs = random_integer(2, 10) * 2; auto const Nfilters = random_integer(2, 5); auto const Nhead = Ncoeffs / 2; auto const Ntail = Ncoeffs - Nhead; auto const coeffs = random_ivector(Ncoeffs, -10, 10); auto const low = random_ivector(Nfilters, -10, 10); auto const high = random_ivector(Nfilters, -10, 10); t_iVector actual(Ncoeffs), expected(Ncoeffs); // does all in go, more complicated but compuationally less intensive up_convolve_sum(actual, coeffs, even(low), odd(low), even(high), odd(high)); // first up-samples, then does convolve: conceptually simpler but does unnecessary operations convolve_sum(expected, upsample(coeffs.head(Nhead)), low, upsample(coeffs.tail(Ntail)), high); CHECK((actual == expected).all()); } } } TEST_CASE("1D wavelet transform with floating point data", "[wavelet]") { using namespace sopt; using namespace sopt::wavelets; Image<> const data = Image<>::Random(16, 16); auto const &wavelet = daubechies_data(4); // Condition on input fixture data REQUIRE((data.rows() % 2 == 0 and (data.cols() == 1 or data.cols() % 2 == 0))); SECTION("Direct transform == two downsample + convolution") { auto const actual = direct_transform(data.row(0), 1, wavelet); Array<> high(data.cols() / 2), low(data.cols() / 2); down_convolve(high, data.row(0), wavelet.direct_filter.high); down_convolve(low, data.row(0), wavelet.direct_filter.low); CHECK(low.transpose().isApprox(actual.head(data.row(0).size() / 2))); CHECK(high.transpose().isApprox(actual.tail(data.row(0).size() / 2))); } SECTION("Indirect transform == two upsample + convolution") { auto const actual = indirect_transform(data.row(0).transpose(), 1, wavelet); auto const low = upsample(data.row(0).transpose().head(data.rows() / 2)); auto const high = upsample(data.row(0).transpose().tail(data.rows() / 2)); auto expected = copy(data.row(0).transpose()); convolve_sum(expected, low, wavelet.direct_filter.low.reverse(), high, wavelet.direct_filter.high.reverse()); CAPTURE(expected.transpose()); CAPTURE(actual.transpose()); CHECK(expected.isApprox(actual)); } SECTION("Round-trip test for single level") { for(t_int i(0); i < 20; ++i) { check_round_trip(Array<>::Random(random_integer(2, 100) * 2), random_integer(1, 38), 1); } } SECTION("Round-trip test for two levels") { check_round_trip(Array<>::Random(8), 1, 2); check_round_trip(Array<>::Random(8), 2, 2); check_round_trip(Array<>::Random(16), 4, 2); check_round_trip(Array<>::Random(52), 10, 2); } t_uint nlevels = 5; SECTION("Round-trip test for multiple levels") { for(t_int i(0); i < 10; ++i) { auto const n = random_integer(2, nlevels); check_round_trip(Array<>::Random(random_integer(2, 100) * (1u << n)), random_integer(1, 38), n); } } } TEST_CASE("1D wavelet transform with complex data", "[wavelet]") { using namespace sopt; using namespace sopt::wavelets; SECTION("Round-trip test for complex data") { auto input = Array::Random(random_integer(2, 100) * 2).eval(); auto const &dbwave = daubechies_data(random_integer(1, 38)); auto const actual = indirect_transform(direct_transform(input, 1, dbwave), 1, dbwave); CHECK(input.isApprox(actual, 1e-14)); CHECK(not input.isApprox(direct_transform(input, 1, dbwave), 1e-4)); } } TEST_CASE("2D wavelet transform with real data", "[wavelet]") { using namespace sopt; using namespace sopt::wavelets; SECTION("Single level round-trip test for square matrix") { auto N = random_integer(2, 100) * 2; check_round_trip(Image<>::Random(N, N), random_integer(1, 38), 1); } SECTION("Single level round-trip test for non-square matrix") { auto Nx = random_integer(2, 5) * 2; auto Ny = Nx + 5 * 2; check_round_trip(Image<>::Random(Nx, Ny), random_integer(1, 38), 1); } SECTION("Round-trip test for multiple levels") { for(t_int i(0); i < 10; ++i) { auto const n = random_integer(2, 5); auto const Nx = random_integer(2, 5) * (1u << n); auto const Ny = random_integer(2, 5) * (1u << n); check_round_trip(Image<>::Random(Nx, Ny), random_integer(1, 38), n); } } } TEST_CASE("Functor implementation", "[wavelet]") { using namespace sopt; auto const wavelet = wavelets::factory("DB3", 4); auto const input = Image::Random(256, 128).eval(); SECTION("Normal instances") { auto const transform = wavelet.direct(input); CHECK(transform.isApprox(wavelets::direct_transform(input, wavelet.levels(), wavelet))); CHECK(input.isApprox(wavelet.indirect(transform))); } SECTION("Expression instances") { Image output(2, input.cols()); wavelet.direct(output.row(0).transpose(), input.row(0).transpose()); wavelet.indirect(output.row(0).transpose(), output.row(1).transpose()); CHECK(input.row(0).isApprox(output.row(1))); } } TEST_CASE("Automatic input resizing", "[wavelet]") { using namespace sopt; auto const wavelet = wavelets::factory("DB3", 4); auto const input = Image::Random(256, 128).eval(); Image output(1, 1); wavelet.direct(output, input); CHECK(output.rows() == input.rows()); CHECK(output.cols() == input.cols()); output.resize(1, 1); wavelet.indirect(input, output); CHECK(output.rows() == input.rows()); CHECK(output.cols() == input.cols()); } TEST_CASE("Dirac wavelets") { using namespace sopt; auto const wavelet = wavelets::factory("Dirac"); Image const input = Image::Random(256, 128); Image output(1, 1); wavelet.direct(output, input); CHECK(output.isApprox(input)); output = Image::Zero(1, 1); wavelet.indirect(input, output); CHECK(output.isApprox(input)); } sopt-2.0.0/cpp/tests/wrapper.cc000066400000000000000000000032111277570055300164210ustar00rootroot00000000000000#include #include #include "sopt/wrapper.h" TEST_CASE("Function wrappers", "[utility]") { using namespace sopt; typedef Array t_Array; typedef t_Array &t_RefArray; typedef t_Array const t_ConstRefArray; SECTION("Square function") { auto func = [](t_RefArray output, t_ConstRefArray const &input) { output = input * 2 + 1; }; t_Array const x = t_Array::Random(5); auto const A = details::wrap(func); // Expected result t_Array const expected = (x * 2 + 1).eval(); CHECK((A * x).matrix() == expected.matrix()); CHECK(A(x).matrix() == expected.matrix()); } SECTION("Rectangular function") { auto func = [](t_RefArray output, t_ConstRefArray const &input) { output = input.head(input.size() / 2) * 2 + 1; }; t_Array const x = t_Array::Random(5); auto const A = details::wrap(func, {{1, 2, 0}}); // Expected result t_Array const expected = (x.head(x.size() / 2) * 2 + 1).eval(); CHECK((A * x).cols() == 1); CHECK((A * x).rows() == 2); CHECK((A * x).matrix() == expected.matrix()); CHECK(A(x).matrix() == expected.matrix()); } SECTION("Fixed output-size functions") { auto func = [](t_RefArray output, t_ConstRefArray const &input) { output = input.head(3) * 2 + 1; }; t_Array const x = t_Array::Random(5); auto const A = details::wrap(func, {{0, 1, 3}}); // Expected result t_Array const expected = (x.head(3) * 2 + 1).eval(); CHECK((A * x).cols() == 1); CHECK((A * x).rows() == 3); CHECK((A * x).matrix() == expected.matrix()); CHECK(A(x).matrix() == expected.matrix()); } } sopt-2.0.0/cpp/tools_for_tests/000077500000000000000000000000001277570055300165235ustar00rootroot00000000000000sopt-2.0.0/cpp/tools_for_tests/CMakeLists.txt000066400000000000000000000017711277570055300212710ustar00rootroot00000000000000add_library(tools_for_tests STATIC tiffwrappers.cc tiffwrappers.h) target_link_libraries(tools_for_tests ${TIFF_LIBRARY}) target_include_directories(tools_for_tests PUBLIC "${CMAKE_CURRENT_SOURCE_DIR}/.." "${PROJECT_BINARY_DIR}/include/" ) target_include_directories(tools_for_tests SYSTEM PUBLIC ${TIFF_INCLUDE_DIR}) if(SPDLOG_INCLUDE_DIR) target_include_directories(tools_for_tests SYSTEM PUBLIC ${SPDLOG_INCLUDE_DIR}) endif() if(EIGEN3_INCLUDE_DIR) target_include_directories(tools_for_tests SYSTEM PUBLIC ${EIGEN3_INCLUDE_DIR}) endif() add_dependencies(tools_for_tests lookup_dependencies) # Simple manual tester of read/write tiff add_executable(copy_tiff copy_tiff.cc) target_link_libraries(copy_tiff tools_for_tests sopt) configure_file("directories.in.h" "${PROJECT_BINARY_DIR}/include/tools_for_tests/directories.h") file(MAKE_DIRECTORY "${PROJECT_BINARY_DIR}/outputs") file(MAKE_DIRECTORY "${PROJECT_BINARY_DIR}/outputs/sdmm") file(MAKE_DIRECTORY "${PROJECT_BINARY_DIR}/outputs/sdmm/regressions") sopt-2.0.0/cpp/tools_for_tests/cdata.h000066400000000000000000000022131277570055300177460ustar00rootroot00000000000000#ifndef SOPT_TOOLS_FOR_TESTS_CDATA #define SOPT_TOOLS_FOR_TESTS_CDATA #include "sopt/config.h" #include #include "sopt/linear_transform.h" namespace sopt { // Wraps calls to sampling and wavelets to C style template struct CData { typedef Eigen::Matrix t_Vector; typename t_Vector::Index nin, nout; sopt::LinearTransform const &transform; t_uint direct_calls, adjoint_calls; }; template void direct_transform(void *out, void *in, void **data) { CData const &cdata = *(CData *)data; typedef Eigen::Matrix t_Vector; t_Vector const eval = cdata.transform * t_Vector::Map((T *)in, cdata.nin); ++(((CData *)data)->direct_calls); t_Vector::Map((T *)out, cdata.nout) = eval; } template void adjoint_transform(void *out, void *in, void **data) { CData const &cdata = *(CData *)data; typedef Eigen::Matrix t_Vector; t_Vector const eval = cdata.transform.adjoint() * t_Vector::Map((T *)in, cdata.nout); ++(((CData *)data)->adjoint_calls); t_Vector::Map((T *)out, cdata.nin) = eval; } } /* sopt */ #endif sopt-2.0.0/cpp/tools_for_tests/copy_tiff.cc000066400000000000000000000016611277570055300210200ustar00rootroot00000000000000#include #include #include #include #include #include #include #include #include #include // \min_{x} ||\Psi^\dagger x||_1 \quad \mbox{s.t.} \quad ||y - x||_2 < \epsilon and x \geq 0 int main(int argc, char const **argv) { if(argc != 3) { std::cout << "Expects two arguments:\n" "- path to the image to clean (or name of standard SOPT image)\n" "- path to output image\n"; exit(0); } // Initializes and sets logger (if compiled with logging) // See set_level function for levels. sopt::logging::initialize(); // Read input file auto const image = sopt::notinstalled::read_standard_tiff(argv[1]); sopt::utilities::write_tiff(image, argv[2]); auto const reread = sopt::utilities::read_tiff(argv[2]); return 0; } sopt-2.0.0/cpp/tools_for_tests/directories.in.h000066400000000000000000000006041277570055300216150ustar00rootroot00000000000000#ifndef SOPT_DATA_DIR_H #define SOPT_DATA_DIR_H #include "sopt/config.h" #include namespace sopt { namespace notinstalled { //! Holds images and such inline std::string data_directory() { return "@PROJECT_SOURCE_DIR@/images"; } //! Output artefacts from tests inline std::string output_directory() { return "@PROJECT_BINARY_DIR@/outputs"; } } } /* sopt::notinstalled */ #endif sopt-2.0.0/cpp/tools_for_tests/inpainting.h000066400000000000000000000027651277570055300210460ustar00rootroot00000000000000#ifndef SOPT_TOOLS_FOR_TESTS_INPAINTING_H #define SOPT_TOOLS_FOR_TESTS_INPAINTING_H #include "sopt/config.h" #include #include #include namespace sopt { template Vector target(sopt::LinearTransform> const &sampling, sopt::Image const &image) { return sampling * Vector::Map(image.data(), image.size()); } template typename real_type::type sigma(sopt::LinearTransform> const &sampling, sopt::Image const &image) { auto const snr = 30.0; auto const y0 = target(sampling, image); return y0.stableNorm() / std::sqrt(y0.size()) * std::pow(10.0, -(snr / 20.0)); } template Vector dirty(sopt::LinearTransform> const &sampling, sopt::Image const &image, RANDOM &mersenne) { // values near the mean are the most likely // standard deviation affects the dispersion of generated values from the mean auto const y0 = target(sampling, image); std::normal_distribution<> gaussian_dist(0, sigma(sampling, image)); Vector y(y0.size()); for(t_int i = 0; i < y0.size(); i++) y(i) = y0(i) + gaussian_dist(mersenne); return y; } template typename real_type::type epsilon(sopt::LinearTransform> const &sampling, sopt::Image const &image) { auto const y0 = target(sampling, image); auto const nmeasure = y0.size(); return std::sqrt(nmeasure + 2 * std::sqrt(nmeasure)) * sigma(sampling, image); } } /* sopt */ #endif sopt-2.0.0/cpp/tools_for_tests/tiffwrappers.cc000066400000000000000000000007521277570055300215520ustar00rootroot00000000000000#include #include "sopt/types.h" #include "sopt/utilities.h" #include "tools_for_tests/directories.h" #include "tools_for_tests/tiffwrappers.h" namespace sopt { namespace notinstalled { Image<> read_standard_tiff(std::string const &name) { std::string const stdname = sopt::notinstalled::data_directory() + "/" + name + ".tiff"; bool const is_std = std::ifstream(stdname).good(); return sopt::utilities::read_tiff(is_std ? stdname : name); } } } /* sopt::notinstalled */ sopt-2.0.0/cpp/tools_for_tests/tiffwrappers.h000066400000000000000000000005051277570055300214100ustar00rootroot00000000000000#ifndef SOPT_TIFF_WRAPPER_H #define SOPT_TIFF_WRAPPER_H #include "sopt/config.h" #include #include "sopt/types.h" namespace sopt { namespace notinstalled { //! 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A random Fourier sampling % measurement operator is considered. Daubechies 8 wavelets are used for % the sparsifying operator. %% Clear workspace clc clear; %% Define paths addpath misc/ addpath prox_operators/ %% Define parameters % Coverages (half the plane for Fourier sampling) p = [0.50]; % Noise level (on the measurements) input_snr = 30; %% Read image % Load image im = im2double(imread('cameraman.tif')); % Normalise im = im/max(max(im)); % Enforce positivity im(im<0) = 0; %% Define sparsity operator dwtmode('per'); % Decomposition level of the wavelet transform. nlevel = 4; % Daubechies 8 wavelet operator [C,S] = wavedec2(im,nlevel,'db8'); Psit = @(x) wavedec2(x, nlevel, 'db8'); Psi = @(x) waverec2(x, S, 'db8'); %% Run simulations % Random Fourier sampling example % Define mask % Uniform sampling of the half Fourier plane mask = rand(size(im)) < p; mask(:,1:floor(size(im,2)/2))=0; mask = ifftshift(mask); mask(1,1)=0; mask(floor(size(im,1)/2):end,1)=0; ind = find(mask==1); Ma = sparse(1:numel(ind), ind, ... ones(numel(ind), 1), numel(ind), numel(im)); % Composition (Masking o Fourier) A = @(x) Ma*reshape(fft2(x)/sqrt(numel(ind)), numel(x), 1); At = @(x) (ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(ind)))); % Apply measurement operator y = A(im); % Add Gaussian i.i.d. noise sigma_noise = 10^(-input_snr/20)*std(im(:)); noise = (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2); y = y + noise; % Tolerance on noise epsilon = sqrt(numel(y) + 2*sqrt(numel(y)))*sigma_noise; epsilon_up = sqrt(numel(y) + 2.1*sqrt(numel(y)))*sigma_noise; tol_B2 = (epsilon_up/epsilon)-1; % Tolerance for the projection onto the L2-ball % Solve optimisation problem % Parameters for BPDN param.verbose = 1; % Print log or not param.gamma = 1e-1; % Convergence parameter param.rel_obj = 5e-4; % Stopping criterion for the L1 problem param.max_iter = 200; % Max. number of iterations for the L1 problem param.nu_B2 = 1; % Bound on the norm of the operator A param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0) param.max_iter_B2 = 200; %Max. number of iterations of the L2-ball projection param.pos_B2 = 1; %Positivity flag param.tol_B2 = tol_B2; % Tolerance for the projection onto the L2-ball param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0) %Optional parameters when Psit is not a tight frame: %param.nu_L1 = 1; % Bound on the norm of the operator Psit %param.max_iter_L1 = 200; %Max. number of iterations of the L1 prox %param.rel_obj_L1 = 1e-3; % Tolerance for the prox L1 % Solve BPDN problem with positivity constraint sol1 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param); % Compute SNR RSNR1 = 20*log10(norm(im,'fro') ... / norm(im-sol1,'fro')); % Example with only reality constraint param.pos_B2 = 0; %Positivity flag param.real_B2 = 1; %Reality flag % Solve BPDN problem sol2 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param); % Compute SNR RSNR2 = 20*log10(norm(im,'fro') ... / norm(im-sol2,'fro')); %% Show results figure imagesc(sol1), axis off, axis image, colorbar title(['Rec. with positivity const. SNR=',num2str(RSNR1), 'dB']) colormap gray figure imagesc(sol2), axis off, axis image, colorbar title(['Rec. with reality const. SNR=',num2str(RSNR2), 'dB']) colormap gray sopt-2.0.0/matlab/Example_TVDN.m000066400000000000000000000054001277570055300163540ustar00rootroot00000000000000%% Example_TVDN % Example to demonstrate use of TVDN solver. A random Fourier sampling % measurement operator is considered. %% Clear workspace clc clear; %% Define paths addpath misc/ addpath prox_operators/ %% Define parameters % Coverages (half the plane for Fourier sampling) p = [0.50]; % Noise level (on the measurements) input_snr = 30; %% Read image % Load image im = im2double(imread('cameraman.tif')); % Normalise im = im/max(max(im)); % Enforce positivity im(im<0) = 0; %% Run simulations %Random Fourier sampling example % Define mask % Uniform sampling of the half Fourier plane mask = rand(size(im)) < p; mask(:,1:floor(size(im,2)/2))=0; mask = ifftshift(mask); mask(1,1)=0; mask(floor(size(im,1)/2):end,1)=0; ind = find(mask==1); Ma = sparse(1:numel(ind), ind, ... ones(numel(ind), 1), numel(ind), numel(im)); % Composition (Masking o Fourier) A = @(x) Ma*reshape(fft2(x)/sqrt(numel(ind)), numel(x), 1); At = @(x) (ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(ind)))); % Apply measurement operator y = A(im); % Add Gaussian i.i.d. noise sigma_noise = 10^(-input_snr/20)*std(im(:)); noise = (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2); y = y + noise; % Tolerance on noise epsilon = sqrt(numel(y) + 2*sqrt(numel(y)))*sigma_noise; epsilon_up = sqrt(numel(y) + 2.1*sqrt(numel(y)))*sigma_noise; tol_B2 = (epsilon_up/epsilon)-1; % Tolerance for the projection onto the L2-ball % Solve optimisation problem % Parameters for TVDN param.verbose = 1; % Print log or not param.gamma = 1e-1; % Converge parameter param.rel_obj = 5e-4; % Stopping criterion for the TVDN problem param.max_iter = 200; % Max. number of iterations for the TVDN problem param_TV.max_iter_TV = 200; % Max. nb. of iter. for the sub-problem (proximal TV operator) param.nu_B2 = 1; % Bound on the norm of the operator A param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0) param.max_iter_B2 = 200; %Max. number of iterations of the L2-ball projection param.pos_B2 = 1; %Positivity flag param.tol_B2 = tol_B2; % Tolerance for the projection onto the L2-ball % Solve TVDN problem with positivity constraint sol1 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param); % Compute SNR RSNR1 = 20*log10(norm(im,'fro') ... / norm(im-sol1,'fro')); % Example with only reality constraint param.pos_B2 = 0; %Positivity flag param.real_B2 = 1; %Reality flag % Solve TVDN problem sol2 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param); % Compute SNR RSNR2 = 20*log10(norm(im,'fro') ... / norm(im-sol2,'fro')); %% Show results figure imagesc(sol1), axis off, axis image, colorbar title(['Rec. with positivity const. SNR=',num2str(RSNR1), 'dB']) colormap gray figure imagesc(sol2), axis off, axis image, colorbar title(['Rec. with reality const. SNR=',num2str(RSNR2), 'dB']) colormap gray sopt-2.0.0/matlab/Example_TVDN2.m000066400000000000000000000054461277570055300164500ustar00rootroot00000000000000%% Example_TVDN2 % Two examples to demonstrate use of TVDN solver. Simple inpainting and % Fourier measurement examples are included. %% Clear workspace clc; clear; %% Define paths addpath misc/ addpath prox_operators/ %% Parameters input_snr = 30; % Noise level (on the measurements) %% Load an image im = im2double(imread('cameraman.tif')); % figure(1); imagesc(im); axis image; axis off; colormap gray; title('Original image'); drawnow; %% Create a mask with 33% of 1 (the rest is set to 0) % Mask mask = rand(size(im)) < 0.33; ind = find(mask==1); % Masking matrix (sparse matrix in matlab) Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im)); % Masking operator A = @(x) Ma*x(:); % Select 33% of the values in x; At = @(x) reshape(Ma'*x(:), size(im)); % Adjoint operator + reshape image %% First problem: Inpainting problem % Select 33% of pixels y = A(im); % Add Gaussian i.i.d. noise sigma_noise = 10^(-input_snr/20)*std(im(:)); y = y + randn(size(y))*sigma_noise; % Display the downsampled image figure(2); clf; subplot(121); imagesc(At(y)); axis image; axis off; colormap gray; title('Measured image'); drawnow; % Parameters for TVDN param.verbose = 1; % Print log or not param.gamma = 1e-1; % Converge parameter param.rel_obj = 1e-4; % Stopping criterion for the TVDN problem param.max_iter = 200; % Max. number of iterations for the TVDN problem param_TV.max_iter_TV = 100; % Max. nb. of iter. for the sub-problem (proximal TV operator) param.nu_B2 = 1; % Bound on the norm of the operator A param.tol_B2 = 1e-4; % Tolerance for the projection onto the L2-ball param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0) param.max_iter_B2 = 500; % Tolerance on noise epsilon = sqrt(chi2inv(0.99, numel(ind)))*sigma_noise; % Solve TVDN problem sol = sopt_mltb_solve_TVDN(y, epsilon, A, At, param); % Show reconstructed image figure(2); subplot(122); imagesc(sol); axis image; axis off; colormap gray; title('Reconstructed image'); drawnow; %% Second problem: Reconstruct from 33% of Fourier measurements % Composition (Masking o Fourier) A = @(x) Ma*reshape(fft2(x)/sqrt(numel(im)), numel(x), 1); At = @(x) ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(im))); % Select 33% of Fourier coefficients y = A(im); % Add Gaussian i.i.d. noise sigma_noise = 10^(-input_snr/20)*std(im(:)); y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2); % Display the downsampled image figure(3); clf; subplot(121); imagesc(real(At(y))); axis image; axis off; colormap gray; title('Measured image'); drawnow; % Tolerance on noise epsilon = sqrt(chi2inv(0.99, 2*numel(ind))/2)*sigma_noise; % Solve TVDN problem sol = sopt_mltb_solve_TVDN(y, epsilon, A, At, param); % Show reconstructed image figure(3); subplot(122); imagesc(real(sol)); axis image; axis off; colormap gray; title('Reconstructed image'); drawnow; sopt-2.0.0/matlab/Example_TVDNoA.m000066400000000000000000000032031277570055300166330ustar00rootroot00000000000000%% Example TVDNoA % Example to demonstrate TVoA_B2 solver, where an additional operator is % included in the TV norm of the TVDN problem. %% Clear workspace clc; clear; %% Define paths addpath misc/ addpath prox_operators/ %% Parameters N = 32; input_snr = 1; % Noise level (on the measurements) randn('seed', 1); %% Load an image im_ref = phantom(N); % figure(1); imagesc(im_ref); axis image; axis off; colormap gray; title('Original image'); drawnow; %% Create an artifial operator to test prox_TVoS S = randn(N^2, N^2)/N^2; im = reshape(S\im_ref(:), N, N); % S*im is thus sparse in TV %% Add Gaussian i.i.d. noise to im y = im; sigma_noise = 10^(-input_snr/20)*std(im_ref(:)); y = y + randn(size(y))*sigma_noise; %% Solving modified ROF % Parameters for TVDN param.verbose = 2; % Print log or not param.gamma = 1; % Converge parameter param.rel_obj = 1e-4; % Stopping criterion for the TVDN problem param.max_iter = 100; % Max. number of iterations for the TVDN problem param.max_iter_TV = 100; % Max. nb. of iter. for the sub-problem (proximal TV operator) param.nu_B2 = 1; % Bound on the norm of the measurement operator (Id here) param.tight_B2 = 1; % Indicate that A is a tight frame (1) or not (0) param.nu_TV = norm(S)^2; % Bound on the norm of the operator S func_S = @(x) reshape(S*x(:), N, N); func_St = @(x) reshape(S'*x(:), N, N); % Tolerance on noise epsilon = sqrt(chi2inv(0.99, numel(im)))*sigma_noise; % Identity A = @(x) x; % Solve sol = sopt_mltb_solve_TVDNoA(y, epsilon, A, A, func_S, func_St, param); % Show reconstructed image figure(2); imagesc(func_S(sol)); axis image; axis off; colormap gray; title('Reconstructed image'); drawnow; sopt-2.0.0/matlab/Example_weighted_BPDN.m000066400000000000000000000051271277570055300202120ustar00rootroot00000000000000%% Exampled_weighted_L1 % Example to demonstrate use of BPDN solver when incorporating weights % (performs one re-weighting of previous solution). %% Clear workspace clc; clear; %% Define paths addpath misc/ addpath prox_operators/ %% Parameters N = 64; input_snr = 30; % Noise level (on the measurements) randn('seed', 1); rand('seed', 1); %% Create an image with few spikes im = zeros(N); ind = randperm(N^2); im(ind(1:100)) = 1; % figure(1); subplot(221), imagesc(im); axis image; axis off; colormap gray; title('Original image'); drawnow; %% Create a mask % Mask mask = rand(size(im)) < 0.095; ind = find(mask==1); % Masking matrix (sparse matrix in matlab) Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im)); % Masking operator A = @(x) Ma*x(:); At = @(x) reshape(Ma'*x(:), size(im)); %% Reconstruct from a few Fourier measurements % Composition (Masking o Fourier) A = @(x) Ma*reshape(fft2(x)/sqrt(numel(im)), numel(x), 1); At = @(x) ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(im))); % Sparsity operator Psit = @(x) x; Psi = Psit; % Select 33% of Fourier coefficients y = A(im); % Add Gaussian i.i.d. noise sigma_noise = 10^(-input_snr/20)*std(im(:)); y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2); % Display the downsampled image figure(1); subplot(222); imagesc(real(At(y))); axis image; axis off; colormap gray; title('Measured image'); drawnow; % Tolerance on noise epsilon = sqrt(chi2inv(0.99, 2*numel(ind))/2)*sigma_noise; % Parameters for BPDN param.verbose = 1; % Print log or not param.gamma = 1e-1; % Converge parameter param.rel_obj = 1e-4; % Stopping criterion for the TVDN problem param.max_iter = 300; % Max. number of iterations for the TVDN problem param.nu_B2 = 1; % Bound on the norm of the operator A param.tol_B2 = 1e-4; % Tolerance for the projection onto the L2-ball param.tight_B2 = 1; % Indicate if A is a tight frame (1) or not (0) param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0) param.pos_l1 = 1; % % Solve BPDN problem (without weights) sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param); % Show reconstructed image figure(1); subplot(223); imagesc(real(sol)); axis image; axis off; colormap gray; title(['First estimate - ', ... num2str(sopt_mltb_SNR(im, real(sol))), 'dB']); drawnow; % Refine the estimate param.weights = 1./(abs(sol)+1e-5); sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param); % Show reconstructed image figure(1); subplot(224); imagesc(real(sol)); axis image; axis off; colormap gray; title(['Second estimate - ', ... num2str(sopt_mltb_SNR(im, real(sol))), 'dB']); drawnow; sopt-2.0.0/matlab/Example_weighted_TVDN.m000066400000000000000000000055341277570055300202440ustar00rootroot00000000000000%% Exampled_weighted_TVDN % Example to demonstrate use of TVDN solver when incorporating weights % (performs one re-weighting of previous solution). %% Clear workspace clc; clear; %% Define paths addpath misc/ addpath prox_operators/ %% Parameters N = 64; input_snr = 30; % Noise level (on the measurements) randn('seed', 1); rand('seed', 1); %% Load image im = phantom(N); % figure(1); clf; subplot(141), imagesc(im); axis image; axis off; colormap gray; title('Original image'); drawnow; %% Create a mask % Mask mask = rand(size(im)) < 0.33; ind = find(mask==1); % Masking matrix (sparse matrix in matlab) Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im)); %% Measure a few Fourier measurements % Composition (Masking o Fourier) A = @(x) Ma*reshape(fft2(x)/sqrt(numel(im)), numel(x), 1); At = @(x) ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(im))); % TV sparsity operator Psit = @(x) x; Psi = Psit; % Select 33% of Fourier coefficients y = A(im); % Add Gaussian i.i.d. noise sigma_noise = 10^(-input_snr/20)*std(im(:)); y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2); % Display the downsampled image figure(1); subplot(142); imagesc(real(At(y))); axis image; axis off; colormap gray; title('Measured image'); drawnow; %% Reconstruct with TV % Tolerance on noise epsilon = sqrt(chi2inv(0.99, 2*numel(ind))/2)*sigma_noise; % Parameters for TVDN param.verbose = 1; % Print log or not param.rel_obj = 1e-4; % Stopping criterion for the TVDN problem param.max_iter = 200; % Max. nb. of iterations for the TVDN problem param.gamma = 1e-1; % Converge parameter param.nu_B2 = 1; % Bound on the norm of the operator A param.tol_B2 = 1e-4; % Tolerance for the projection onto the L2-ball param.tight_B2 = 1; % Indicate if A is a tight frame (1) or not (0) param.max_iter_TV = 500; % param.zero_weights_flag_TV = 0; % param.identical_weights_flag_TV = 1; % % Solve TVDN problem (without weights) sol_1 = sopt_mltb_solve_TVDNoA(y, epsilon, A, At, Psi, Psit, param); % Show first reconstructed image figure(1); subplot(143); imagesc(real(sol_1)); axis image; axis off; colormap gray; title(['First estimate: ', ... num2str(sopt_mltb_SNR(im, real(sol_1))), 'dB']); drawnow; clc; %% Re-fine the estimate with weighted TV % Weights [param.weights_dx_TV param.weights_dy_TV] = sopt_mltb_gradient_op(real(sol_1)); param.weights_dx_TV = 1./(abs(param.weights_dx_TV)+1e-3); param.weights_dy_TV = 1./(abs(param.weights_dy_TV)+1e-3); param.identical_weights_flag_TV = 0; param.gamma = 1e-3; % First reconstruction with weights in the gradient param.zero_weights_flag_TV = 1; sol_2 = sopt_mltb_solve_TVDNoA(y, epsilon, A, At, Psi, Psit, param); % Show second reconstructed image figure(1); subplot(144); imagesc(real(sol_2)); axis image; axis off; colormap gray; title(['Second estimate: ', ... num2str(sopt_mltb_SNR(im, real(sol_2))), 'dB']); drawnow; sopt-2.0.0/matlab/Experiment1.m000066400000000000000000000173511277570055300163370ustar00rootroot00000000000000%% Experiment1 % In this experiment we evaluate the performance of SARA for spread % spectrum acquisition. We use a 256x256 version of Lena as a test image. % Number of measurements is M = 0.2N and input SNR is set to 30 dB. These % parameters can be changed by modifying the variables p (for the % undersampling ratio) and input_snr (for the input SNR). %% Clear workspace clc clear; %% Define paths addpath misc/ addpath prox_operators/ addpath test_images/ %% Read image imagename = 'lena_256.tiff'; % Load image im = im2double(imread(imagename)); % Normalise im = im/max(max(im)); % Enforce positivity im(im<0) = 0; %% Parameters input_snr = 30; % Noise level (on the measurements) %Undersampling ratio M/N p=0.2; %% Sparsity operators %Wavelet decomposition depth nlevel=4; dwtmode('per'); [C,S]=wavedec2(im,nlevel,'db8'); ncoef=length(C); [C1,S1]=wavedec2(im,nlevel,'db1'); ncoef1=length(C1); [C2,S2]=wavedec2(im,nlevel,'db2'); ncoef2=length(C2); [C3,S3]=wavedec2(im,nlevel,'db3'); ncoef3=length(C3); [C4,S4]=wavedec2(im,nlevel,'db4'); ncoef4=length(C4); [C5,S5]=wavedec2(im,nlevel,'db5'); ncoef5=length(C5); [C6,S6]=wavedec2(im,nlevel,'db6'); ncoef6=length(C6); [C7,S7]=wavedec2(im,nlevel,'db7'); ncoef7=length(C7); %SARA Psit = @(x) [wavedec2(x,nlevel,'db1')'; wavedec2(x,nlevel,'db2')';wavedec2(x,nlevel,'db3')';... wavedec2(x,nlevel,'db4')'; wavedec2(x,nlevel,'db5')'; wavedec2(x,nlevel,'db6')';... wavedec2(x,nlevel,'db7')';wavedec2(x,nlevel,'db8')']/sqrt(8); Psi = @(x) (waverec2(x(1:ncoef1),S1,'db1')+waverec2(x(ncoef1+1:ncoef1+ncoef2),S2,'db2')+... waverec2(x(ncoef1+ncoef2+1:ncoef1+ncoef2+ncoef3),S3,'db3')+... waverec2(x(ncoef1+ncoef2+ncoef3+1:ncoef1+ncoef2+ncoef3+ncoef4),S4,'db4')+... waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5),S5,'db5')+... waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6),S6,'db6')+... waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7),S7,'db7')+... waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+ncoef),S,'db8'))/sqrt(8); %Db8 wavelet basis Psit2 = @(x) wavedec2(x, nlevel,'db8'); Psi2 = @(x) waverec2(x,S,'db8'); %Curvelet %CurveLab needs to be installed to run Curvelet simulations realv = 1; Cv = fdct_usfft(im,realv); Mod = sopt_mltb_struct2size(Cv); Psit3 = @(x) sopt_mltb_fwdcurvelet(x,realv); Psi3 = @(x) sopt_mltb_adjcurvelet(x,Mod,realv); %% Spread spectrum operator % Mask mask = rand(size(im)) < p; ind = find(mask==1); % Masking matrix (sparse matrix in matlab) Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im)); %Spread spectrum sequence ss=rand(size(im)); C=(2*(ss<0.5)-1); A = @(x) Ma*reshape(fft2(C.*x)/sqrt(numel(ind)), numel(x), 1); At = @(x) C.*(ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(ind)))); % Sampling y = A(im); % Add Gaussian i.i.d. noise sigma_noise = 10^(-input_snr/20)*std(im(:)); y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2); % Tolerance on noise epsilon = sqrt(numel(y)+2*sqrt(numel(y)))*sigma_noise; epsilon_up = sqrt(numel(y)+2.1*sqrt(numel(y)))*sigma_noise; % Parameters for BPDN param.verbose = 1; % Print log or not param.gamma = 1e-1; % Converge parameter param.rel_obj = 5e-4; % Stopping criterion for the L1 problem param.max_iter = 200; % Max. number of iterations for the L1 problem param.nu_B2 = 1; % Bound on the norm of the operator A param.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0) param.pos_B2 = 1; %Positivity constraint: (1) active, (0) not active param.max_iter_B2=300; param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0) param.nu_L1 = 1; param.max_iter_L1 = 20; param.rel_obj_L1 = 1e-2; % Solve BPSA problem sol1 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param); RSNR1=20*log10(norm(im,'fro')/norm(im-sol1,'fro')); % SARA % It uses the solution to BPSA as a warm start maxiter=10; sigma=sigma_noise*sqrt(numel(y)/(numel(im)*8)); tol=1e-3; sol2 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi, Psit, param, sigma, tol, maxiter, sol1); RSNR2=20*log10(norm(im,'fro')/norm(im-sol2,'fro')); % Solve BPBb8 problem sol3 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi2, Psit2, param); RSNR3=20*log10(norm(im,'fro')/norm(im-sol3,'fro')); % RWBPDb8 % It uses the solution to BPDBb8 as a warm start maxiter=10; sigma=sigma_noise*sqrt(numel(y)/(numel(im))); tol=1e-3; sol4 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi2, Psit2, param, sigma, tol, maxiter, sol3); RSNR4=20*log10(norm(im,'fro')/norm(im-sol4,'fro')); % Parameters for Curvelet % Parameters for BPDN param3.verbose = 1; % Print log or not param3.gamma = 1e-1; % Converge parameter param3.rel_obj = 5e-4; % Stopping criterion for the L1 problem param3.max_iter = 200; % Max. number of iterations for the L1 problem param3.nu_B2 = 1; % Bound on the norm of the operator A param3.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball param3.tight_B2 = 1; % Indicate if A is a tight frame (1) or not (0) param3.pos_B2 = 1; % Positivity constraint flag. (1) active (0) otherwise param3.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0) % Solve BP Curvelet problem sol5 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi3, Psit3, param3); RSNR5=20*log10(norm(im,'fro')/norm(im-sol5,'fro')); % RW-Curvelet % It uses the solution to BPDBb8 as a warm start maxiter=10; sigma=sigma_noise*sqrt(numel(y)/(numel(im))); tol=1e-3; sol6 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi3, Psit3, param3, sigma, tol, maxiter, sol5); RSNR6=20*log10(norm(im,'fro')/norm(im-sol6,'fro')); % Parameters for TVDN param1.verbose = 1; % Print log or not param1.gamma = 1e-1; % Converge parameter param1.rel_obj = 5e-4; % Stopping criterion for the TVDN problem param1.max_iter = 200; % Max. number of iterations for the TVDN problem param1.max_iter_TV = 200; % Max. nb. of iter. for the sub-problem (proximal TV operator) param1.nu_B2 = 1; % Bound on the norm of the operator A param1.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball param1.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0) param1.max_iter_B2 = 300; param1.pos_B2 = 1; % Positivity constraint flag. (1) active (0) otherwise % Solve TV problem sol7 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param1); RSNR7=20*log10(norm(im,'fro')/norm(im-sol7,'fro')); % RWTV % It uses the solution to TV as a warm start maxiter=10; sigma=sigma_noise*sqrt(numel(y)/(numel(im))); tol=1e-3; sol8 = sopt_mltb_solve_rwTVDN(y, epsilon, A, At, param1,sigma, tol, maxiter, sol7); RSNR8=20*log10(norm(im,'fro')/norm(im-sol8,'fro')); %Show reconstructed images figure, imagesc(sol1,[0 1]); axis image; axis off; colormap gray; title(['BPSA, SNR=',num2str(RSNR1), 'dB']) figure, imagesc(sol2,[0 1]); axis image; axis off; colormap gray; title(['SARA, SNR=',num2str(RSNR2), 'dB']) figure, imagesc(sol3,[0 1]); axis image; axis off; colormap gray; title(['BPDb8, SNR=',num2str(RSNR3), 'dB']) figure, imagesc(sol4,[0 1]); axis image; axis off; colormap gray; title(['RW- BPDb8, SNR=',num2str(RSNR4), 'dB']) figure, imagesc(sol5,[0 1]); axis image; axis off; colormap gray; title(['Curvelet, SNR=',num2str(RSNR5), 'dB']) figure, imagesc(sol6,[0 1]); axis image; axis off; colormap gray; title(['RW-Curvelet, SNR=',num2str(RSNR6), 'dB']) figure, imagesc(sol7,[0 1]); axis image; axis off; colormap gray; title(['TV, SNR=',num2str(RSNR7), 'dB']) figure, imagesc(sol8,[0 1]); axis image; axis off; colormap gray; title(['RW-TV, SNR=',num2str(RSNR8), 'dB']) sopt-2.0.0/matlab/Experiment2.m000066400000000000000000000172431277570055300163400ustar00rootroot00000000000000%% Experiment2 % In this experiment we study the performance of SARA with Gaussian random % matrices as measurements operators. Due to computational limitations for % the use of a dense sensing matrix, for this experiment we use a cropped % version of Lena, around the head, of dimension 128x128 as a test image. % Number of measurements is M = 0.3N and input SNR is set to 30 dB. These % parameters can be changed by modifying the variables p (for the % undersampling ratio) and input_snr (for the input SNR). %% Clear workspace clc clear; %% Define paths addpath misc/ addpath prox_operators/ addpath test_images/ %% Read image imagename = 'lena_256.tiff'; % Load image im1 = im2double(imread(imagename)); %Crope image a=70; b=96; im=im1(a+1:a+128,b+1:b+128); % Enforce positivity im(im<0) = 0; %% Parameters input_snr = 30; % Noise level (on the measurements) %Undersampling ratio M/N p=0.3; %% Sparsity operators %Wavelet decomposition depth nlevel=4; dwtmode('per'); [C,S]=wavedec2(im,nlevel,'db8'); ncoef=length(C); [C1,S1]=wavedec2(im,nlevel,'db1'); ncoef1=length(C1); [C2,S2]=wavedec2(im,nlevel,'db2'); ncoef2=length(C2); [C3,S3]=wavedec2(im,nlevel,'db3'); ncoef3=length(C3); [C4,S4]=wavedec2(im,nlevel,'db4'); ncoef4=length(C4); [C5,S5]=wavedec2(im,nlevel,'db5'); ncoef5=length(C5); [C6,S6]=wavedec2(im,nlevel,'db6'); ncoef6=length(C6); [C7,S7]=wavedec2(im,nlevel,'db7'); ncoef7=length(C7); %Sparsity averaging operator Psit = @(x) [wavedec2(x,nlevel,'db1')'; wavedec2(x,nlevel,'db2')';wavedec2(x,nlevel,'db3')';... wavedec2(x,nlevel,'db4')'; wavedec2(x,nlevel,'db5')'; wavedec2(x,nlevel,'db6')';... wavedec2(x,nlevel,'db7')';wavedec2(x,nlevel,'db8')']/sqrt(8); Psi = @(x) (waverec2(x(1:ncoef1),S1,'db1')+waverec2(x(ncoef1+1:ncoef1+ncoef2),S2,'db2')+... waverec2(x(ncoef1+ncoef2+1:ncoef1+ncoef2+ncoef3),S3,'db3')+... waverec2(x(ncoef1+ncoef2+ncoef3+1:ncoef1+ncoef2+ncoef3+ncoef4),S4,'db4')+... waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5),S5,'db5')+... waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6),S6,'db6')+... waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7),S7,'db7')+... waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+ncoef),S,'db8'))/sqrt(8); %Db8 wavelet basis Psit2 = @(x) wavedec2(x, nlevel,'db8'); Psi2 = @(x) waverec2(x,S,'db8'); %Curvelet %CurveLab needs to be installed to run Curvelet simulations realv = 1; Cv = fdct_usfft(im,realv); Mod = sopt_mltb_struct2size(Cv); Psit3 = @(x) sopt_mltb_fwdcurvelet(x,realv); Psi3 = @(x) sopt_mltb_adjcurvelet(x,Mod,realv); %% Gaussian matrix operator num_meas=floor(p*numel(im)); Phi = randn(num_meas,numel(im))/sqrt(num_meas); bound=svds(Phi,1)^2; A = @(x) Phi*x(:); At = @(x) reshape(Phi'*x(:),size(im)); % Sampling y = A(im); % Add Gaussian i.i.d. noise sigma_noise = 10^(-input_snr/20)*std(im(:)); y = y + (randn(size(y)))*sigma_noise; % Tolerance on noise epsilon = sqrt(numel(y)+2*sqrt(numel(y)))*sigma_noise; epsilon_up = sqrt(numel(y)+2.1*sqrt(numel(y)))*sigma_noise; % Parameters for BPDN param.verbose = 1; % Print log or not param.gamma = 1e-1; % Converge parameter param.rel_obj = 5e-4; % Stopping criterion for the L1 problem param.max_iter = 200; % Max. number of iterations for the L1 problem param.nu_B2 = bound; % Bound on the norm of the operator A param.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0) param.pos_B2 = 1; %Positivity constraint: (1) active, (0) not active param.max_iter_B2=300; param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0) param.nu_L1 = 1; param.max_iter_L1 = 20; param.rel_obj_L1 = 1e-2; % Solve BPSA problem sol1 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param); RSNR1=20*log10(norm(im,'fro')/norm(im-sol1,'fro')); % SARA % It uses the solution to BPSA as a warm start maxiter=10; sigma=sigma_noise*sqrt(numel(y)/(numel(im)*8)); tol=1e-3; sol2 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi, Psit, param, sigma, tol, maxiter, sol1); RSNR2=20*log10(norm(im,'fro')/norm(im-sol2,'fro')); % Solve BPBb8 problem sol3 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi2, Psit2, param); RSNR3=20*log10(norm(im,'fro')/norm(im-sol3,'fro')); % RWBPDb8 % It uses the solution to BPDBb8 as a warm start maxiter=10; sigma=sigma_noise*sqrt(numel(y)/(numel(im))); tol=1e-3; sol4 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi2, Psit2, param, sigma, tol, maxiter, sol3); RSNR4=20*log10(norm(im,'fro')/norm(im-sol4,'fro')); % Parameters for Curvelet % Parameters for BPDN param3.verbose = 1; % Print log or not param3.gamma = 1e-1; % Converge parameter param3.rel_obj = 5e-4; % Stopping criterion for the L1 problem param3.max_iter = 200; % Max. number of iterations for the L1 problem param3.nu_B2 = bound; % Bound on the norm of the operator A param3.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball param3.tight_B2 = 1; % Indicate if A is a tight frame (1) or not (0) param3.pos_B2 = 1; % Positivity constraint flag. (1) active (0) otherwise param3.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0) % Solve BP Curvelet problem sol5 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi3, Psit3, param3); RSNR5=20*log10(norm(im,'fro')/norm(im-sol5,'fro')); % RW-Curvelet % It uses the solution to BPDBb8 as a warm start maxiter=10; sigma=sigma_noise*sqrt(numel(y)/(numel(im))); tol=1e-3; sol6 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi3, Psit3, param3, sigma, tol, maxiter, sol5); RSNR6=20*log10(norm(im,'fro')/norm(im-sol6,'fro')); % Parameters for TVDN param1.verbose = 1; % Print log or not param1.gamma = 1e-1; % Converge parameter param1.rel_obj = 5e-4; % Stopping criterion for the TVDN problem param1.max_iter = 200; % Max. number of iterations for the TVDN problem param1.max_iter_TV = 200; % Max. nb. of iter. for the sub-problem (proximal TV operator) param1.nu_B2 = bound; % Bound on the norm of the operator A param1.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball param1.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0) param1.max_iter_B2 = 300; param1.pos_B2 = 1; % Solve TV problem sol7 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param1); RSNR7=20*log10(norm(im,'fro')/norm(im-sol7,'fro')); % RWTV % It uses the solution to TV as a warm start maxiter=10; sigma=sigma_noise*sqrt(numel(y)/(numel(im))); tol=1e-3; sol8 = sopt_mltb_solve_rwTVDN(y, epsilon, A, At, param1,sigma, tol, maxiter, sol7); RSNR8=20*log10(norm(im,'fro')/norm(im-sol8,'fro')); %Show reconstructed images figure, imagesc(sol1,[0 1]); axis image; axis off; colormap gray; title(['BPSA, SNR=',num2str(RSNR1), 'dB']) figure, imagesc(sol2,[0 1]); axis image; axis off; colormap gray; title(['SARA, SNR=',num2str(RSNR2), 'dB']) figure, imagesc(sol3,[0 1]); axis image; axis off; colormap gray; title(['BPDb8, SNR=',num2str(RSNR3), 'dB']) figure, imagesc(sol4,[0 1]); axis image; axis off; colormap gray; title(['RW- BPDb8, SNR=',num2str(RSNR4), 'dB']) figure, imagesc(sol5,[0 1]); axis image; axis off; colormap gray; title(['Curvelet, SNR=',num2str(RSNR5), 'dB']) figure, imagesc(sol6,[0 1]); axis image; axis off; colormap gray; title(['RW-Curvelet, SNR=',num2str(RSNR6), 'dB']) figure, imagesc(sol7,[0 1]); axis image; axis off; colormap gray; title(['TV, SNR=',num2str(RSNR7), 'dB']) figure, imagesc(sol8,[0 1]); axis image; axis off; colormap gray; title(['RW-TV, SNR=',num2str(RSNR8), 'dB']) sopt-2.0.0/matlab/Experiment3.m000066400000000000000000000123541277570055300163370ustar00rootroot00000000000000%% Experiement3 % In this experiment we illustrate the application of SARA to Fourier % imaging. We reconstruct a 224x168 positive brain image from standard % variable density sampling. Fourier measurements, for an undersampling % ratio of M = 0.05N and input SNR set to 30 dB. %% Clear workspace clc clear; %% Define paths addpath misc/ addpath prox_operators/ addpath test_images/ %% Read image % Load image load Brain_low_res im=abs(map_ref); % Normalise im = im/max(max(im)); % Enforce positivity im(im<0) = 0; %% Parameters input_snr = 30; % Noise level (on the measurements) %Undersampling ratio M/N %The range of p is 0 < p <= 0.5 since the vdsmask %function samples only half plane. Therefore, full %sampling is p = 0.5 p=0.05; %% Sparsity operators %Wavelet decomposition depth nlevel=4; dwtmode('per'); [C,S]=wavedec2(im,nlevel,'db8'); ncoef=length(C); [C1,S1]=wavedec2(im,nlevel,'db1'); ncoef1=length(C1); [C2,S2]=wavedec2(im,nlevel,'db2'); ncoef2=length(C2); [C3,S3]=wavedec2(im,nlevel,'db3'); ncoef3=length(C3); [C4,S4]=wavedec2(im,nlevel,'db4'); ncoef4=length(C4); [C5,S5]=wavedec2(im,nlevel,'db5'); ncoef5=length(C5); [C6,S6]=wavedec2(im,nlevel,'db6'); ncoef6=length(C6); [C7,S7]=wavedec2(im,nlevel,'db7'); ncoef7=length(C7); %Sparsity averaging operator Psit = @(x) [wavedec2(x,nlevel,'db1')'; wavedec2(x,nlevel,'db2')';wavedec2(x,nlevel,'db3')';... wavedec2(x,nlevel,'db4')'; wavedec2(x,nlevel,'db5')'; wavedec2(x,nlevel,'db6')';... wavedec2(x,nlevel,'db7')';wavedec2(x,nlevel,'db8')']/sqrt(8); Psi = @(x) (waverec2(x(1:ncoef1),S1,'db1')+waverec2(x(ncoef1+1:ncoef1+ncoef2),S2,'db2')+... waverec2(x(ncoef1+ncoef2+1:ncoef1+ncoef2+ncoef3),S3,'db3')+... waverec2(x(ncoef1+ncoef2+ncoef3+1:ncoef1+ncoef2+ncoef3+ncoef4),S4,'db4')+... waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5),S5,'db5')+... waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6),S6,'db6')+... waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7),S7,'db7')+... waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+ncoef),S,'db8'))/sqrt(8); %Db8 wavelet basis Psit2 = @(x) wavedec2(x, nlevel,'db8'); Psi2 = @(x) waverec2(x,S,'db8'); % Variable density mask mask = sopt_mltb_vdsmask(size(im,1),size(im,2),p); ind = find(mask==1); % Masking matrix (sparse matrix in matlab) Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im)); % Composition (Masking o Fourier) A = @(x) Ma*reshape(fft2(x)/sqrt(numel(ind)), numel(x), 1); At = @(x) (ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(ind)))); % Select p% of Fourier coefficients y = A(im); % Add Gaussian i.i.d. noise sigma_noise = 10^(-input_snr/20)*std(im(:)); y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2); % Tolerance on noise epsilon = sqrt(numel(y)+2*sqrt(numel(y)))*sigma_noise; epsilon_up = sqrt(numel(y)+2.1*sqrt(numel(y)))*sigma_noise; %% Parameters for BPDN param.verbose = 1; % Print log or not param.gamma = 1e-1; % Converge parameter param.rel_obj = 4e-4; % Stopping criterion for the L1 problem param.max_iter = 200; % Max. number of iterations for the L1 problem param.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball param.nu_B2 = 1; % Bound on the norm of the operator A param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0) param.pos_B2 = 1; % Positivity constraint flag. (1) active (0) otherwise param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0) param.nu_L1 = 1; param.max_iter_L1 = 20; param.rel_obj_L1 = 1e-2; % Solve SARA maxiter=10; sigma=sigma_noise*sqrt(numel(y)/(numel(im)*9)); tol=1e-3; sol1 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi, Psit, param, sigma, tol, maxiter); RSNR1=20*log10(norm(im,'fro')/norm(im-sol1,'fro')); %% Parameters for TVDN param1.verbose = 1; % Print log or not param1.gamma = 3e-1; % Converge parameter param1.rel_obj = 5e-4; % Stopping criterion for the TVDN problem param1.max_iter = 200; % Max. number of iterations for the TVDN problem param1.max_iter_TV = 200; % Max. nb. of iter. for the sub-problem (proximal TV operator) param1.nu_B2 = 1; % Bound on the norm of the operator A param1.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball param1.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0) param1.max_iter_B2 = 200; param1.pos_B2 = 1; % Positivity constraint flag. (1) active (0) otherwise % Solve TVDN problem sol2 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param1); RSNR2=20*log10(norm(im,'fro')/norm(im-sol2,'fro')); % Solve BPDN problem sol3 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi2, Psit2, param); RSNR3=20*log10(norm(im,'fro')/norm(im-sol3,'fro')); figure, imagesc(sol1,[0 1]);axis image;axis off; colormap gray; title(['SARA, SNR=',num2str(RSNR1), 'dB']) figure, imagesc(sol2,[0 1]);axis image;axis off; colormap gray; title(['TV, SNR=',num2str(RSNR2), 'dB']) figure, imagesc(sol3,[0 1]);axis image;axis off; colormap gray; title(['BPDb8, SNR=',num2str(RSNR3), 'dB']) figure, imagesc(im, [0 1]); axis image; axis off; colormap gray; title('Original image') sopt-2.0.0/matlab/README.txt000066400000000000000000000054051277570055300154530ustar00rootroot00000000000000 SOPT: Sparse OPTimisation SARA algorithm for sparsity averaging ---------------------------------------------------------------- DESCRIPTION This is a Matlab implementation of the SARA algorithm presented in: [1] R. E. Carrillo, J. D. McEwen, D. Van De Ville, J.-P. Thiran, and Y. Wiaux. "Sparsity averaging for compressive imaging", IEEE Signal Processing Letters, Vol. 20, No. 6, pp. 591-594, 2013 (arXiv:1208.2330). AUTHORS R. E. Carillo (http://people.epfl.ch/rafael.carrillo) G. Puy (http://people.epfl.ch/gilles.puy) J. D. McEwen (http://www.jasonmcewen.org) Y. Wiaux (http://basp.eps.hw.ac.uk) EXPERIMENTS To test a reconstruction with spread spectrum measurements, run the script Experiment1.m. If you want to run the same experiment with the cameraman test image just change the variable image name to 'cameraman_256.tiff'. To test a reconstruction with random Gaussian measurements, run the script Experiment2.m. To run the MRI example in the paper, run the script Experiment3.m. REFERENCES When referencing this code, please cite our related paper: [1] R. E. Carrillo, J. D. McEwen, D. Van De Ville, J.-P. Thiran, and Y. Wiaux. "Sparsity averaging for compressive imaging", IEEE Signal Processing Letters, Vol. 20, No. 6, pp. 591-594, 2013 (arXiv:1208.2330). DOCUMENTATION See doc/index.html INSTALLATION To run the experiments the CurveLab toolbox (www.curvelet.org) must be installed. Otherwise the SOPT Matlab code should run as is, without any additional installation. DOWNLOAD https://github.com/basp-group/sopt SUPPORT If you have any questions or comments, feel free to contact Rafael Carrillo at: rafael {DOT} carrillo {AT} epfl {DOT} ch. NOTES The code is not optimized and is given for educational purpose. This is a first Matlab implementation of the code. A more general and optimised C code is found in the root folder. LICENSE SOPT: Sparse OPTimisation package Copyright (C) 2013 Rafael Carrillo, Gilles Puy, Jason McEwen, Yves Wiaux This program is free software; you can redistribute it and/or modify it under the terms of the GNU General Public License as published by the Free Software Foundation; either version 2 of the License, or (at your option) any later version. This program is distributed in the hope that it will be useful, but WITHOUT ANY WARRANTY; without even the implied warranty of MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE. See the GNU General Public License for more details (LICENSE.txt). 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\n"; } function matches_text($num) { if ($num==0) { return 'Sorry, no documents matching your query.'; } else if ($num==1) { return 'Found 1 document matching your query.'; } else // $num>1 { return 'Found '.$num.' documents matching your query. Showing best matches first.'; } } function main($idxfile) { if(strcmp('4.1.0', phpversion()) > 0) { die("Error: PHP version 4.1.0 or above required!"); } if (!($file=fopen($idxfile,"rb"))) { die("Error: Search index file could NOT be opened!"); } if (readHeader($file)!="DOXS") { die("Error: Header of index file is invalid!"); } $query=""; if (array_key_exists("query", $_GET)) { $query=$_GET["query"]; } $results = array(); $requiredWords = array(); $forbiddenWords = array(); $foundWords = array(); $word=strtolower(strtok($query," ")); while ($word) // for each word in the search query { if (($word{0}=='+')) { $word=substr($word,1); $requiredWords[]=$word; } if (($word{0}=='-')) { $word=substr($word,1); $forbiddenWords[]=$word; } if (!in_array($word,$foundWords)) { $foundWords[]=$word; search($file,$word,$results); } $word=strtolower(strtok(" ")); } $docs = array(); combine_results($results,$docs); // filter out documents with forbidden word or that do not contain // required words $filteredDocs = filter_results($docs,$requiredWords,$forbiddenWords); // normalize rankings so they are in the range [0-100] normalize_ranking($filteredDocs); // sort the results based on rank $sorted = array(); sort_results($filteredDocs,$sorted); // report results to the user report_results($sorted); fclose($file); } ?> sopt-2.0.0/matlab/doc/fortran.png000066400000000000000000000004111277570055300166730ustar00rootroot00000000000000PNG  IHDRPIDATxQ q*p&z{`mڴ.K5@[GD0 0nKB1\øNB*hm8Vb]<'!].=U1PB]S{BTZY@49 lJK*\li69g|jжe'Dd{סul -˸#IENDB`sopt-2.0.0/matlab/doc/hp.png000066400000000000000000000003771277570055300156420ustar00rootroot00000000000000PNG  IHDRPIDATx @Mh ɬbvWD?!-O9X!`Bhu'V"q~L!4Ηpmugpƙ<9 }G9 21Ke[̼ٶCSضfLz0m5RYj«aT'+Vosm?KcKՊven;e$&>!IENDB`sopt-2.0.0/matlab/doc/index.html000066400000000000000000000107651277570055300165240ustar00rootroot00000000000000 Matlab Index

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Matlab Files found in these Directories

Example_BPDN sopt_mltb_SNR sopt_mltb_gradient_op_sphere sopt_mltb_solve_TVDNoA
Example_TVDN sopt_mltb_TV_norm sopt_mltb_modifypdf sopt_mltb_solve_rwBPDN
Example_TVDN2 sopt_mltb_adjcurvelet sopt_mltb_proj_B2 sopt_mltb_solve_rwTVDN
Example_TVDNoA sopt_mltb_div_op sopt_mltb_prox_L1 sopt_mltb_struct2size
Example_weighted_BPDN sopt_mltb_div_op_sphere sopt_mltb_prox_TV sopt_mltb_vdsmask
Example_weighted_TVDN sopt_mltb_fast_proj_B2 sopt_mltb_prox_TVoA
Experiment1 sopt_mltb_fwdcurvelet sopt_mltb_solve_BPDN
Experiment2 sopt_mltb_genmask sopt_mltb_solve_L2DN
Experiment3 sopt_mltb_gradient_op sopt_mltb_solve_TVDN
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sopt-2.0.0/matlab/doc/left.png000066400000000000000000000002101277570055300161470ustar00rootroot00000000000000PNG  IHDR PLTEfs pHYsHHFk>(IDATxc```B5*0aH`ap``?P ^/IENDB`sopt-2.0.0/matlab/doc/linux.png000066400000000000000000000004201277570055300163570ustar00rootroot00000000000000PNG  IHDRPIDATxQ U3իԻ:B@SC%_JH@DN"1dtв,;5<*ǫ=O"D}v*l 99x4 UOTl< I\ᒧQ֗J=WmKZzM+a3#b%ݸt3ݙ 03,']~Yn'F|S2IENDB`sopt-2.0.0/matlab/doc/m2html.css000066400000000000000000000022441277570055300164350ustar00rootroot00000000000000body { background: white; color: black; font-family: arial,sans-serif; margin: 0; padding: 1ex; } div.fragment { width: 98%; border: 1px solid #CCCCCC; background-color: #f5f5f5; padding-left: 4px; margin: 4px; } div.box { width: 98%; background-color: #f5f5f5; border: 1px solid #CCCCCC; color: black; padding: 4px; } .comment { color: #228B22; } .string { color: #B20000; } .keyword { color: #0000FF; } .keywordtype { color: #604020; } .keywordflow { color: #e08000; } .preprocessor { color: #806020; } .stringliteral { color: #002080; } .charliteral { color: #008080; } a { text-decoration: none; } a:hover { background-color: #006699; color:#FFFFFF; } a.code { font-weight: normal; color: #A020F0; } a.code:hover { background-color: #FF0000; color: #FFFFFF; } h1 { background: transparent; color: #006699; font-size: x-large; text-align: center; } h2 { background: transparent; color: #006699; font-size: large; } address { font-size:small; } form.search { margin-bottom: 0px; margin-top: 0px; } input.search { font-size: 75%; color: #000080; font-weight: normal; background-color: #eeeeff; } li { padding-left:5px; }sopt-2.0.0/matlab/doc/matlab/000077500000000000000000000000001277570055300157565ustar00rootroot00000000000000sopt-2.0.0/matlab/doc/matlab/.DS_Store000066400000000000000000000140041277570055300174400ustar00rootroot00000000000000Bud1bwspblobmiscbwspblobbplist00  \WindowBounds[ShowSidebar]ShowStatusBar[ShowPathbar[ShowToolbar\SidebarWidth_{{1493, 398}, {1119, 778}}  ". Description of Example_BPDN
Home > matlab > Example_BPDN.m

Example_BPDN

PURPOSE ^

% Example_BPDN

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Example_BPDN
 Example to demonstrate use of BPDN solver.  A random Fourier sampling
 measurement operator is considered.  Daubechies 8 wavelets are used for
 the sparsifying operator.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Example_BPDN
0002 % Example to demonstrate use of BPDN solver.  A random Fourier sampling
0003 % measurement operator is considered.  Daubechies 8 wavelets are used for
0004 % the sparsifying operator.
0005 
0006 
0007 %% Clear workspace
0008 
0009 clc
0010 clear;
0011 
0012 
0013 %% Define paths
0014 
0015 addpath misc/
0016 addpath prox_operators/
0017 
0018 
0019 %% Define parameters
0020 
0021 % Coverages (half the plane for Fourier sampling)
0022 p = [0.50];
0023 
0024 % Noise level (on the measurements)
0025 input_snr = 30;
0026 
0027 
0028 %% Read image
0029 
0030 % Load image
0031 im = im2double(imread('cameraman.tif'));
0032 
0033 % Normalise
0034 im = im/max(max(im));
0035 
0036 % Enforce positivity
0037 im(im<0) = 0;
0038 
0039 
0040 %% Define sparsity operator
0041 
0042 dwtmode('per');
0043 
0044 % Decomposition level of the wavelet transform.
0045 nlevel = 4;
0046 
0047 % Daubechies 8 wavelet operator
0048 [C,S] = wavedec2(im,nlevel,'db8');
0049 
0050 Psit = @(x) wavedec2(x, nlevel, 'db8');
0051 Psi = @(x) waverec2(x, S, 'db8');
0052 
0053 
0054 %% Run simulations
0055 
0056 % Random Fourier sampling example
0057 %  Define mask
0058 %  Uniform sampling of the half Fourier plane
0059 mask = rand(size(im)) < p;
0060 mask(:,1:floor(size(im,2)/2))=0;
0061 mask = ifftshift(mask);
0062 mask(1,1)=0;
0063 mask(floor(size(im,1)/2):end,1)=0;
0064 
0065 ind = find(mask==1);
0066 Ma = sparse(1:numel(ind), ind, ...
0067   ones(numel(ind), 1), numel(ind), numel(im));
0068 
0069 % Composition (Masking o Fourier)
0070 A = @(x) Ma*reshape(fft2(x)/sqrt(numel(ind)), numel(x), 1);
0071 At = @(x) (ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(ind))));
0072 
0073 % Apply measurement operator
0074 y = A(im);
0075 
0076 % Add Gaussian i.i.d. noise
0077 sigma_noise = 10^(-input_snr/20)*std(im(:));
0078 noise = (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0079 y = y + noise;
0080 
0081 % Tolerance on noise
0082 epsilon = sqrt(numel(y) + 2*sqrt(numel(y)))*sigma_noise;
0083 epsilon_up = sqrt(numel(y) + 2.1*sqrt(numel(y)))*sigma_noise;
0084 tol_B2 = (epsilon_up/epsilon)-1; % Tolerance for the projection onto the L2-ball
0085 
0086 % Solve optimisation problem
0087 
0088 % Parameters for BPDN
0089 param.verbose = 1; % Print log or not
0090 param.gamma = 1e-1; % Convergence parameter
0091 param.rel_obj = 5e-4; % Stopping criterion for the L1 problem
0092 param.max_iter = 200; % Max. number of iterations for the L1 problem
0093 param.nu_B2 = 1; % Bound on the norm of the operator A
0094 param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0095 param.max_iter_B2 = 200; %Max. number of iterations of the L2-ball projection
0096 param.pos_B2 = 1; %Positivity flag
0097 param.tol_B2 = tol_B2; % Tolerance for the projection onto the L2-ball
0098 param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0099 %Optional parameters when Psit is not a tight frame:
0100 %param.nu_L1 = 1; % Bound on the norm of the operator Psit
0101 %param.max_iter_L1 = 200; %Max. number of iterations of the L1 prox
0102 %param.rel_obj_L1 = 1e-3; % Tolerance for the prox L1
0103 
0104 % Solve BPDN problem with positivity constraint
0105 sol1 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0106 
0107 % Compute SNR
0108 RSNR1 = 20*log10(norm(im,'fro') ...
0109   / norm(im-sol1,'fro'));
0110 
0111 % Example with only reality constraint
0112 param.pos_B2 = 0; %Positivity flag
0113 param.real_B2 = 1; %Reality flag
0114 
0115 % Solve BPDN problem
0116 sol2 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0117 
0118 % Compute SNR
0119 RSNR2 = 20*log10(norm(im,'fro') ...
0120   / norm(im-sol2,'fro'));
0121 
0122 
0123 %% Show results
0124 
0125 figure
0126 imagesc(sol1), axis off, axis image, colorbar
0127 title(['Rec. with positivity const. SNR=',num2str(RSNR1), 'dB'])
0128 colormap gray
0129 
0130 figure
0131 imagesc(sol2), axis off, axis image, colorbar
0132 title(['Rec. with reality const. SNR=',num2str(RSNR2), 'dB'])
0133 colormap gray
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/.svn/text-base/Example_TVDN.html.svn-base000066400000000000000000000167721277570055300254040ustar00rootroot00000000000000 Description of Example_TVDN
Home > matlab > Example_TVDN.m

Example_TVDN

PURPOSE ^

% Example_TVDN

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Example_TVDN
 Example to demonstrate use of TVDN solver.  A random Fourier sampling
 measurement operator is considered.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Example_TVDN
0002 % Example to demonstrate use of TVDN solver.  A random Fourier sampling
0003 % measurement operator is considered.
0004 
0005 
0006 %% Clear workspace
0007 
0008 clc
0009 clear;
0010 
0011 
0012 %% Define paths
0013 
0014 addpath misc/
0015 addpath prox_operators/
0016 
0017 
0018 %% Define parameters
0019 
0020 % Coverages (half the plane for Fourier sampling)
0021 p = [0.50];
0022 
0023 % Noise level (on the measurements)
0024 input_snr = 30;
0025 
0026 
0027 %% Read image
0028 
0029 % Load image
0030 im = im2double(imread('cameraman.tif'));
0031 
0032 % Normalise
0033 im = im/max(max(im));
0034 
0035 % Enforce positivity
0036 im(im<0) = 0;
0037 
0038 
0039 %% Run simulations
0040 
0041 %Random Fourier sampling example
0042 % Define mask
0043 % Uniform sampling of the half Fourier plane
0044 mask = rand(size(im)) < p;
0045 mask(:,1:floor(size(im,2)/2))=0;
0046 mask = ifftshift(mask);
0047 mask(1,1)=0;
0048 mask(floor(size(im,1)/2):end,1)=0;
0049 
0050 ind = find(mask==1);
0051 Ma = sparse(1:numel(ind), ind, ...
0052   ones(numel(ind), 1), numel(ind), numel(im));
0053 
0054 % Composition (Masking o Fourier)
0055 A = @(x) Ma*reshape(fft2(x)/sqrt(numel(ind)), numel(x), 1);
0056 At = @(x) (ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(ind))));
0057 
0058 % Apply measurement operator
0059 y = A(im);
0060 
0061 % Add Gaussian i.i.d. noise
0062 sigma_noise = 10^(-input_snr/20)*std(im(:));
0063 noise = (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0064 y = y + noise;
0065 
0066 % Tolerance on noise
0067 epsilon = sqrt(numel(y) + 2*sqrt(numel(y)))*sigma_noise;
0068 epsilon_up = sqrt(numel(y) + 2.1*sqrt(numel(y)))*sigma_noise;
0069 tol_B2 = (epsilon_up/epsilon)-1; % Tolerance for the projection onto the L2-ball
0070 
0071 % Solve optimisation problem
0072 
0073 % Parameters for TVDN
0074 param.verbose = 1; % Print log or not
0075 param.gamma = 1e-1; % Converge parameter
0076 param.rel_obj = 5e-4; % Stopping criterion for the TVDN problem
0077 param.max_iter = 200; % Max. number of iterations for the TVDN problem
0078 param_TV.max_iter_TV = 200; % Max. nb. of iter. for the sub-problem (proximal TV operator)
0079 param.nu_B2 = 1; % Bound on the norm of the operator A
0080 param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0081 param.max_iter_B2 = 200; %Max. number of iterations of the L2-ball projection
0082 param.pos_B2 = 1; %Positivity flag
0083 param.tol_B2 = tol_B2; % Tolerance for the projection onto the L2-ball
0084 
0085 % Solve TVDN problem with positivity constraint
0086 sol1 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param);
0087 
0088 % Compute SNR
0089 RSNR1 = 20*log10(norm(im,'fro') ...
0090   / norm(im-sol1,'fro'));
0091 
0092 % Example with only reality constraint
0093 param.pos_B2 = 0; %Positivity flag
0094 param.real_B2 = 1; %Reality flag
0095 
0096 % Solve TVDN problem
0097 sol2 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param);
0098 
0099 % Compute SNR
0100 RSNR2 = 20*log10(norm(im,'fro') ...
0101   / norm(im-sol2,'fro'));
0102 
0103 
0104 %% Show results
0105 
0106 figure
0107 imagesc(sol1), axis off, axis image, colorbar
0108 title(['Rec. with positivity const. SNR=',num2str(RSNR1), 'dB'])
0109 colormap gray
0110 
0111 figure
0112 imagesc(sol2), axis off, axis image, colorbar
0113 title(['Rec. with reality const. SNR=',num2str(RSNR2), 'dB'])
0114 colormap gray
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/.svn/text-base/Example_TVDN2.html.svn-base000066400000000000000000000162141277570055300254550ustar00rootroot00000000000000 Description of Example_TVDN2
Home > matlab > Example_TVDN2.m

Example_TVDN2

PURPOSE ^

% Example_TVDN2

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Example_TVDN2
 Two examples to demonstrate use of TVDN solver.  Simple inpainting and
 Fourier measurement examples are included.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Example_TVDN2
0002 % Two examples to demonstrate use of TVDN solver.  Simple inpainting and
0003 % Fourier measurement examples are included.
0004 
0005 
0006 %% Clear workspace
0007 clc;
0008 clear;
0009 
0010 %% Define paths
0011 addpath misc/
0012 addpath prox_operators/
0013 
0014 %% Parameters
0015 input_snr = 30; % Noise level (on the measurements)
0016 
0017 %% Load an image
0018 im = im2double(imread('cameraman.tif'));
0019 %
0020 figure(1);
0021 imagesc(im); axis image; axis off;
0022 colormap gray; title('Original image'); drawnow;
0023 
0024 %% Create a mask with 33% of 1 (the rest is set to 0)
0025 % Mask
0026 mask = rand(size(im)) < 0.33; ind = find(mask==1);
0027 % Masking matrix (sparse matrix in matlab)
0028 Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im));
0029 % Masking operator
0030 A = @(x) Ma*x(:); % Select 33% of the values in x;
0031 At = @(x) reshape(Ma'*x(:), size(im)); % Adjoint operator + reshape image
0032 
0033 %% First problem: Inpainting problem
0034 % Select 33% of pixels
0035 y = A(im);
0036 % Add Gaussian i.i.d. noise
0037 sigma_noise = 10^(-input_snr/20)*std(im(:));
0038 y = y + randn(size(y))*sigma_noise;
0039 % Display the downsampled image
0040 figure(2); clf;
0041 subplot(121); imagesc(At(y)); axis image; axis off;
0042 colormap gray; title('Measured image'); drawnow;
0043 % Parameters for TVDN
0044 param.verbose = 1; % Print log or not
0045 param.gamma = 1e-1; % Converge parameter
0046 param.rel_obj = 1e-4; % Stopping criterion for the TVDN problem
0047 param.max_iter = 200; % Max. number of iterations for the TVDN problem
0048 param_TV.max_iter_TV = 100; % Max. nb. of iter. for the sub-problem (proximal TV operator)
0049 param.nu_B2 = 1; % Bound on the norm of the operator A
0050 param.tol_B2 = 1e-4; % Tolerance for the projection onto the L2-ball
0051 param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0052 param.max_iter_B2 = 500;
0053 % Tolerance on noise
0054 epsilon = sqrt(chi2inv(0.99, numel(ind)))*sigma_noise;
0055 % Solve TVDN problem
0056 sol = sopt_mltb_solve_TVDN(y, epsilon, A, At, param);
0057 % Show reconstructed image
0058 figure(2);
0059 subplot(122); imagesc(sol); axis image; axis off;
0060 colormap gray; title('Reconstructed image'); drawnow;
0061 
0062 %% Second problem: Reconstruct from 33% of Fourier measurements
0063 % Composition (Masking o Fourier)
0064 A = @(x) Ma*reshape(fft2(x)/sqrt(numel(im)), numel(x), 1);
0065 At = @(x) ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(im)));
0066 % Select 33% of Fourier coefficients
0067 y = A(im);
0068 % Add Gaussian i.i.d. noise
0069 sigma_noise = 10^(-input_snr/20)*std(im(:));
0070 y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0071 % Display the downsampled image
0072 figure(3); clf;
0073 subplot(121); imagesc(real(At(y))); axis image; axis off;
0074 colormap gray; title('Measured image'); drawnow;
0075 % Tolerance on noise
0076 epsilon = sqrt(chi2inv(0.99, 2*numel(ind))/2)*sigma_noise;
0077 % Solve TVDN problem
0078 sol = sopt_mltb_solve_TVDN(y, epsilon, A, At, param);
0079 % Show reconstructed image
0080 figure(3);
0081 subplot(122); imagesc(real(sol)); axis image; axis off;
0082 colormap gray; title('Reconstructed image'); drawnow;
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/.svn/text-base/Example_TVDNoA.html.svn-base000066400000000000000000000125111277570055300256470ustar00rootroot00000000000000 Description of Example_TVDNoA
Home > matlab > Example_TVDNoA.m

Example_TVDNoA

PURPOSE ^

% Example TVDNoA

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Example TVDNoA
 Example to demonstrate TVoA_B2 solver, where an additional operator is
 included in the TV norm of the TVDN problem.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Example TVDNoA
0002 % Example to demonstrate TVoA_B2 solver, where an additional operator is
0003 % included in the TV norm of the TVDN problem.
0004 
0005 
0006 %% Clear workspace
0007 clc;
0008 clear;
0009 
0010 %% Define paths
0011 addpath misc/
0012 addpath prox_operators/
0013 
0014 %% Parameters
0015 N = 32;
0016 input_snr = 1; % Noise level (on the measurements)
0017 randn('seed', 1);
0018 
0019 %% Load an image
0020 im_ref = phantom(N);
0021 %
0022 figure(1);
0023 imagesc(im_ref); axis image; axis off;
0024 colormap gray; title('Original image'); drawnow;
0025 
0026 %% Create an artifial operator to test prox_TVoS
0027 S = randn(N^2, N^2)/N^2;
0028 im = reshape(S\im_ref(:), N, N); % S*im is thus sparse in TV
0029 
0030 %% Add Gaussian i.i.d. noise to im
0031 y = im;
0032 sigma_noise = 10^(-input_snr/20)*std(im_ref(:));
0033 y = y + randn(size(y))*sigma_noise;
0034 
0035 %% Solving modified ROF
0036 % Parameters for TVDN
0037 param.verbose = 2; % Print log or not
0038 param.gamma = 1; % Converge parameter
0039 param.rel_obj = 1e-4; % Stopping criterion for the TVDN problem
0040 param.max_iter = 100; % Max. number of iterations for the TVDN problem
0041 param.max_iter_TV = 100; % Max. nb. of iter. for the sub-problem (proximal TV operator)
0042 param.nu_B2 = 1; % Bound on the norm of the measurement operator (Id here)
0043 param.tight_B2 = 1; % Indicate that A is a tight frame (1) or not (0)
0044 param.nu_TV = norm(S)^2; % Bound on the norm of the operator S
0045 func_S = @(x) reshape(S*x(:), N, N);
0046 func_St = @(x) reshape(S'*x(:), N, N);
0047 % Tolerance on noise
0048 epsilon = sqrt(chi2inv(0.99, numel(im)))*sigma_noise;
0049 % Identity
0050 A = @(x) x;
0051 % Solve
0052 sol = sopt_mltb_solve_TVDNoA(y, epsilon, A, A, func_S, func_St, param);
0053 % Show reconstructed image
0054 figure(2);
0055 imagesc(func_S(sol)); axis image; axis off;
0056 colormap gray; title('Reconstructed image'); drawnow;
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/.svn/text-base/Example_weighted_BPDN.html.svn-base000066400000000000000000000160221277570055300272200ustar00rootroot00000000000000 Description of Example_weighted_BPDN
Home > matlab > Example_weighted_BPDN.m

Example_weighted_BPDN

PURPOSE ^

% Exampled_weighted_L1

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Exampled_weighted_L1
 Example to demonstrate use of BPDN solver when incorporating weights
 (performs one re-weighting of previous solution).

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Exampled_weighted_L1
0002 % Example to demonstrate use of BPDN solver when incorporating weights
0003 % (performs one re-weighting of previous solution).
0004 
0005 
0006 %% Clear workspace
0007 clc;
0008 clear;
0009 
0010 %% Define paths
0011 addpath misc/
0012 addpath prox_operators/
0013 
0014 %% Parameters
0015 N = 64;
0016 input_snr = 30; % Noise level (on the measurements)
0017 randn('seed', 1); rand('seed', 1);
0018 
0019 %% Create an image with few spikes
0020 im = zeros(N); ind = randperm(N^2); im(ind(1:100)) = 1;
0021 %
0022 figure(1);
0023 subplot(221), imagesc(im); axis image; axis off;
0024 colormap gray; title('Original image'); drawnow;
0025 
0026 %% Create a mask
0027 % Mask
0028 mask = rand(size(im)) < 0.095; ind = find(mask==1);
0029 % Masking matrix (sparse matrix in matlab)
0030 Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im));
0031 % Masking operator
0032 A = @(x) Ma*x(:);
0033 At = @(x) reshape(Ma'*x(:), size(im));
0034 
0035 %% Reconstruct from a few Fourier measurements
0036 
0037 % Composition (Masking o Fourier)
0038 A = @(x) Ma*reshape(fft2(x)/sqrt(numel(im)), numel(x), 1);
0039 At = @(x) ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(im)));
0040 
0041 % Sparsity operator
0042 Psit = @(x) x; Psi = Psit;
0043 
0044 % Select 33% of Fourier coefficients
0045 y = A(im);
0046 
0047 % Add Gaussian i.i.d. noise
0048 sigma_noise = 10^(-input_snr/20)*std(im(:));
0049 y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0050 
0051 % Display the downsampled image
0052 figure(1);
0053 subplot(222); imagesc(real(At(y))); axis image; axis off;
0054 colormap gray; title('Measured image'); drawnow;
0055 
0056 % Tolerance on noise
0057 epsilon = sqrt(chi2inv(0.99, 2*numel(ind))/2)*sigma_noise;
0058 
0059 % Parameters for BPDN
0060 param.verbose = 1; % Print log or not
0061 param.gamma = 1e-1; % Converge parameter
0062 param.rel_obj = 1e-4; % Stopping criterion for the TVDN problem
0063 param.max_iter = 300; % Max. number of iterations for the TVDN problem
0064 param.nu_B2 = 1; % Bound on the norm of the operator A
0065 param.tol_B2 = 1e-4; % Tolerance for the projection onto the L2-ball
0066 param.tight_B2 = 1; % Indicate if A is a tight frame (1) or not (0)
0067 param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0068 param.pos_l1 = 1; %
0069 
0070 % Solve BPDN problem (without weights)
0071 sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0072 
0073 % Show reconstructed image
0074 figure(1);
0075 subplot(223); imagesc(real(sol)); axis image; axis off;
0076 colormap gray; title(['First estimate - ', ...
0077     num2str(sopt_mltb_SNR(im, real(sol))), 'dB']); drawnow;
0078 
0079 % Refine the estimate
0080 param.weights = 1./(abs(sol)+1e-5);
0081 sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0082 
0083 % Show reconstructed image
0084 figure(1);
0085 subplot(224); imagesc(real(sol)); axis image; axis off;
0086 colormap gray; title(['Second estimate - ', ...
0087     num2str(sopt_mltb_SNR(im, real(sol))), 'dB']); drawnow;
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/.svn/text-base/Example_weighted_TVDN.html.svn-base000066400000000000000000000166741277570055300272650ustar00rootroot00000000000000 Description of Example_weighted_TVDN
Home > matlab > Example_weighted_TVDN.m

Example_weighted_TVDN

PURPOSE ^

% Exampled_weighted_TVDN

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Exampled_weighted_TVDN
 Example to demonstrate use of TVDN solver when incorporating weights
 (performs one re-weighting of previous solution).

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Exampled_weighted_TVDN
0002 % Example to demonstrate use of TVDN solver when incorporating weights
0003 % (performs one re-weighting of previous solution).
0004 
0005 
0006 %% Clear workspace
0007 clc;
0008 clear;
0009 
0010 %% Define paths
0011 addpath misc/
0012 addpath prox_operators/
0013 
0014 %% Parameters
0015 N = 64;
0016 input_snr = 30; % Noise level (on the measurements)
0017 randn('seed', 1); rand('seed', 1);
0018 
0019 %% Load image
0020 im = phantom(N);
0021 %
0022 figure(1); clf;
0023 subplot(141), imagesc(im); axis image; axis off;
0024 colormap gray; title('Original image'); drawnow;
0025 
0026 %% Create a mask
0027 % Mask
0028 mask = rand(size(im)) < 0.33; ind = find(mask==1);
0029 % Masking matrix (sparse matrix in matlab)
0030 Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im));
0031 
0032 %% Measure a few Fourier measurements
0033 
0034 % Composition (Masking o Fourier)
0035 A = @(x) Ma*reshape(fft2(x)/sqrt(numel(im)), numel(x), 1);
0036 At = @(x) ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(im)));
0037 
0038 % TV sparsity operator
0039 Psit = @(x) x; Psi = Psit;
0040 
0041 % Select 33% of Fourier coefficients
0042 y = A(im);
0043 
0044 % Add Gaussian i.i.d. noise
0045 sigma_noise = 10^(-input_snr/20)*std(im(:));
0046 y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0047 
0048 % Display the downsampled image
0049 figure(1);
0050 subplot(142); imagesc(real(At(y))); axis image; axis off;
0051 colormap gray; title('Measured image'); drawnow;
0052 
0053 %% Reconstruct with TV
0054 
0055 % Tolerance on noise
0056 epsilon = sqrt(chi2inv(0.99, 2*numel(ind))/2)*sigma_noise;
0057 
0058 % Parameters for TVDN
0059 param.verbose = 1; % Print log or not
0060 param.rel_obj = 1e-4; % Stopping criterion for the TVDN problem
0061 param.max_iter = 200; % Max. nb. of iterations for the TVDN problem
0062 param.gamma = 1e-1; % Converge parameter
0063 param.nu_B2 = 1; % Bound on the norm of the operator A
0064 param.tol_B2 = 1e-4; % Tolerance for the projection onto the L2-ball
0065 param.tight_B2 = 1; % Indicate if A is a tight frame (1) or not (0)
0066 param.max_iter_TV = 500; %
0067 param.zero_weights_flag_TV = 0; %
0068 param.identical_weights_flag_TV = 1; %
0069 
0070 % Solve TVDN problem (without weights)
0071 sol_1 = sopt_mltb_solve_TVDNoA(y, epsilon, A, At, Psi, Psit, param);
0072 
0073 % Show first reconstructed image
0074 figure(1);
0075 subplot(143); imagesc(real(sol_1)); axis image; axis off;
0076 colormap gray; 
0077 title(['First estimate: ', ...
0078     num2str(sopt_mltb_SNR(im, real(sol_1))), 'dB']);
0079 drawnow;
0080 clc;
0081 
0082 %% Re-fine the estimate with weighted TV
0083 % Weights
0084 [param.weights_dx_TV param.weights_dy_TV] = sopt_mltb_gradient_op(real(sol_1));
0085 param.weights_dx_TV = 1./(abs(param.weights_dx_TV)+1e-3);
0086 param.weights_dy_TV = 1./(abs(param.weights_dy_TV)+1e-3);
0087 param.identical_weights_flag_TV = 0;
0088 param.gamma = 1e-3;
0089 
0090 % First reconstruction with weights in the gradient
0091 param.zero_weights_flag_TV = 1;
0092 sol_2 = sopt_mltb_solve_TVDNoA(y, epsilon, A, At, Psi, Psit, param);
0093 % Show second reconstructed image
0094 figure(1);
0095 subplot(144); imagesc(real(sol_2)); axis image; axis off;
0096 colormap gray; 
0097 title(['Second estimate: ', ...
0098     num2str(sopt_mltb_SNR(im, real(sol_2))), 'dB']); 
0099 drawnow;
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/.svn/text-base/Experiment1.html.svn-base000066400000000000000000000426341277570055300253530ustar00rootroot00000000000000 Description of Experiment1
Home > matlab > Experiment1.m

Experiment1

PURPOSE ^

% Experiment1

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Experiment1
 In this experiment we evaluate the performance of SARA for spread 
 spectrum acquisition. We use a 256x256 version of Lena as a test image. 
 Number of measurements is M = 0.2N and input SNR is set to 30 dB. These
 parameters can be changed by modifying the variables p (for the
 undersampling ratio) and input_snr (for the input SNR).

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Experiment1
0002 % In this experiment we evaluate the performance of SARA for spread
0003 % spectrum acquisition. We use a 256x256 version of Lena as a test image.
0004 % Number of measurements is M = 0.2N and input SNR is set to 30 dB. These
0005 % parameters can be changed by modifying the variables p (for the
0006 % undersampling ratio) and input_snr (for the input SNR).
0007 
0008 
0009 %% Clear workspace
0010 
0011 clc
0012 clear;
0013 
0014 
0015 %% Define paths
0016 
0017 addpath misc/
0018 addpath prox_operators/
0019 addpath test_images/
0020 
0021 
0022 
0023 %% Read image
0024 
0025 imagename = 'lena_256.tiff';
0026 
0027 % Load image
0028 im = im2double(imread(imagename));
0029 
0030 % Normalise
0031 im = im/max(max(im));
0032 
0033 % Enforce positivity
0034 im(im<0) = 0;
0035 
0036 %% Parameters
0037 
0038 input_snr = 30; % Noise level (on the measurements)
0039 
0040 %Undersampling ratio M/N
0041 p=0.2;
0042 
0043 
0044 %% Sparsity operators
0045 
0046 %Wavelet decomposition depth
0047 nlevel=4;
0048 
0049 dwtmode('per');
0050 [C,S]=wavedec2(im,nlevel,'db8'); 
0051 ncoef=length(C);
0052 [C1,S1]=wavedec2(im,nlevel,'db1'); 
0053 ncoef1=length(C1);
0054 [C2,S2]=wavedec2(im,nlevel,'db2'); 
0055 ncoef2=length(C2);
0056 [C3,S3]=wavedec2(im,nlevel,'db3'); 
0057 ncoef3=length(C3);
0058 [C4,S4]=wavedec2(im,nlevel,'db4'); 
0059 ncoef4=length(C4);
0060 [C5,S5]=wavedec2(im,nlevel,'db5'); 
0061 ncoef5=length(C5);
0062 [C6,S6]=wavedec2(im,nlevel,'db6'); 
0063 ncoef6=length(C6);
0064 [C7,S7]=wavedec2(im,nlevel,'db7'); 
0065 ncoef7=length(C7);
0066 
0067 %SARA
0068 
0069 Psit = @(x) [wavedec2(x,nlevel,'db1')'; wavedec2(x,nlevel,'db2')';wavedec2(x,nlevel,'db3')';...
0070     wavedec2(x,nlevel,'db4')'; wavedec2(x,nlevel,'db5')'; wavedec2(x,nlevel,'db6')';...
0071     wavedec2(x,nlevel,'db7')';wavedec2(x,nlevel,'db8')']/sqrt(8); 
0072 
0073 Psi = @(x) (waverec2(x(1:ncoef1),S1,'db1')+waverec2(x(ncoef1+1:ncoef1+ncoef2),S2,'db2')+...
0074     waverec2(x(ncoef1+ncoef2+1:ncoef1+ncoef2+ncoef3),S3,'db3')+...
0075     waverec2(x(ncoef1+ncoef2+ncoef3+1:ncoef1+ncoef2+ncoef3+ncoef4),S4,'db4')+...
0076     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5),S5,'db5')+...
0077     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6),S6,'db6')+...
0078     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7),S7,'db7')+...
0079     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+ncoef),S,'db8'))/sqrt(8);
0080 
0081 %Db8 wavelet basis
0082 Psit2 = @(x) wavedec2(x, nlevel,'db8'); 
0083 Psi2 = @(x) waverec2(x,S,'db8');
0084 
0085 %Curvelet
0086 %CurveLab needs to be installed to run Curvelet simulations
0087 realv = 1;
0088 Cv = fdct_usfft(im,realv);
0089 Mod = sopt_mltb_struct2size(Cv);
0090 
0091 Psit3 = @(x) sopt_mltb_fwdcurvelet(x,realv); 
0092 Psi3 = @(x) sopt_mltb_adjcurvelet(x,Mod,realv);
0093 
0094 %% Spread spectrum operator
0095 % Mask
0096 mask = rand(size(im)) < p; 
0097 ind = find(mask==1);
0098 % Masking matrix (sparse matrix in matlab)
0099 Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im));
0100     
0101 %Spread spectrum sequence
0102     
0103 ss=rand(size(im));
0104 C=(2*(ss<0.5)-1);
0105 
0106 A = @(x) Ma*reshape(fft2(C.*x)/sqrt(numel(ind)), numel(x), 1);
0107 At = @(x) C.*(ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(ind))));
0108     
0109 % Sampling
0110 y = A(im);
0111 % Add Gaussian i.i.d. noise
0112 sigma_noise = 10^(-input_snr/20)*std(im(:));
0113 y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0114     
0115     
0116 % Tolerance on noise
0117 epsilon = sqrt(numel(y)+2*sqrt(numel(y)))*sigma_noise;
0118 epsilon_up = sqrt(numel(y)+2.1*sqrt(numel(y)))*sigma_noise;
0119     
0120     
0121 % Parameters for BPDN
0122 param.verbose = 1; % Print log or not
0123 param.gamma = 1e-1; % Converge parameter
0124 param.rel_obj = 5e-4; % Stopping criterion for the L1 problem
0125 param.max_iter = 200; % Max. number of iterations for the L1 problem
0126 param.nu_B2 = 1; % Bound on the norm of the operator A
0127 param.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0128 param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0129 param.pos_B2 = 1; %Positivity constraint: (1) active, (0) not active
0130 param.max_iter_B2=300;
0131 param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0132 param.nu_L1 = 1;
0133 param.max_iter_L1 = 20;
0134 param.rel_obj_L1 = 1e-2;
0135     
0136     
0137 % Solve BPSA problem
0138     
0139 sol1 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0140     
0141 RSNR1=20*log10(norm(im,'fro')/norm(im-sol1,'fro'));
0142     
0143 % SARA
0144 % It uses the solution to BPSA as a warm start
0145 maxiter=10;
0146 sigma=sigma_noise*sqrt(numel(y)/(numel(im)*8));
0147 tol=1e-3;
0148   
0149 sol2 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi, Psit, param, sigma, tol, maxiter, sol1);
0150 
0151 RSNR2=20*log10(norm(im,'fro')/norm(im-sol2,'fro'));
0152 
0153 % Solve BPBb8 problem
0154     
0155 sol3 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi2, Psit2, param);
0156     
0157 RSNR3=20*log10(norm(im,'fro')/norm(im-sol3,'fro'));
0158     
0159 % RWBPDb8
0160 % It uses the solution to BPDBb8 as a warm start
0161 maxiter=10;
0162 sigma=sigma_noise*sqrt(numel(y)/(numel(im)));
0163 tol=1e-3;
0164   
0165 sol4 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi2, Psit2, param, sigma, tol, maxiter, sol3);
0166       
0167 RSNR4=20*log10(norm(im,'fro')/norm(im-sol4,'fro'));
0168 
0169 % Parameters for Curvelet
0170 
0171 % Parameters for BPDN
0172 param3.verbose = 1; % Print log or not
0173 param3.gamma = 1e-1; % Converge parameter
0174 param3.rel_obj = 5e-4; % Stopping criterion for the L1 problem
0175 param3.max_iter = 200; % Max. number of iterations for the L1 problem
0176 param3.nu_B2 = 1; % Bound on the norm of the operator A
0177 param3.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0178 param3.tight_B2 = 1; % Indicate if A is a tight frame (1) or not (0)
0179 param3.pos_B2 = 1; % Positivity constraint flag. (1) active (0) otherwise
0180 param3.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0181 
0182     
0183 
0184 
0185 % Solve BP Curvelet problem
0186     
0187 sol5 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi3, Psit3, param3);
0188     
0189 RSNR5=20*log10(norm(im,'fro')/norm(im-sol5,'fro'));
0190     
0191 % RW-Curvelet
0192 % It uses the solution to BPDBb8 as a warm start
0193 maxiter=10;
0194 sigma=sigma_noise*sqrt(numel(y)/(numel(im)));
0195 tol=1e-3;
0196   
0197 sol6 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi3, Psit3, param3, sigma, tol, maxiter, sol5);
0198      
0199 RSNR6=20*log10(norm(im,'fro')/norm(im-sol6,'fro'));
0200 
0201     
0202 % Parameters for TVDN
0203 param1.verbose = 1; % Print log or not
0204 param1.gamma = 1e-1; % Converge parameter
0205 param1.rel_obj = 5e-4; % Stopping criterion for the TVDN problem
0206 param1.max_iter = 200; % Max. number of iterations for the TVDN problem
0207 param1.max_iter_TV = 200; % Max. nb. of iter. for the sub-problem (proximal TV operator)
0208 param1.nu_B2 = 1; % Bound on the norm of the operator A
0209 param1.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0210 param1.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0211 param1.max_iter_B2 = 300;
0212 param1.pos_B2 = 1; % Positivity constraint flag. (1) active (0) otherwise
0213     
0214 % Solve TV problem
0215     
0216 sol7 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param1);
0217     
0218 RSNR7=20*log10(norm(im,'fro')/norm(im-sol7,'fro'));
0219     
0220 % RWTV
0221 % It uses the solution to TV as a warm start
0222 maxiter=10;
0223 sigma=sigma_noise*sqrt(numel(y)/(numel(im)));
0224 tol=1e-3;
0225   
0226 sol8 = sopt_mltb_solve_rwTVDN(y, epsilon, A, At, param1,sigma, tol, maxiter, sol7);
0227     
0228 RSNR8=20*log10(norm(im,'fro')/norm(im-sol8,'fro'));
0229 
0230 
0231 %Show reconstructed images
0232 
0233 figure, imagesc(sol1,[0 1]); axis image; axis off; colormap gray;
0234 title(['BPSA, SNR=',num2str(RSNR1), 'dB'])
0235 figure, imagesc(sol2,[0 1]); axis image; axis off; colormap gray;
0236 title(['SARA, SNR=',num2str(RSNR2), 'dB'])
0237 
0238 figure, imagesc(sol3,[0 1]); axis image; axis off; colormap gray;
0239 title(['BPDb8, SNR=',num2str(RSNR3), 'dB'])
0240 figure, imagesc(sol4,[0 1]); axis image; axis off; colormap gray;
0241 title(['RW- BPDb8, SNR=',num2str(RSNR4), 'dB'])
0242 
0243 figure, imagesc(sol5,[0 1]); axis image; axis off; colormap gray;
0244 title(['Curvelet, SNR=',num2str(RSNR5), 'dB'])
0245 figure, imagesc(sol6,[0 1]); axis image; axis off; colormap gray;
0246 title(['RW-Curvelet, SNR=',num2str(RSNR6), 'dB'])
0247 
0248 figure, imagesc(sol7,[0 1]); axis image; axis off; colormap gray;
0249 title(['TV, SNR=',num2str(RSNR7), 'dB'])
0250 figure, imagesc(sol8,[0 1]); axis image; axis off; colormap gray;
0251 title(['RW-TV, SNR=',num2str(RSNR8), 'dB'])
0252 
0253 
0254 
0255 
0256 
0257 
0258 
0259 
0260 
0261 
0262
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/.svn/text-base/Experiment2.html.svn-base000066400000000000000000000426531277570055300253550ustar00rootroot00000000000000 Description of Experiment2
Home > matlab > Experiment2.m

Experiment2

PURPOSE ^

% Experiment2

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Experiment2
 In this experiment we study the performance of SARA with Gaussian random 
 matrices as measurements operators. Due to computational limitations for 
 the use of a dense sensing matrix, for this experiment we use a cropped 
 version of Lena, around the head, of dimension 128x128 as a test image. 
 Number of measurements is M = 0.3N and input SNR is set to 30 dB. These
 parameters can be changed by modifying the variables p (for the
 undersampling ratio) and input_snr (for the input SNR).

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Experiment2
0002 % In this experiment we study the performance of SARA with Gaussian random
0003 % matrices as measurements operators. Due to computational limitations for
0004 % the use of a dense sensing matrix, for this experiment we use a cropped
0005 % version of Lena, around the head, of dimension 128x128 as a test image.
0006 % Number of measurements is M = 0.3N and input SNR is set to 30 dB. These
0007 % parameters can be changed by modifying the variables p (for the
0008 % undersampling ratio) and input_snr (for the input SNR).
0009 
0010 
0011 %% Clear workspace
0012 
0013 clc
0014 clear;
0015 
0016 
0017 %% Define paths
0018 
0019 addpath misc/
0020 addpath prox_operators/
0021 addpath test_images/
0022 
0023 
0024 %% Read image
0025 
0026 imagename = 'lena_256.tiff';
0027 
0028 % Load image
0029 im1 = im2double(imread(imagename));
0030 
0031 %Crope image
0032 a=70;
0033 b=96;
0034 im=im1(a+1:a+128,b+1:b+128);
0035 
0036 % Enforce positivity
0037 im(im<0) = 0;
0038 
0039 %% Parameters
0040 
0041 input_snr = 30; % Noise level (on the measurements)
0042 
0043 %Undersampling ratio M/N
0044 p=0.3;
0045 
0046 %% Sparsity operators
0047 
0048 %Wavelet decomposition depth
0049 nlevel=4;
0050 
0051 dwtmode('per');
0052 [C,S]=wavedec2(im,nlevel,'db8'); 
0053 ncoef=length(C);
0054 [C1,S1]=wavedec2(im,nlevel,'db1'); 
0055 ncoef1=length(C1);
0056 [C2,S2]=wavedec2(im,nlevel,'db2'); 
0057 ncoef2=length(C2);
0058 [C3,S3]=wavedec2(im,nlevel,'db3'); 
0059 ncoef3=length(C3);
0060 [C4,S4]=wavedec2(im,nlevel,'db4'); 
0061 ncoef4=length(C4);
0062 [C5,S5]=wavedec2(im,nlevel,'db5'); 
0063 ncoef5=length(C5);
0064 [C6,S6]=wavedec2(im,nlevel,'db6'); 
0065 ncoef6=length(C6);
0066 [C7,S7]=wavedec2(im,nlevel,'db7'); 
0067 ncoef7=length(C7);
0068 
0069 %Sparsity averaging operator
0070 
0071 Psit = @(x) [wavedec2(x,nlevel,'db1')'; wavedec2(x,nlevel,'db2')';wavedec2(x,nlevel,'db3')';...
0072     wavedec2(x,nlevel,'db4')'; wavedec2(x,nlevel,'db5')'; wavedec2(x,nlevel,'db6')';...
0073     wavedec2(x,nlevel,'db7')';wavedec2(x,nlevel,'db8')']/sqrt(8); 
0074 
0075 Psi = @(x) (waverec2(x(1:ncoef1),S1,'db1')+waverec2(x(ncoef1+1:ncoef1+ncoef2),S2,'db2')+...
0076     waverec2(x(ncoef1+ncoef2+1:ncoef1+ncoef2+ncoef3),S3,'db3')+...
0077     waverec2(x(ncoef1+ncoef2+ncoef3+1:ncoef1+ncoef2+ncoef3+ncoef4),S4,'db4')+...
0078     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5),S5,'db5')+...
0079     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6),S6,'db6')+...
0080     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7),S7,'db7')+...
0081     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+ncoef),S,'db8'))/sqrt(8);
0082 
0083 %Db8 wavelet basis
0084 Psit2 = @(x) wavedec2(x, nlevel,'db8'); 
0085 Psi2 = @(x) waverec2(x,S,'db8');
0086 
0087 %Curvelet
0088 %CurveLab needs to be installed to run Curvelet simulations
0089 realv = 1;
0090 Cv = fdct_usfft(im,realv);
0091 Mod = sopt_mltb_struct2size(Cv);
0092 
0093 Psit3 = @(x) sopt_mltb_fwdcurvelet(x,realv); 
0094 Psi3 = @(x) sopt_mltb_adjcurvelet(x,Mod,realv);
0095 
0096 %% Gaussian matrix operator
0097 
0098 num_meas=floor(p*numel(im));
0099 
0100 Phi = randn(num_meas,numel(im))/sqrt(num_meas);
0101         
0102 bound=svds(Phi,1)^2;
0103      
0104 A = @(x) Phi*x(:);
0105     
0106 At = @(x) reshape(Phi'*x(:),size(im));
0107     
0108    
0109 % Sampling
0110 y = A(im);
0111 % Add Gaussian i.i.d. noise
0112 sigma_noise = 10^(-input_snr/20)*std(im(:));
0113 y = y + (randn(size(y)))*sigma_noise;
0114     
0115     
0116 % Tolerance on noise
0117 epsilon = sqrt(numel(y)+2*sqrt(numel(y)))*sigma_noise;
0118 epsilon_up = sqrt(numel(y)+2.1*sqrt(numel(y)))*sigma_noise;
0119     
0120     
0121 % Parameters for BPDN
0122 param.verbose = 1; % Print log or not
0123 param.gamma = 1e-1; % Converge parameter
0124 param.rel_obj = 5e-4; % Stopping criterion for the L1 problem
0125 param.max_iter = 200; % Max. number of iterations for the L1 problem
0126 param.nu_B2 = bound; % Bound on the norm of the operator A
0127 param.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0128 param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0129 param.pos_B2 = 1; %Positivity constraint: (1) active, (0) not active
0130 param.max_iter_B2=300;
0131 param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0132 param.nu_L1 = 1;
0133 param.max_iter_L1 = 20;
0134 param.rel_obj_L1 = 1e-2;
0135     
0136     
0137 % Solve BPSA problem
0138     
0139 sol1 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0140     
0141 RSNR1=20*log10(norm(im,'fro')/norm(im-sol1,'fro'));
0142     
0143 % SARA
0144 % It uses the solution to BPSA as a warm start
0145 maxiter=10;
0146 sigma=sigma_noise*sqrt(numel(y)/(numel(im)*8));
0147 tol=1e-3;
0148   
0149 sol2 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi, Psit, param, sigma, tol, maxiter, sol1);
0150 
0151 RSNR2=20*log10(norm(im,'fro')/norm(im-sol2,'fro'));
0152 
0153 % Solve BPBb8 problem
0154     
0155 sol3 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi2, Psit2, param);
0156     
0157 RSNR3=20*log10(norm(im,'fro')/norm(im-sol3,'fro'));
0158     
0159 % RWBPDb8
0160 % It uses the solution to BPDBb8 as a warm start
0161 maxiter=10;
0162 sigma=sigma_noise*sqrt(numel(y)/(numel(im)));
0163 tol=1e-3;
0164   
0165 sol4 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi2, Psit2, param, sigma, tol, maxiter, sol3);
0166       
0167 RSNR4=20*log10(norm(im,'fro')/norm(im-sol4,'fro'));
0168 
0169 % Parameters for Curvelet
0170 
0171 % Parameters for BPDN
0172 param3.verbose = 1; % Print log or not
0173 param3.gamma = 1e-1; % Converge parameter
0174 param3.rel_obj = 5e-4; % Stopping criterion for the L1 problem
0175 param3.max_iter = 200; % Max. number of iterations for the L1 problem
0176 param3.nu_B2 = bound; % Bound on the norm of the operator A
0177 param3.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0178 param3.tight_B2 = 1; % Indicate if A is a tight frame (1) or not (0)
0179 param3.pos_B2 = 1; % Positivity constraint flag. (1) active (0) otherwise
0180 param3.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0181 
0182     
0183 
0184 
0185 % Solve BP Curvelet problem
0186     
0187 sol5 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi3, Psit3, param3);
0188     
0189 RSNR5=20*log10(norm(im,'fro')/norm(im-sol5,'fro'));
0190     
0191 % RW-Curvelet
0192 % It uses the solution to BPDBb8 as a warm start
0193 maxiter=10;
0194 sigma=sigma_noise*sqrt(numel(y)/(numel(im)));
0195 tol=1e-3;
0196   
0197 sol6 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi3, Psit3, param3, sigma, tol, maxiter, sol5);
0198      
0199 RSNR6=20*log10(norm(im,'fro')/norm(im-sol6,'fro'));
0200 
0201     
0202 % Parameters for TVDN
0203 param1.verbose = 1; % Print log or not
0204 param1.gamma = 1e-1; % Converge parameter
0205 param1.rel_obj = 5e-4; % Stopping criterion for the TVDN problem
0206 param1.max_iter = 200; % Max. number of iterations for the TVDN problem
0207 param1.max_iter_TV = 200; % Max. nb. of iter. for the sub-problem (proximal TV operator)
0208 param1.nu_B2 = bound; % Bound on the norm of the operator A
0209 param1.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0210 param1.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0211 param1.max_iter_B2 = 300;
0212 param1.pos_B2 = 1;
0213     
0214 % Solve TV problem
0215     
0216 sol7 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param1);
0217     
0218 RSNR7=20*log10(norm(im,'fro')/norm(im-sol7,'fro'));
0219     
0220 % RWTV
0221 % It uses the solution to TV as a warm start
0222 maxiter=10;
0223 sigma=sigma_noise*sqrt(numel(y)/(numel(im)));
0224 tol=1e-3;
0225   
0226 sol8 = sopt_mltb_solve_rwTVDN(y, epsilon, A, At, param1,sigma, tol, maxiter, sol7);
0227     
0228 RSNR8=20*log10(norm(im,'fro')/norm(im-sol8,'fro'));
0229 
0230 %Show reconstructed images
0231 
0232 figure, imagesc(sol1,[0 1]); axis image; axis off; colormap gray;
0233 title(['BPSA, SNR=',num2str(RSNR1), 'dB'])
0234 figure, imagesc(sol2,[0 1]); axis image; axis off; colormap gray;
0235 title(['SARA, SNR=',num2str(RSNR2), 'dB'])
0236 
0237 figure, imagesc(sol3,[0 1]); axis image; axis off; colormap gray;
0238 title(['BPDb8, SNR=',num2str(RSNR3), 'dB'])
0239 figure, imagesc(sol4,[0 1]); axis image; axis off; colormap gray;
0240 title(['RW- BPDb8, SNR=',num2str(RSNR4), 'dB'])
0241 
0242 figure, imagesc(sol5,[0 1]); axis image; axis off; colormap gray;
0243 title(['Curvelet, SNR=',num2str(RSNR5), 'dB'])
0244 figure, imagesc(sol6,[0 1]); axis image; axis off; colormap gray;
0245 title(['RW-Curvelet, SNR=',num2str(RSNR6), 'dB'])
0246 
0247 figure, imagesc(sol7,[0 1]); axis image; axis off; colormap gray;
0248 title(['TV, SNR=',num2str(RSNR7), 'dB'])
0249 figure, imagesc(sol8,[0 1]); axis image; axis off; colormap gray;
0250 title(['RW-TV, SNR=',num2str(RSNR8), 'dB'])
0251 
0252 
0253 
0254 
0255 
0256 
0257 
0258 
0259 
0260 
0261
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/.svn/text-base/Experiment3.html.svn-base000066400000000000000000000304561277570055300253540ustar00rootroot00000000000000 Description of Experiment3
Home > matlab > Experiment3.m

Experiment3

PURPOSE ^

% Experiement3

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Experiement3
 In this experiment we illustrate the application of SARA to Fourier 
 imaging. We reconstruct a 224x168 positive brain image from standard 
 variable density sampling. Fourier measurements, for an undersampling 
 ratio of M = 0.05N and input SNR set to 30 dB.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Experiement3
0002 % In this experiment we illustrate the application of SARA to Fourier
0003 % imaging. We reconstruct a 224x168 positive brain image from standard
0004 % variable density sampling. Fourier measurements, for an undersampling
0005 % ratio of M = 0.05N and input SNR set to 30 dB.
0006 
0007 
0008 %% Clear workspace
0009 
0010 clc
0011 clear;
0012 
0013 
0014 %% Define paths
0015 
0016 addpath misc/
0017 addpath prox_operators/
0018 addpath test_images/
0019 
0020 
0021 %% Read image
0022 
0023 % Load image
0024 load Brain_low_res
0025 im=abs(map_ref);
0026 
0027 % Normalise
0028 im = im/max(max(im));
0029 
0030 % Enforce positivity
0031 im(im<0) = 0;
0032 
0033 %% Parameters
0034 
0035 input_snr = 30; % Noise level (on the measurements)
0036 
0037 %Undersampling ratio M/N
0038 %The range of p is 0 < p <= 0.5 since the vdsmask
0039 %function samples only half plane. Therefore, full
0040 %sampling is p = 0.5
0041 p=0.05;
0042 
0043 
0044 %% Sparsity operators
0045 
0046 %Wavelet decomposition depth
0047 nlevel=4;
0048 
0049 dwtmode('per');
0050 [C,S]=wavedec2(im,nlevel,'db8'); 
0051 ncoef=length(C);
0052 [C1,S1]=wavedec2(im,nlevel,'db1'); 
0053 ncoef1=length(C1);
0054 [C2,S2]=wavedec2(im,nlevel,'db2'); 
0055 ncoef2=length(C2);
0056 [C3,S3]=wavedec2(im,nlevel,'db3'); 
0057 ncoef3=length(C3);
0058 [C4,S4]=wavedec2(im,nlevel,'db4'); 
0059 ncoef4=length(C4);
0060 [C5,S5]=wavedec2(im,nlevel,'db5'); 
0061 ncoef5=length(C5);
0062 [C6,S6]=wavedec2(im,nlevel,'db6'); 
0063 ncoef6=length(C6);
0064 [C7,S7]=wavedec2(im,nlevel,'db7'); 
0065 ncoef7=length(C7);
0066 
0067 %Sparsity averaging operator
0068 
0069 Psit = @(x) [wavedec2(x,nlevel,'db1')'; wavedec2(x,nlevel,'db2')';wavedec2(x,nlevel,'db3')';...
0070     wavedec2(x,nlevel,'db4')'; wavedec2(x,nlevel,'db5')'; wavedec2(x,nlevel,'db6')';...
0071     wavedec2(x,nlevel,'db7')';wavedec2(x,nlevel,'db8')']/sqrt(8); 
0072 
0073 Psi = @(x) (waverec2(x(1:ncoef1),S1,'db1')+waverec2(x(ncoef1+1:ncoef1+ncoef2),S2,'db2')+...
0074     waverec2(x(ncoef1+ncoef2+1:ncoef1+ncoef2+ncoef3),S3,'db3')+...
0075     waverec2(x(ncoef1+ncoef2+ncoef3+1:ncoef1+ncoef2+ncoef3+ncoef4),S4,'db4')+...
0076     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5),S5,'db5')+...
0077     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6),S6,'db6')+...
0078     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7),S7,'db7')+...
0079     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+ncoef),S,'db8'))/sqrt(8);
0080 
0081 %Db8 wavelet basis
0082 Psit2 = @(x) wavedec2(x, nlevel,'db8'); 
0083 Psi2 = @(x) waverec2(x,S,'db8');
0084 
0085 
0086 % Variable density mask
0087 mask = sopt_mltb_vdsmask(size(im,1),size(im,2),p);
0088 ind = find(mask==1);
0089 % Masking matrix (sparse matrix in matlab)
0090 Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im));
0091 
0092   
0093 % Composition (Masking o Fourier)
0094     
0095 A = @(x) Ma*reshape(fft2(x)/sqrt(numel(ind)), numel(x), 1);
0096 At = @(x) (ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(ind))));
0097       
0098 % Select p% of Fourier coefficients
0099 y = A(im);
0100 % Add Gaussian i.i.d. noise
0101 sigma_noise = 10^(-input_snr/20)*std(im(:));
0102 y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0103 
0104 
0105 % Tolerance on noise
0106 epsilon = sqrt(numel(y)+2*sqrt(numel(y)))*sigma_noise;
0107 epsilon_up = sqrt(numel(y)+2.1*sqrt(numel(y)))*sigma_noise;
0108 
0109 
0110 %% Parameters for BPDN
0111 param.verbose = 1; % Print log or not
0112 param.gamma = 1e-1; % Converge parameter
0113 param.rel_obj = 4e-4; % Stopping criterion for the L1 problem
0114 param.max_iter = 200; % Max. number of iterations for the L1 problem
0115 param.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0116 param.nu_B2 = 1; % Bound on the norm of the operator A
0117 param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0118 param.pos_B2 = 1;  % Positivity constraint flag. (1) active (0) otherwise
0119 param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0120 param.nu_L1 = 1;
0121 param.max_iter_L1 = 20;
0122 param.rel_obj_L1 = 1e-2;
0123     
0124 % Solve SARA
0125 maxiter=10;
0126 sigma=sigma_noise*sqrt(numel(y)/(numel(im)*9));
0127 tol=1e-3;
0128 sol1 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi, Psit, param, sigma, tol, maxiter);
0129     
0130 RSNR1=20*log10(norm(im,'fro')/norm(im-sol1,'fro'));
0131     
0132     
0133 %% Parameters for TVDN
0134 param1.verbose = 1; % Print log or not
0135 param1.gamma = 3e-1; % Converge parameter
0136 param1.rel_obj = 5e-4; % Stopping criterion for the TVDN problem
0137 param1.max_iter = 200; % Max. number of iterations for the TVDN problem
0138 param1.max_iter_TV = 200; % Max. nb. of iter. for the sub-problem (proximal TV operator)
0139 param1.nu_B2 = 1; % Bound on the norm of the operator A
0140 param1.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0141 param1.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0142 param1.max_iter_B2 = 200;
0143 param1.pos_B2 = 1; % Positivity constraint flag. (1) active (0) otherwise
0144 
0145 % Solve TVDN problem
0146 sol2 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param1);
0147 
0148 RSNR2=20*log10(norm(im,'fro')/norm(im-sol2,'fro'));
0149    
0150     
0151 % Solve BPDN problem
0152 sol3 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi2, Psit2, param);
0153 
0154 
0155 RSNR3=20*log10(norm(im,'fro')/norm(im-sol3,'fro'));
0156 
0157 figure, imagesc(sol1,[0 1]);axis image;axis off; colormap gray;
0158 title(['SARA, SNR=',num2str(RSNR1), 'dB'])
0159 figure, imagesc(sol2,[0 1]);axis image;axis off; colormap gray;
0160 title(['TV, SNR=',num2str(RSNR2), 'dB'])
0161 figure, imagesc(sol3,[0 1]);axis image;axis off; colormap gray;
0162 title(['BPDb8, SNR=',num2str(RSNR3), 'dB'])
0163 figure, imagesc(im, [0 1]); axis image; axis off; colormap gray;
0164 title('Original image')
0165     
0166     
0167     
0168     
0169     
0170     
0171     
0172 
0173 
0174 
0175 
0176 
0177 
0178 
0179 
0180 
0181 
0182 
0183
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/.svn/text-base/index.html.svn-base000066400000000000000000000073541277570055300242610ustar00rootroot00000000000000 Index for Directory matlab
< Master index Index for matlab >

Index for matlab

Matlab files in this directory:

 Example_BPDN% Example_BPDN
 Example_TVDN% Example_TVDN
 Example_TVDN2% Example_TVDN2
 Example_TVDNoA% Example TVDNoA
 Example_weighted_BPDN% Exampled_weighted_L1
 Example_weighted_TVDN% Exampled_weighted_TVDN
 Experiment1% Experiment1
 Experiment2% Experiment2
 Experiment3% Experiement3
 sopt_mltb_solve_BPDNsopt_mltb_solve_BPDN - Solve BPDN problem
 sopt_mltb_solve_L2DNsopt_mltb_solve_L2DN - Solve L2DN problem.
 sopt_mltb_solve_TVDNsopt_mltb_solve_TVDN - Solve TVDN problem
 sopt_mltb_solve_TVDNoAsopt_mltb_solve_TVDNoA - Solve augmented TVDN problem
 sopt_mltb_solve_rwBPDNsopt_mltb_solve_rwBPDN - Solve reweighted BPDN problem
 sopt_mltb_solve_rwTVDNsopt_mltb_solve_rwTVDN - Solve reweighted TVDN problem

Subsequent directories:

Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/.svn/text-base/sopt_mltb_solve_BPDN.html.svn-base000066400000000000000000000411461277570055300271650ustar00rootroot00000000000000 Description of sopt_mltb_solve_BPDN
Home > matlab > sopt_mltb_solve_BPDN.m

sopt_mltb_solve_BPDN

PURPOSE ^

sopt_mltb_solve_BPDN - Solve BPDN problem

SYNOPSIS ^

function sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param)

DESCRIPTION ^

 sopt_mltb_solve_BPDN - Solve BPDN problem

 Solve the Basis Pursuit Denoising (BPDN) problem

   min ||Psit x||_1   s.t.  ||y-A x||_2 < epsilon

 where y contains the measurements, A is the forward measurement operator 
 and At the associated adjoint operator, Psit is a sparfying transform and
 Psi its adjoint. The structure param should contain the following fields:

   General parameters:
 
   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - max_iter: Maximum number of iterations (default = 200).

   - rel_obj: Minimum relative change of the objective value 
       (default = 1e-4).  The algorithm stops if
           | ||x(t)||_1 - ||x(t-1)||_1 | / ||x(t)||_1 < rel_obj,
       where x(t) is the estimate of the solution at iteration t.

   - gamma: Convergence speed (weighting of L1 norm when solving for
       L1 proximal operator) (default = 1e-1).

   - initsol: Initial solution for a warmstart.
 
   Projection onto the L2-ball:

   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
 
   - nu_B2: Bound on the norm of the operator A, i.e.
       ||A x||^2 <= nu * ||x||^2 (default = 1).

   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
       stops if
         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
       (default = 1e-3).

   - max_iter_B2: Maximum number of iterations for the projection onto the
       L2 ball (default = 200).

   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
       default = 0).

   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
       default = 0).
 
   Proximal L1 operator:

   - rel_obj_L1: Used as stopping criterion for the proximal L1
       operator. Minimum relative change of the objective value between
       two successive estimates.

   - max_iter_L1: Used as stopping criterion for the proximal L1
       operator. Maximun number of iterations.
 
   - param.nu_L1: Bound on the norm^2 of the operator Psi, i.e.
       ||Psi x||^2 <= nu * ||x||^2 (default = 1).
 
   - param.tight_L1: 1 if Psit is a tight frame, 0 otherwise 
       (default = 1).
 
   - param.weights: Weights for a weighted L1-norm defined
       by sum_i{weights_i.*abs(x_i)} (default = 1). 

   - pos_l1: Positivity flag (1 to impose positivity, 0 otherwise;
       default = 0).

 References:
 [1] P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting 
 Approach to Nonsmooth Convex Variational Signal Recovery", IEEE Journal
 of Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param)
0002 % sopt_mltb_solve_BPDN - Solve BPDN problem
0003 %
0004 % Solve the Basis Pursuit Denoising (BPDN) problem
0005 %
0006 %   min ||Psit x||_1   s.t.  ||y-A x||_2 < epsilon
0007 %
0008 % where y contains the measurements, A is the forward measurement operator
0009 % and At the associated adjoint operator, Psit is a sparfying transform and
0010 % Psi its adjoint. The structure param should contain the following fields:
0011 %
0012 %   General parameters:
0013 %
0014 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0015 %       2 = print main steps; default = 1).
0016 %
0017 %   - max_iter: Maximum number of iterations (default = 200).
0018 %
0019 %   - rel_obj: Minimum relative change of the objective value
0020 %       (default = 1e-4).  The algorithm stops if
0021 %           | ||x(t)||_1 - ||x(t-1)||_1 | / ||x(t)||_1 < rel_obj,
0022 %       where x(t) is the estimate of the solution at iteration t.
0023 %
0024 %   - gamma: Convergence speed (weighting of L1 norm when solving for
0025 %       L1 proximal operator) (default = 1e-1).
0026 %
0027 %   - initsol: Initial solution for a warmstart.
0028 %
0029 %   Projection onto the L2-ball:
0030 %
0031 %   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
0032 %
0033 %   - nu_B2: Bound on the norm of the operator A, i.e.
0034 %       ||A x||^2 <= nu * ||x||^2 (default = 1).
0035 %
0036 %   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
0037 %       stops if
0038 %         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
0039 %       (default = 1e-3).
0040 %
0041 %   - max_iter_B2: Maximum number of iterations for the projection onto the
0042 %       L2 ball (default = 200).
0043 %
0044 %   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
0045 %       default = 0).
0046 %
0047 %   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
0048 %       default = 0).
0049 %
0050 %   Proximal L1 operator:
0051 %
0052 %   - rel_obj_L1: Used as stopping criterion for the proximal L1
0053 %       operator. Minimum relative change of the objective value between
0054 %       two successive estimates.
0055 %
0056 %   - max_iter_L1: Used as stopping criterion for the proximal L1
0057 %       operator. Maximun number of iterations.
0058 %
0059 %   - param.nu_L1: Bound on the norm^2 of the operator Psi, i.e.
0060 %       ||Psi x||^2 <= nu * ||x||^2 (default = 1).
0061 %
0062 %   - param.tight_L1: 1 if Psit is a tight frame, 0 otherwise
0063 %       (default = 1).
0064 %
0065 %   - param.weights: Weights for a weighted L1-norm defined
0066 %       by sum_i{weights_i.*abs(x_i)} (default = 1).
0067 %
0068 %   - pos_l1: Positivity flag (1 to impose positivity, 0 otherwise;
0069 %       default = 0).
0070 %
0071 % References:
0072 % [1] P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting
0073 % Approach to Nonsmooth Convex Variational Signal Recovery", IEEE Journal
0074 % of Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.
0075 
0076 % Optional input arguments
0077 if ~isfield(param, 'verbose'), param.verbose = 1; end
0078 if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end
0079 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0080 if ~isfield(param, 'gamma'), param.gamma = 1e-2; end
0081 if ~isfield(param, 'pos_l1'), param.pos_l1 = 0; end
0082 
0083 % Input arguments for projection onto the L2 ball
0084 param_B2.A = A; param_B2.At = At;
0085 param_B2.y = y; param_B2.epsilon = epsilon;
0086 param_B2.verbose = param.verbose;
0087 if isfield(param, 'nu_B2'), param_B2.nu = param.nu_B2; end
0088 if isfield(param, 'tol_B2'), param_B2.tol = param.tol_B2; end
0089 if isfield(param, 'tight_B2'), param_B2.tight = param.tight_B2; end
0090 if isfield(param, 'max_iter_B2')
0091     param_B2.max_iter = param.max_iter_B2;
0092 end
0093 if isfield(param,'pos_B2'), param_B2.pos=param.pos_B2; end
0094 if isfield(param,'real_B2'), param_B2.real=param.real_B2; end
0095 
0096 % Input arguments for prox L1
0097 param_L1.Psi = Psi; param_L1.Psit = Psit; param_L1.pos = param.pos_l1;
0098 param_L1.verbose = param.verbose; 
0099 %param_L1.verbose = 2;
0100 param_L1.rel_obj = param.rel_obj;
0101 if isfield(param, 'nu_L1')
0102     param_L1.nu = param.nu_L1;
0103 end
0104 if isfield(param, 'tight_L1')
0105     param_L1.tight = param.tight_L1;
0106 end
0107 if isfield(param, 'max_iter_L1')
0108     param_L1.max_iter = param.max_iter_L1;
0109 end
0110 if isfield(param, 'rel_obj_L1')
0111     param_L1.rel_obj = param.rel_obj_L1;
0112 end
0113 if isfield(param, 'weights')
0114     param_L1.weights = param.weights;
0115 else
0116     param_L1.weights = 1;
0117 end
0118 
0119 % Initialization
0120 if isfield(param,'initsol')
0121     xhat = param.initsol;
0122 else
0123     xhat = At(y); 
0124 end
0125 
0126 iter = 1; prev_norm = 0;
0127 
0128 % Main loop
0129 while 1
0130     
0131     if param.verbose >= 1
0132         fprintf('Iteration %i:\n', iter);
0133     end
0134     
0135     % Projection onto the L2-ball
0136     [sol, param_B2.u] = sopt_mltb_fast_proj_B2(xhat, param_B2);
0137     
0138     % Global stopping criterion
0139     dummy = Psit(sol);
0140     curr_norm = sum(param_L1.weights(:).*abs(dummy(:)));    
0141     rel_norm = abs(curr_norm - prev_norm)/curr_norm;
0142     if param.verbose >= 1
0143         fprintf('  ||x||_1 = %e, rel_norm = %e\n', ...
0144             curr_norm, rel_norm);
0145     end
0146     if (rel_norm < param.rel_obj)
0147         crit_BPDN = 'REL_NORM';
0148         break;
0149     elseif iter >= param.max_iter
0150         crit_BPDN = 'MAX_IT';
0151         break;
0152     end
0153     
0154     % Proximal L1 operator
0155     xhat = 2*sol - xhat;
0156     temp = sopt_mltb_prox_L1(xhat, param.gamma, param_L1);
0157     xhat = temp + sol - xhat;
0158     
0159     % Update variables
0160     iter = iter + 1;
0161     prev_norm = curr_norm;
0162     
0163 end
0164 
0165 % Log
0166 if param.verbose >= 1
0167   
0168     % L1 norm
0169     fprintf('\n Solution found:\n');
0170     fprintf(' Final L1 norm: %e\n', curr_norm);
0171     
0172     % Residual
0173     dummy = A(sol); res = norm(y(:)-dummy(:), 2);
0174     fprintf(' epsilon = %e, ||y-Ax||_2=%e\n', epsilon, res);
0175     
0176     % Stopping criterion
0177     fprintf(' %i iterations\n', iter);
0178     fprintf(' Stopping criterion: %s \n\n', crit_BPDN);
0179     
0180 end
0181 
0182 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/.svn/text-base/sopt_mltb_solve_L2DN.html.svn-base000066400000000000000000000314111277570055300271330ustar00rootroot00000000000000 Description of sopt_mltb_solve_L2DN
Home > matlab > sopt_mltb_solve_L2DN.m

sopt_mltb_solve_L2DN

PURPOSE ^

sopt_mltb_solve_L2DN - Solve L2DN problem.

SYNOPSIS ^

function sol = sopt_mltb_solve_L2DN(y, epsilon, A, At, param)

DESCRIPTION ^

 sopt_mltb_solve_L2DN - Solve L2DN problem.

 Solve the L2 denoising problem

   min ||x||_2   s.t.  ||y-A x||_2 < epsilon

 where y contains the measurements, A is the forward measurement operator 
 and At the associated adjoint operator. The structure param should 
 contain the following fields:

   General parameters:
 
   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - max_iter: Maximum number of iterations (default = 200).

   - rel_obj: Minimum relative change of the objective value 
       (default = 1e-4).  The algorithm stops if
           | ||x(t)||_2 - ||x(t-1)||_2 | / ||x(t)||_2 < rel_obj,
       where x(t) is the estimate of the solution at iteration t.

   - gamma: Convergence speed (weighting of L1 norm when solving for
       L1 proximal operator) (default = 1e-1).

   - param.weights: weightsfor a weighted L2-norm defined
       by norm(weights_i.*x_i,2) (default = 1).

   - initsol: Initial solution for a warmstart.

   Projection onto the L2-ball:

   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
 
   - nu_B2: Bound on the norm of the operator A, i.e.
       ||A x||^2 <= nu * ||x||^2 (default = 1).

   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
       stops if
         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
       (default = 1e-3).

   - max_iter_B2: Maximum number of iterations for the projection onto the
       L2 ball (default = 200).

   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
       default = 0).

   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
       default = 0).

 References:
 [1] P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting 
 Approach to Nonsmooth Convex Variational Signal Recovery", IEEE Journal
 of Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_solve_L2DN(y, epsilon, A, At, param)
0002 % sopt_mltb_solve_L2DN - Solve L2DN problem.
0003 %
0004 % Solve the L2 denoising problem
0005 %
0006 %   min ||x||_2   s.t.  ||y-A x||_2 < epsilon
0007 %
0008 % where y contains the measurements, A is the forward measurement operator
0009 % and At the associated adjoint operator. The structure param should
0010 % contain the following fields:
0011 %
0012 %   General parameters:
0013 %
0014 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0015 %       2 = print main steps; default = 1).
0016 %
0017 %   - max_iter: Maximum number of iterations (default = 200).
0018 %
0019 %   - rel_obj: Minimum relative change of the objective value
0020 %       (default = 1e-4).  The algorithm stops if
0021 %           | ||x(t)||_2 - ||x(t-1)||_2 | / ||x(t)||_2 < rel_obj,
0022 %       where x(t) is the estimate of the solution at iteration t.
0023 %
0024 %   - gamma: Convergence speed (weighting of L1 norm when solving for
0025 %       L1 proximal operator) (default = 1e-1).
0026 %
0027 %   - param.weights: weightsfor a weighted L2-norm defined
0028 %       by norm(weights_i.*x_i,2) (default = 1).
0029 %
0030 %   - initsol: Initial solution for a warmstart.
0031 %
0032 %   Projection onto the L2-ball:
0033 %
0034 %   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
0035 %
0036 %   - nu_B2: Bound on the norm of the operator A, i.e.
0037 %       ||A x||^2 <= nu * ||x||^2 (default = 1).
0038 %
0039 %   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
0040 %       stops if
0041 %         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
0042 %       (default = 1e-3).
0043 %
0044 %   - max_iter_B2: Maximum number of iterations for the projection onto the
0045 %       L2 ball (default = 200).
0046 %
0047 %   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
0048 %       default = 0).
0049 %
0050 %   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
0051 %       default = 0).
0052 %
0053 % References:
0054 % [1] P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting
0055 % Approach to Nonsmooth Convex Variational Signal Recovery", IEEE Journal
0056 % of Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.
0057 
0058 % Optional input arguments
0059 if ~isfield(param, 'verbose'), param.verbose = 1; end
0060 if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end
0061 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0062 if ~isfield(param, 'gamma'), param.gamma = 1e-2; end
0063 if ~isfield(param, 'weights'), param.weights = 1; end
0064 
0065 % Input arguments for projection onto the L2 ball
0066 param_B2.A = A; param_B2.At = At;
0067 param_B2.y = y; param_B2.epsilon = epsilon;
0068 param_B2.verbose = param.verbose;
0069 if isfield(param, 'nu_B2'), param_B2.nu = param.nu_B2; end
0070 if isfield(param, 'tol_B2'), param_B2.tol = param.tol_B2; end
0071 if isfield(param, 'tight_B2'), param_B2.tight = param.tight_B2; end
0072 if isfield(param, 'max_iter_B2')
0073     param_B2.max_iter = param.max_iter_B2;
0074 end
0075 if isfield(param,'pos_B2'), param_B2.pos=param.pos_B2; end
0076 if isfield(param,'real_B2'), param_B2.real=param.real_B2; end
0077 
0078 % Initialization
0079 if isfield(param,'initsol')
0080     xhat = param.initsol;
0081 else
0082     xhat = At(y); 
0083 end
0084 
0085 iter = 1; prev_norm = 0;
0086 
0087 % Main loop
0088 while 1
0089     
0090     %
0091     if param.verbose >= 1
0092         fprintf('Iteration %i:\n', iter);
0093     end
0094     
0095     % Projection onto the L2-ball
0096     [sol, param_B2.u] = sopt_mltb_fast_proj_B2(xhat, param_B2);
0097     
0098     % Global stopping criterion
0099     dummy = sol;
0100     curr_norm = norm(param.weights(:).*dummy(:));    
0101     rel_norm = abs(curr_norm - prev_norm)/curr_norm;
0102     if param.verbose >= 1
0103         fprintf('  ||x||_1 = %e, rel_norm = %e\n', ...
0104             curr_norm, rel_norm);
0105     end
0106     if (rel_norm < param.rel_obj)
0107         crit_BPDN = 'REL_NORM';
0108         break;
0109     elseif iter >= param.max_iter
0110         crit_BPDN = 'MAX_IT';
0111         break;
0112     end
0113     
0114     % Proximal L2 operator
0115     xhat = 2*sol - xhat;   
0116     temp = xhat ./ (1 + 2*param.weights);    
0117     xhat = temp + sol - xhat;
0118     
0119     % Update variables
0120     iter = iter + 1;
0121     prev_norm = curr_norm;
0122     
0123 end
0124 
0125 % Log
0126 if param.verbose >= 1
0127   
0128     % L1 norm
0129     fprintf('\n Solution found:\n');
0130     fprintf(' Final L1 norm: %e\n', curr_norm);
0131     
0132     % Residual
0133     dummy = A(sol); res = norm(y(:)-dummy(:), 2);
0134     fprintf(' epsilon = %e, ||y-Ax||_2=%e\n', epsilon, res);
0135     
0136     % Stopping criterion
0137     fprintf(' %i iterations\n', iter);
0138     fprintf(' Stopping criterion: %s \n\n', crit_BPDN);
0139     
0140 end
0141 
0142 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/.svn/text-base/sopt_mltb_solve_TVDN.html.svn-base000066400000000000000000000330451277570055300272140ustar00rootroot00000000000000 Description of sopt_mltb_solve_TVDN
Home > matlab > sopt_mltb_solve_TVDN.m

sopt_mltb_solve_TVDN

PURPOSE ^

sopt_mltb_solve_TVDN - Solve TVDN problem

SYNOPSIS ^

function sol = sopt_mltb_solve_TVDN(y, epsilon, A, At, param)

DESCRIPTION ^

 sopt_mltb_solve_TVDN - Solve TVDN problem

 Solve the total variation denoising (TVDN) problem

   min ||x||_TV   s.t.  ||y - A x||_2 < epsilon

 where y contains the measurements, A is the forward measurement operator 
 and At the associated adjoint operator. The structure param should
 contain the following fields:

   General parameters:
 
   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - max_iter: Maximum number of iterations (default = 200).

   - rel_obj: Minimum relative change of the objective value 
       (default = 1e-4).  The algorithm stops if
           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj,
       where x(t) is the estimate of the solution at iteration t.

   - gamma: Convergence speed (weighting of TV norm when solving for
       TV proximal operator) (default = 1e-1).

   - initsol: Initial solution for a warmstart.
 
   Projection onto the L2-ball:

   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
 
   - nu_B2: Bound on the norm of the operator A, i.e.
       ||A x||^2 <= nu * ||x||^2 (default = 1).

   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
       stops if
         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
       (default = 1e-3).

   - max_iter_B2: Maximum number of iterations for the projection onto the
       L2 ball (default = 200).

   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
       default = 0).

   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
       default = 0).
 
   Proximal L1 operator:

   - max_iter_TV: Used as stopping criterion for the proximal TV
       operator. Maximun number of iterations (default = 200).

 References:
 P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting Approach
 to Nonsmooth Convex Variational Signal Recovery", IEEE Journal of
 Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_solve_TVDN(y, epsilon, A, At, param)
0002 % sopt_mltb_solve_TVDN - Solve TVDN problem
0003 %
0004 % Solve the total variation denoising (TVDN) problem
0005 %
0006 %   min ||x||_TV   s.t.  ||y - A x||_2 < epsilon
0007 %
0008 % where y contains the measurements, A is the forward measurement operator
0009 % and At the associated adjoint operator. The structure param should
0010 % contain the following fields:
0011 %
0012 %   General parameters:
0013 %
0014 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0015 %       2 = print main steps; default = 1).
0016 %
0017 %   - max_iter: Maximum number of iterations (default = 200).
0018 %
0019 %   - rel_obj: Minimum relative change of the objective value
0020 %       (default = 1e-4).  The algorithm stops if
0021 %           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj,
0022 %       where x(t) is the estimate of the solution at iteration t.
0023 %
0024 %   - gamma: Convergence speed (weighting of TV norm when solving for
0025 %       TV proximal operator) (default = 1e-1).
0026 %
0027 %   - initsol: Initial solution for a warmstart.
0028 %
0029 %   Projection onto the L2-ball:
0030 %
0031 %   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
0032 %
0033 %   - nu_B2: Bound on the norm of the operator A, i.e.
0034 %       ||A x||^2 <= nu * ||x||^2 (default = 1).
0035 %
0036 %   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
0037 %       stops if
0038 %         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
0039 %       (default = 1e-3).
0040 %
0041 %   - max_iter_B2: Maximum number of iterations for the projection onto the
0042 %       L2 ball (default = 200).
0043 %
0044 %   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
0045 %       default = 0).
0046 %
0047 %   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
0048 %       default = 0).
0049 %
0050 %   Proximal L1 operator:
0051 %
0052 %   - max_iter_TV: Used as stopping criterion for the proximal TV
0053 %       operator. Maximun number of iterations (default = 200).
0054 %
0055 % References:
0056 % P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting Approach
0057 % to Nonsmooth Convex Variational Signal Recovery", IEEE Journal of
0058 % Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.
0059 
0060 % Optional input arguments
0061 if ~isfield(param, 'verbose'), param.verbose = 1; end
0062 if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end
0063 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0064 if ~isfield(param, 'gamma'), param.gamma = 1e-2; end
0065 
0066 % Input arguments for projection onto the L2 ball
0067 param_B2.A = A; param_B2.At = At;
0068 param_B2.y = y; param_B2.epsilon = epsilon;
0069 param_B2.verbose = param.verbose;
0070 if isfield(param, 'nu_B2'), param_B2.nu = param.nu_B2; end
0071 if isfield(param, 'tol_B2'), param_B2.tol = param.tol_B2; end
0072 if isfield(param, 'tight_B2'), param_B2.tight = param.tight_B2; end
0073 if isfield(param, 'max_iter_B2')
0074     param_B2.max_iter = param.max_iter_B2;
0075 end
0076 if isfield(param,'pos_B2'), param_B2.pos=param.pos_B2; end
0077 if isfield(param,'real_B2'), param_B2.real=param.real_B2; end
0078 
0079 % Input arguments for prox TV
0080 param_TV.verbose = param.verbose; param_TV.rel_obj = param.rel_obj;
0081 if isfield(param, 'max_iter_TV')
0082     param_TV.max_iter= param.max_iter_TV;
0083 end
0084 
0085 % Initialization
0086 if isfield(param,'initsol')
0087     xhat = param.initsol;
0088 else
0089     xhat = At(y); 
0090 end 
0091 
0092 iter = 1; prev_norm = 0;
0093 
0094 % Main loop
0095 while 1
0096     
0097     if param.verbose>=1
0098         fprintf('Iteration %i:\n', iter);
0099     end
0100     
0101     % Projection onto the L2-ball
0102     [sol, param_B2.u] = sopt_mltb_fast_proj_B2(xhat, param_B2);
0103     
0104     % Global stopping criterion
0105     curr_norm = sopt_mltb_TV_norm(sol, 0);
0106     rel_norm = abs(curr_norm - prev_norm)/curr_norm;
0107     if param.verbose >= 1
0108         fprintf('  ||x||_TV = %e, rel_norm = %e\n', ...
0109             curr_norm, rel_norm);
0110     end
0111     if (rel_norm < param.rel_obj)
0112         crit_BPDN = 'REL_NORM';
0113         break;
0114     elseif iter >= param.max_iter
0115         crit_BPDN = 'MAX_IT';
0116         break;
0117     end
0118     
0119     % Proximal L1 operator
0120     xhat = 2*sol - xhat;
0121     temp = sopt_mltb_prox_TV(xhat, param.gamma, param_TV);
0122     xhat = temp + sol - xhat;
0123     
0124     % Update variables
0125     iter = iter + 1;
0126     prev_norm = curr_norm;
0127     
0128 end
0129 
0130 % Log
0131 if param.verbose >= 1
0132   
0133     % L1 norm
0134     fprintf('\n Solution found:\n');
0135     fprintf(' Final TV norm: %e\n', curr_norm);
0136     
0137     % Residual
0138     temp = A(sol);
0139     fprintf(' epsilon = %e, ||y-Ax||_2=%e\n', epsilon, ...
0140         norm(y(:)-temp(:)));
0141     
0142     % Stopping criterion
0143     fprintf(' %i iterations\n', iter);
0144     fprintf(' Stopping criterion: %s \n\n', crit_BPDN);
0145     
0146 end
0147 
0148 end
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sopt-2.0.0/matlab/doc/matlab/.svn/text-base/sopt_mltb_solve_TVDNoA.html.svn-base000066400000000000000000000411271277570055300274740ustar00rootroot00000000000000 Description of sopt_mltb_solve_TVDNoA
Home > matlab > sopt_mltb_solve_TVDNoA.m

sopt_mltb_solve_TVDNoA

PURPOSE ^

sopt_mltb_solve_TVDNoA - Solve augmented TVDN problem

SYNOPSIS ^

function sol = sopt_mltb_solve_TVDNoA(y, epsilon, A, At, S, St, param)

DESCRIPTION ^

 sopt_mltb_solve_TVDNoA - Solve augmented TVDN problem

 Solve the total variation denoising (TVDN) problem when an additional 
 linear operator S is incorporated in the TV norm, i.e. solve

   min ||S x||_TV   s.t.  ||y - A x||_2 < epsilon

 where y contains the measurements, A is the forward measurement operator 
 and At the associated adjoint operator, S is the operator appearing in
 the TV norm and St the associated adjoint operator.  The structure param 
 should contain the following fields:

   General parameters:
 
   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - max_iter: Maximum number of iterations (default = 200).

   - rel_obj: Minimum relative change of the objective value 
       (default = 1e-4).  The algorithm stops if
           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj,
       where x(t) is the estimate of the solution at iteration t.

   - gamma: Convergence speed (weighting of TV norm when solving for
       TV proximal operator) (default = 1e-1).
 
   - nu_TV: Bound on the norm of the operator S, i.e.
       ||S x||^2 <= nu * ||x||^2 (default = 1).

   - initsol: Initial solution for a warmstart.
 
   Projection onto the L2-ball:

   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
 
   - nu_B2: Bound on the norm of the operator A, i.e.
       ||A x||^2 <= nu * ||x||^2 (default = 1).

   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
       stops if
         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
       (default = 1e-3).

   - max_iter_B2: Maximum number of iterations for the projection onto the
       L2 ball (default = 200).

   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
       default = 0).

   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
       default = 0).
 
   Proximal L1 operator:

   - max_iter_TV: Used as stopping criterion for the proximal TV
       operator. Maximun number of iterations (default = 200).

   - param.weights_dx_TV: Weights for a weighted TV-norm in the x
       direction (default = 1).

   - param.weights_dy_TV: Weights for a weighted TV-norm in the y
       direction (default = 1).

 References:
 P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting Approach 
 to Nonsmooth Convex Variational Signal Recovery", IEEE Journal of 
 Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_solve_TVDNoA(y, epsilon, A, At, S, St, param)
0002 % sopt_mltb_solve_TVDNoA - Solve augmented TVDN problem
0003 %
0004 % Solve the total variation denoising (TVDN) problem when an additional
0005 % linear operator S is incorporated in the TV norm, i.e. solve
0006 %
0007 %   min ||S x||_TV   s.t.  ||y - A x||_2 < epsilon
0008 %
0009 % where y contains the measurements, A is the forward measurement operator
0010 % and At the associated adjoint operator, S is the operator appearing in
0011 % the TV norm and St the associated adjoint operator.  The structure param
0012 % should contain the following fields:
0013 %
0014 %   General parameters:
0015 %
0016 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0017 %       2 = print main steps; default = 1).
0018 %
0019 %   - max_iter: Maximum number of iterations (default = 200).
0020 %
0021 %   - rel_obj: Minimum relative change of the objective value
0022 %       (default = 1e-4).  The algorithm stops if
0023 %           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj,
0024 %       where x(t) is the estimate of the solution at iteration t.
0025 %
0026 %   - gamma: Convergence speed (weighting of TV norm when solving for
0027 %       TV proximal operator) (default = 1e-1).
0028 %
0029 %   - nu_TV: Bound on the norm of the operator S, i.e.
0030 %       ||S x||^2 <= nu * ||x||^2 (default = 1).
0031 %
0032 %   - initsol: Initial solution for a warmstart.
0033 %
0034 %   Projection onto the L2-ball:
0035 %
0036 %   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
0037 %
0038 %   - nu_B2: Bound on the norm of the operator A, i.e.
0039 %       ||A x||^2 <= nu * ||x||^2 (default = 1).
0040 %
0041 %   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
0042 %       stops if
0043 %         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
0044 %       (default = 1e-3).
0045 %
0046 %   - max_iter_B2: Maximum number of iterations for the projection onto the
0047 %       L2 ball (default = 200).
0048 %
0049 %   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
0050 %       default = 0).
0051 %
0052 %   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
0053 %       default = 0).
0054 %
0055 %   Proximal L1 operator:
0056 %
0057 %   - max_iter_TV: Used as stopping criterion for the proximal TV
0058 %       operator. Maximun number of iterations (default = 200).
0059 %
0060 %   - param.weights_dx_TV: Weights for a weighted TV-norm in the x
0061 %       direction (default = 1).
0062 %
0063 %   - param.weights_dy_TV: Weights for a weighted TV-norm in the y
0064 %       direction (default = 1).
0065 %
0066 % References:
0067 % P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting Approach
0068 % to Nonsmooth Convex Variational Signal Recovery", IEEE Journal of
0069 % Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.
0070 
0071 % Optional input arguments
0072 if ~isfield(param, 'verbose'), param.verbose = 1; end
0073 if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end
0074 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0075 if ~isfield(param, 'gamma'), param.gamma = 1e-2; end
0076 if ~isfield(param, 'weights_dx_TV'), param.weights_dx_TV = 1.0; end
0077 if ~isfield(param, 'weights_dy_TV'), param.weights_dy_TV = 1.0; end
0078 if ~isfield(param, 'incNP'), param.incNP = false; end
0079 if ~isfield(param, 'sphere_flag'), param.sphere_flag = false; end
0080 
0081 % Input arguments for projection onto the L2 ball
0082 param_B2.A = A; param_B2.At = At;
0083 param_B2.y = y; param_B2.epsilon = epsilon;
0084 param_B2.verbose = param.verbose;
0085 if isfield(param, 'nu_B2'), param_B2.nu = param.nu_B2; end
0086 if isfield(param, 'tol_B2'), param_B2.tol = param.tol_B2; end
0087 if isfield(param, 'tight_B2'), param_B2.tight = param.tight_B2; end
0088 if isfield(param, 'max_iter_B2')
0089     param_B2.max_iter = param.max_iter_B2;
0090 end
0091 if isfield(param,'pos_B2'), param_B2.pos=param.pos_B2; end
0092 if isfield(param,'real_B2'), param_B2.real=param.real_B2; end
0093 
0094 % Input arguments for prox TVoA
0095 param_TV.A = S; param_TV.At = St;
0096 param_TV.verbose = param.verbose; 
0097 param_TV.rel_obj = param.rel_obj;
0098 param_TV.weights_dx = param.weights_dx_TV;
0099 param_TV.weights_dy = param.weights_dy_TV;
0100 if isfield(param, 'nu_TV')
0101    param_TV.nu = param.nu_TV; 
0102 end
0103 if isfield(param, 'max_iter_TV')
0104     param_TV.max_iter= param.max_iter_TV;
0105 end
0106 if isfield(param, 'zero_weights_flag_TV')
0107    param_TV.zero_weights_flag = param.zero_weights_flag_TV; 
0108 end
0109 if isfield(param, 'identical_weights_flag_TV'), 
0110    param_TV.identical_weights_flag = param.identical_weights_flag_TV;
0111 end
0112 if isfield(param, 'sphere_flag'), 
0113    param_TV.sphere_flag = param.sphere_flag;
0114    param_TV.incNP = param.incNP;
0115 end
0116 
0117 
0118 % Initialization
0119 if isfield(param,'initsol')
0120     xhat = param.initsol;
0121 else
0122     xhat = At(y); 
0123 end
0124 
0125 iter = 1; prev_norm = 0;
0126 
0127 % Main loop
0128 while 1
0129     
0130     %
0131     if param.verbose>=1
0132         fprintf('Iteration %i:\n', iter);
0133     end
0134     
0135     % Projection onto the L2-ball
0136     [sol, param_B2.u] = sopt_mltb_fast_proj_B2(xhat, param_B2);    
0137     
0138     % Global stopping criterion
0139     curr_norm = sopt_mltb_TV_norm(S(sol), param_TV.sphere_flag, param_TV.incNP, param_TV.weights_dx, param_TV.weights_dy);
0140     rel_norm = abs(curr_norm - prev_norm)/curr_norm;
0141     if param.verbose >= 1
0142         fprintf('  ||x||_TV = %e, rel_norm = %e\n', ...
0143             curr_norm, rel_norm);
0144     end
0145     if (rel_norm < param.rel_obj)
0146         crit_BPDN = 'REL_NORM';
0147         break;
0148     elseif iter >= param.max_iter
0149         crit_BPDN = 'MAX_IT';
0150         break;
0151     end
0152     
0153     % Proximal L1 operator
0154     xhat = 2*sol - xhat;
0155     temp = sopt_mltb_prox_TVoA(xhat, param.gamma, param_TV);
0156     xhat = temp + sol - xhat;
0157     
0158     % Update variables
0159     iter = iter + 1;
0160     prev_norm = curr_norm;
0161     
0162 end
0163 
0164 % Log
0165 if param.verbose>=1
0166     % L1 norm
0167     fprintf('\n Solution found:\n');
0168     fprintf(' Final TV norm: %e\n', curr_norm);
0169     
0170     % Residual
0171     temp = A(sol);
0172     fprintf(' epsilon = %e, ||y-Ax||_2=%e\n', epsilon, ...
0173         norm(y(:)-temp(:)));
0174     
0175     % Stopping criterion
0176     fprintf(' %i iterations\n', iter);
0177     fprintf(' Stopping criterion: %s \n\n', crit_BPDN);
0178     
0179 end
0180 
0181 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/.svn/text-base/sopt_mltb_solve_rwBPDN.html.svn-base000066400000000000000000000224301277570055300275310ustar00rootroot00000000000000 Description of sopt_mltb_solve_rwBPDN
Home > matlab > sopt_mltb_solve_rwBPDN.m

sopt_mltb_solve_rwBPDN

PURPOSE ^

sopt_mltb_solve_rwBPDN - Solve reweighted BPDN problem

SYNOPSIS ^

function sol = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi, Psit,paramT, sigma, tol, maxiter, initsol)

DESCRIPTION ^

 sopt_mltb_solve_rwBPDN - Solve reweighted BPDN problem

 Solve the reweighted L1 minimization function using an homotopy
 continuation method to approximate the L0 norm.  At each iteration the 
 following problem is solved:

   min_x ||W Psit x||_1   s.t.  ||y-A x||_2 < epsilon

 where W is a diagonal matrix with diagonal elements given by

   [W]_ii = delta(t)/(delta(t)+|[Psit x(t)]_i|).

 The input parameters are defined as follows.

   - y: Input data (measurements).

   - epsilon: Noise bound.

   - A: Forward measurement operator.

   - At: Adjoint measurement operator.

   - Psi: Synthesis sparsity transform.

   - Psit: Analysis sparsity transform.

   - paramT: Structure containing parameters for the L1 solver (see 
       documentation for sopt_mltb_solve_BPDN).  

   - sigma: Noise standard deviation in the analysis domain.

   - tol: Minimum relative change in the solution.
       The algorithm stops if 
           ||x(t)-x(t-1)||_2/||x(t-1)||_2 < tol.
       where x(t) is the estimate of the solution at iteration t.

   - maxiter: Maximum number of iterations of the reweighted algorithm.

   - initsol: Initial solution for a warmstart.

 References:
 [1] R. E. Carrillo, J. D. McEwen, D. Van De Ville, J.-Ph. Thiran, and 
 Y. Wiaux. Sparsity averaging for compressive imaging. IEEE Sig. Proc. 
 Let., in press, 2013.
 [2] R. E. Carrillo, J. D. McEwen, and Y. Wiaux. Sparsity Averaging 
 Reweighted Analysis (SARA): a novel algorithm for radio-interferometric 
 imaging. Mon. Not. Roy. Astron. Soc., 426(2):1223-1234, 2012.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi, Psit, ...
0002   paramT, sigma, tol, maxiter, initsol)
0003 % sopt_mltb_solve_rwBPDN - Solve reweighted BPDN problem
0004 %
0005 % Solve the reweighted L1 minimization function using an homotopy
0006 % continuation method to approximate the L0 norm.  At each iteration the
0007 % following problem is solved:
0008 %
0009 %   min_x ||W Psit x||_1   s.t.  ||y-A x||_2 < epsilon
0010 %
0011 % where W is a diagonal matrix with diagonal elements given by
0012 %
0013 %   [W]_ii = delta(t)/(delta(t)+|[Psit x(t)]_i|).
0014 %
0015 % The input parameters are defined as follows.
0016 %
0017 %   - y: Input data (measurements).
0018 %
0019 %   - epsilon: Noise bound.
0020 %
0021 %   - A: Forward measurement operator.
0022 %
0023 %   - At: Adjoint measurement operator.
0024 %
0025 %   - Psi: Synthesis sparsity transform.
0026 %
0027 %   - Psit: Analysis sparsity transform.
0028 %
0029 %   - paramT: Structure containing parameters for the L1 solver (see
0030 %       documentation for sopt_mltb_solve_BPDN).
0031 %
0032 %   - sigma: Noise standard deviation in the analysis domain.
0033 %
0034 %   - tol: Minimum relative change in the solution.
0035 %       The algorithm stops if
0036 %           ||x(t)-x(t-1)||_2/||x(t-1)||_2 < tol.
0037 %       where x(t) is the estimate of the solution at iteration t.
0038 %
0039 %   - maxiter: Maximum number of iterations of the reweighted algorithm.
0040 %
0041 %   - initsol: Initial solution for a warmstart.
0042 %
0043 % References:
0044 % [1] R. E. Carrillo, J. D. McEwen, D. Van De Ville, J.-Ph. Thiran, and
0045 % Y. Wiaux. Sparsity averaging for compressive imaging. IEEE Sig. Proc.
0046 % Let., in press, 2013.
0047 % [2] R. E. Carrillo, J. D. McEwen, and Y. Wiaux. Sparsity Averaging
0048 % Reweighted Analysis (SARA): a novel algorithm for radio-interferometric
0049 % imaging. Mon. Not. Roy. Astron. Soc., 426(2):1223-1234, 2012.
0050 
0051 param=paramT;
0052 iter=0;
0053 rel_dist=1;
0054 
0055 if nargin<11
0056     fprintf('RW iteration: %i\n', iter);
0057     sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0058 else
0059     sol = initsol;
0060 end
0061 
0062 temp=Psit(sol);
0063 delta=std(temp(:));
0064 
0065 while (rel_dist>tol && iter<maxiter)
0066   
0067     iter=iter+1;
0068     delta=max(sigma/10,delta);
0069     fprintf('RW iteration: %i\n', iter);
0070     fprintf('delta = %e\n', delta);
0071     
0072     % Weights
0073     weights=abs(Psit(sol));
0074     param.weights=delta./(delta+weights);
0075     
0076     % Warm start
0077     param.initsol=sol;
0078     param.gamma=1e-1*max(weights(:));
0079     sol1=sol;
0080     
0081     % Weighted L1 problem
0082     sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0083     
0084     % Relative distance
0085     rel_dist=norm(sol(:)-sol1(:))/norm(sol1(:));
0086     fprintf('Relative distance = %e\n\n', rel_dist);
0087     delta = delta/10;
0088     
0089 end
0090
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/.svn/text-base/sopt_mltb_solve_rwTVDN.html.svn-base000066400000000000000000000217451277570055300275710ustar00rootroot00000000000000 Description of sopt_mltb_solve_rwTVDN
Home > matlab > sopt_mltb_solve_rwTVDN.m

sopt_mltb_solve_rwTVDN

PURPOSE ^

sopt_mltb_solve_rwTVDN - Solve reweighted TVDN problem

SYNOPSIS ^

function sol = sopt_mltb_solve_rwTVDN(y, epsilon, A, At, paramT,sigma, tol, maxiter, initsol)

DESCRIPTION ^

 sopt_mltb_solve_rwTVDN - Solve reweighted TVDN problem

 Solve the reweighted TV minimization function using an homotopy
 continuation method to approximate the L0 norm of the magnitude of the 
 gradient.  At each iteration the following problem is solved:

   min_x ||W x||_TV   s.t.  ||y-A x||_2 < epsilon

 where W is a diagonal matrix with diagonal elements given by esentially
 the inverse of the graident.
   
 The input parameters are defined as follows.

   - y: Input data (measurements).

   - epsilon: Noise bound.

   - A: Forward measurement operator.

   - At: Adjoint measurement operator.

   - paramT: Structure containing parameters for the TV solver (see 
       documentation for sopt_mltb_solve_TVDN).  

   - sigma: Noise standard deviation in the analysis domain.

   - tol: Minimum relative change in the solution.
       The algorithm stops if 
           ||x(t)-x(t-1)||_2/||x(t-1)||_2 < tol.
       where x(t) is the estimate of the solution at iteration t.

   - maxiter: Maximum number of iterations of the reweighted algorithm.

   - initsol: Initial solution for a warmstart.

 References:
 [1] R. E. Carrillo, J. D. McEwen, D. Van De Ville, J.-Ph. Thiran, and 
 Y. Wiaux. Sparsity averaging for compressive imaging. IEEE Sig. Proc. 
 Let., in press, 2013.
 [2] R. E. Carrillo, J. D. McEwen, and Y. Wiaux. Sparsity Averaging 
 Reweighted Analysis (SARA): a novel algorithm for radio-interferometric 
 imaging. Mon. Not. Roy. Astron. Soc., 426(2):1223-1234, 2012.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_solve_rwTVDN(y, epsilon, A, At, paramT, ...
0002   sigma, tol, maxiter, initsol)
0003 % sopt_mltb_solve_rwTVDN - Solve reweighted TVDN problem
0004 %
0005 % Solve the reweighted TV minimization function using an homotopy
0006 % continuation method to approximate the L0 norm of the magnitude of the
0007 % gradient.  At each iteration the following problem is solved:
0008 %
0009 %   min_x ||W x||_TV   s.t.  ||y-A x||_2 < epsilon
0010 %
0011 % where W is a diagonal matrix with diagonal elements given by esentially
0012 % the inverse of the graident.
0013 %
0014 % The input parameters are defined as follows.
0015 %
0016 %   - y: Input data (measurements).
0017 %
0018 %   - epsilon: Noise bound.
0019 %
0020 %   - A: Forward measurement operator.
0021 %
0022 %   - At: Adjoint measurement operator.
0023 %
0024 %   - paramT: Structure containing parameters for the TV solver (see
0025 %       documentation for sopt_mltb_solve_TVDN).
0026 %
0027 %   - sigma: Noise standard deviation in the analysis domain.
0028 %
0029 %   - tol: Minimum relative change in the solution.
0030 %       The algorithm stops if
0031 %           ||x(t)-x(t-1)||_2/||x(t-1)||_2 < tol.
0032 %       where x(t) is the estimate of the solution at iteration t.
0033 %
0034 %   - maxiter: Maximum number of iterations of the reweighted algorithm.
0035 %
0036 %   - initsol: Initial solution for a warmstart.
0037 %
0038 % References:
0039 % [1] R. E. Carrillo, J. D. McEwen, D. Van De Ville, J.-Ph. Thiran, and
0040 % Y. Wiaux. Sparsity averaging for compressive imaging. IEEE Sig. Proc.
0041 % Let., in press, 2013.
0042 % [2] R. E. Carrillo, J. D. McEwen, and Y. Wiaux. Sparsity Averaging
0043 % Reweighted Analysis (SARA): a novel algorithm for radio-interferometric
0044 % imaging. Mon. Not. Roy. Astron. Soc., 426(2):1223-1234, 2012.
0045 
0046 Psit = @(x) x; Psi=@(x) x;
0047 
0048 param=paramT;
0049 iter=0;
0050 rel_dist=1;
0051 
0052 if nargin<9
0053     fprintf('RW iteration: %i\n', iter);
0054     sol = sopt_mltb_solve_TVoA_B2(y, epsilon, A, At, Psi, Psit, param);
0055 else
0056     sol = initsol;
0057 end
0058 
0059 
0060 delta=std(sol(:));
0061 
0062 while (rel_dist>tol && iter<maxiter)
0063   
0064     iter=iter+1;
0065     delta=max(sigma/10,delta);
0066     fprintf('RW iteration: %i\n', iter);
0067     fprintf('delta = %e\n', delta);
0068     
0069     %Warm start
0070     param.initsol=sol;
0071     sol1=sol;
0072     
0073     % Weights
0074     [param.weights_dx_TV param.weights_dy_TV] = sopt_mltb_gradient_op(real(sol));
0075     param.weights_dx_TV = delta./(abs(param.weights_dx_TV)+delta);
0076     param.weights_dy_TV = delta./(abs(param.weights_dy_TV)+delta);
0077     param.identical_weights_flag_TV = 0;
0078     param.gamma=1e-1*max(abs(sol1(:)));
0079     
0080     %Weighted TV problem
0081     sol = sopt_mltb_solve_TVDNoA(y, epsilon, A, At, Psi, Psit, param);
0082 
0083     %Relative distance
0084     rel_dist=norm(sol(:)-sol1(:))/norm(sol1(:)); 
0085     fprintf('relative distance = %e\n\n', rel_dist);
0086     delta = delta/10;
0087     
0088 end
0089
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/Example_BPDN.html000066400000000000000000000210721277570055300210440ustar00rootroot00000000000000 Description of Example_BPDN
Home > matlab > Example_BPDN.m

Example_BPDN

PURPOSE ^

% Example_BPDN

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Example_BPDN
 Example to demonstrate use of BPDN solver.  A random Fourier sampling
 measurement operator is considered.  Daubechies 8 wavelets are used for
 the sparsifying operator.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Example_BPDN
0002 % Example to demonstrate use of BPDN solver.  A random Fourier sampling
0003 % measurement operator is considered.  Daubechies 8 wavelets are used for
0004 % the sparsifying operator.
0005 
0006 
0007 %% Clear workspace
0008 
0009 clc
0010 clear;
0011 
0012 
0013 %% Define paths
0014 
0015 addpath misc/
0016 addpath prox_operators/
0017 
0018 
0019 %% Define parameters
0020 
0021 % Coverages (half the plane for Fourier sampling)
0022 p = [0.50];
0023 
0024 % Noise level (on the measurements)
0025 input_snr = 30;
0026 
0027 
0028 %% Read image
0029 
0030 % Load image
0031 im = im2double(imread('cameraman.tif'));
0032 
0033 % Normalise
0034 im = im/max(max(im));
0035 
0036 % Enforce positivity
0037 im(im<0) = 0;
0038 
0039 
0040 %% Define sparsity operator
0041 
0042 dwtmode('per');
0043 
0044 % Decomposition level of the wavelet transform.
0045 nlevel = 4;
0046 
0047 % Daubechies 8 wavelet operator
0048 [C,S] = wavedec2(im,nlevel,'db8');
0049 
0050 Psit = @(x) wavedec2(x, nlevel, 'db8');
0051 Psi = @(x) waverec2(x, S, 'db8');
0052 
0053 
0054 %% Run simulations
0055 
0056 % Random Fourier sampling example
0057 %  Define mask
0058 %  Uniform sampling of the half Fourier plane
0059 mask = rand(size(im)) < p;
0060 mask(:,1:floor(size(im,2)/2))=0;
0061 mask = ifftshift(mask);
0062 mask(1,1)=0;
0063 mask(floor(size(im,1)/2):end,1)=0;
0064 
0065 ind = find(mask==1);
0066 Ma = sparse(1:numel(ind), ind, ...
0067   ones(numel(ind), 1), numel(ind), numel(im));
0068 
0069 % Composition (Masking o Fourier)
0070 A = @(x) Ma*reshape(fft2(x)/sqrt(numel(ind)), numel(x), 1);
0071 At = @(x) (ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(ind))));
0072 
0073 % Apply measurement operator
0074 y = A(im);
0075 
0076 % Add Gaussian i.i.d. noise
0077 sigma_noise = 10^(-input_snr/20)*std(im(:));
0078 noise = (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0079 y = y + noise;
0080 
0081 % Tolerance on noise
0082 epsilon = sqrt(numel(y) + 2*sqrt(numel(y)))*sigma_noise;
0083 epsilon_up = sqrt(numel(y) + 2.1*sqrt(numel(y)))*sigma_noise;
0084 tol_B2 = (epsilon_up/epsilon)-1; % Tolerance for the projection onto the L2-ball
0085 
0086 % Solve optimisation problem
0087 
0088 % Parameters for BPDN
0089 param.verbose = 1; % Print log or not
0090 param.gamma = 1e-1; % Convergence parameter
0091 param.rel_obj = 5e-4; % Stopping criterion for the L1 problem
0092 param.max_iter = 200; % Max. number of iterations for the L1 problem
0093 param.nu_B2 = 1; % Bound on the norm of the operator A
0094 param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0095 param.max_iter_B2 = 200; %Max. number of iterations of the L2-ball projection
0096 param.pos_B2 = 1; %Positivity flag
0097 param.tol_B2 = tol_B2; % Tolerance for the projection onto the L2-ball
0098 param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0099 %Optional parameters when Psit is not a tight frame:
0100 %param.nu_L1 = 1; % Bound on the norm of the operator Psit
0101 %param.max_iter_L1 = 200; %Max. number of iterations of the L1 prox
0102 %param.rel_obj_L1 = 1e-3; % Tolerance for the prox L1
0103 
0104 % Solve BPDN problem with positivity constraint
0105 sol1 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0106 
0107 % Compute SNR
0108 RSNR1 = 20*log10(norm(im,'fro') ...
0109   / norm(im-sol1,'fro'));
0110 
0111 % Example with only reality constraint
0112 param.pos_B2 = 0; %Positivity flag
0113 param.real_B2 = 1; %Reality flag
0114 
0115 % Solve BPDN problem
0116 sol2 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0117 
0118 % Compute SNR
0119 RSNR2 = 20*log10(norm(im,'fro') ...
0120   / norm(im-sol2,'fro'));
0121 
0122 
0123 %% Show results
0124 
0125 figure
0126 imagesc(sol1), axis off, axis image, colorbar
0127 title(['Rec. with positivity const. SNR=',num2str(RSNR1), 'dB'])
0128 colormap gray
0129 
0130 figure
0131 imagesc(sol2), axis off, axis image, colorbar
0132 title(['Rec. with reality const. SNR=',num2str(RSNR2), 'dB'])
0133 colormap gray
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/Example_TVDN.html000066400000000000000000000167721277570055300211070ustar00rootroot00000000000000 Description of Example_TVDN
Home > matlab > Example_TVDN.m

Example_TVDN

PURPOSE ^

% Example_TVDN

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Example_TVDN
 Example to demonstrate use of TVDN solver.  A random Fourier sampling
 measurement operator is considered.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Example_TVDN
0002 % Example to demonstrate use of TVDN solver.  A random Fourier sampling
0003 % measurement operator is considered.
0004 
0005 
0006 %% Clear workspace
0007 
0008 clc
0009 clear;
0010 
0011 
0012 %% Define paths
0013 
0014 addpath misc/
0015 addpath prox_operators/
0016 
0017 
0018 %% Define parameters
0019 
0020 % Coverages (half the plane for Fourier sampling)
0021 p = [0.50];
0022 
0023 % Noise level (on the measurements)
0024 input_snr = 30;
0025 
0026 
0027 %% Read image
0028 
0029 % Load image
0030 im = im2double(imread('cameraman.tif'));
0031 
0032 % Normalise
0033 im = im/max(max(im));
0034 
0035 % Enforce positivity
0036 im(im<0) = 0;
0037 
0038 
0039 %% Run simulations
0040 
0041 %Random Fourier sampling example
0042 % Define mask
0043 % Uniform sampling of the half Fourier plane
0044 mask = rand(size(im)) < p;
0045 mask(:,1:floor(size(im,2)/2))=0;
0046 mask = ifftshift(mask);
0047 mask(1,1)=0;
0048 mask(floor(size(im,1)/2):end,1)=0;
0049 
0050 ind = find(mask==1);
0051 Ma = sparse(1:numel(ind), ind, ...
0052   ones(numel(ind), 1), numel(ind), numel(im));
0053 
0054 % Composition (Masking o Fourier)
0055 A = @(x) Ma*reshape(fft2(x)/sqrt(numel(ind)), numel(x), 1);
0056 At = @(x) (ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(ind))));
0057 
0058 % Apply measurement operator
0059 y = A(im);
0060 
0061 % Add Gaussian i.i.d. noise
0062 sigma_noise = 10^(-input_snr/20)*std(im(:));
0063 noise = (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0064 y = y + noise;
0065 
0066 % Tolerance on noise
0067 epsilon = sqrt(numel(y) + 2*sqrt(numel(y)))*sigma_noise;
0068 epsilon_up = sqrt(numel(y) + 2.1*sqrt(numel(y)))*sigma_noise;
0069 tol_B2 = (epsilon_up/epsilon)-1; % Tolerance for the projection onto the L2-ball
0070 
0071 % Solve optimisation problem
0072 
0073 % Parameters for TVDN
0074 param.verbose = 1; % Print log or not
0075 param.gamma = 1e-1; % Converge parameter
0076 param.rel_obj = 5e-4; % Stopping criterion for the TVDN problem
0077 param.max_iter = 200; % Max. number of iterations for the TVDN problem
0078 param_TV.max_iter_TV = 200; % Max. nb. of iter. for the sub-problem (proximal TV operator)
0079 param.nu_B2 = 1; % Bound on the norm of the operator A
0080 param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0081 param.max_iter_B2 = 200; %Max. number of iterations of the L2-ball projection
0082 param.pos_B2 = 1; %Positivity flag
0083 param.tol_B2 = tol_B2; % Tolerance for the projection onto the L2-ball
0084 
0085 % Solve TVDN problem with positivity constraint
0086 sol1 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param);
0087 
0088 % Compute SNR
0089 RSNR1 = 20*log10(norm(im,'fro') ...
0090   / norm(im-sol1,'fro'));
0091 
0092 % Example with only reality constraint
0093 param.pos_B2 = 0; %Positivity flag
0094 param.real_B2 = 1; %Reality flag
0095 
0096 % Solve TVDN problem
0097 sol2 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param);
0098 
0099 % Compute SNR
0100 RSNR2 = 20*log10(norm(im,'fro') ...
0101   / norm(im-sol2,'fro'));
0102 
0103 
0104 %% Show results
0105 
0106 figure
0107 imagesc(sol1), axis off, axis image, colorbar
0108 title(['Rec. with positivity const. SNR=',num2str(RSNR1), 'dB'])
0109 colormap gray
0110 
0111 figure
0112 imagesc(sol2), axis off, axis image, colorbar
0113 title(['Rec. with reality const. SNR=',num2str(RSNR2), 'dB'])
0114 colormap gray
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/Example_TVDN2.html000066400000000000000000000162141277570055300211600ustar00rootroot00000000000000 Description of Example_TVDN2
Home > matlab > Example_TVDN2.m

Example_TVDN2

PURPOSE ^

% Example_TVDN2

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Example_TVDN2
 Two examples to demonstrate use of TVDN solver.  Simple inpainting and
 Fourier measurement examples are included.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Example_TVDN2
0002 % Two examples to demonstrate use of TVDN solver.  Simple inpainting and
0003 % Fourier measurement examples are included.
0004 
0005 
0006 %% Clear workspace
0007 clc;
0008 clear;
0009 
0010 %% Define paths
0011 addpath misc/
0012 addpath prox_operators/
0013 
0014 %% Parameters
0015 input_snr = 30; % Noise level (on the measurements)
0016 
0017 %% Load an image
0018 im = im2double(imread('cameraman.tif'));
0019 %
0020 figure(1);
0021 imagesc(im); axis image; axis off;
0022 colormap gray; title('Original image'); drawnow;
0023 
0024 %% Create a mask with 33% of 1 (the rest is set to 0)
0025 % Mask
0026 mask = rand(size(im)) < 0.33; ind = find(mask==1);
0027 % Masking matrix (sparse matrix in matlab)
0028 Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im));
0029 % Masking operator
0030 A = @(x) Ma*x(:); % Select 33% of the values in x;
0031 At = @(x) reshape(Ma'*x(:), size(im)); % Adjoint operator + reshape image
0032 
0033 %% First problem: Inpainting problem
0034 % Select 33% of pixels
0035 y = A(im);
0036 % Add Gaussian i.i.d. noise
0037 sigma_noise = 10^(-input_snr/20)*std(im(:));
0038 y = y + randn(size(y))*sigma_noise;
0039 % Display the downsampled image
0040 figure(2); clf;
0041 subplot(121); imagesc(At(y)); axis image; axis off;
0042 colormap gray; title('Measured image'); drawnow;
0043 % Parameters for TVDN
0044 param.verbose = 1; % Print log or not
0045 param.gamma = 1e-1; % Converge parameter
0046 param.rel_obj = 1e-4; % Stopping criterion for the TVDN problem
0047 param.max_iter = 200; % Max. number of iterations for the TVDN problem
0048 param_TV.max_iter_TV = 100; % Max. nb. of iter. for the sub-problem (proximal TV operator)
0049 param.nu_B2 = 1; % Bound on the norm of the operator A
0050 param.tol_B2 = 1e-4; % Tolerance for the projection onto the L2-ball
0051 param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0052 param.max_iter_B2 = 500;
0053 % Tolerance on noise
0054 epsilon = sqrt(chi2inv(0.99, numel(ind)))*sigma_noise;
0055 % Solve TVDN problem
0056 sol = sopt_mltb_solve_TVDN(y, epsilon, A, At, param);
0057 % Show reconstructed image
0058 figure(2);
0059 subplot(122); imagesc(sol); axis image; axis off;
0060 colormap gray; title('Reconstructed image'); drawnow;
0061 
0062 %% Second problem: Reconstruct from 33% of Fourier measurements
0063 % Composition (Masking o Fourier)
0064 A = @(x) Ma*reshape(fft2(x)/sqrt(numel(im)), numel(x), 1);
0065 At = @(x) ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(im)));
0066 % Select 33% of Fourier coefficients
0067 y = A(im);
0068 % Add Gaussian i.i.d. noise
0069 sigma_noise = 10^(-input_snr/20)*std(im(:));
0070 y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0071 % Display the downsampled image
0072 figure(3); clf;
0073 subplot(121); imagesc(real(At(y))); axis image; axis off;
0074 colormap gray; title('Measured image'); drawnow;
0075 % Tolerance on noise
0076 epsilon = sqrt(chi2inv(0.99, 2*numel(ind))/2)*sigma_noise;
0077 % Solve TVDN problem
0078 sol = sopt_mltb_solve_TVDN(y, epsilon, A, At, param);
0079 % Show reconstructed image
0080 figure(3);
0081 subplot(122); imagesc(real(sol)); axis image; axis off;
0082 colormap gray; title('Reconstructed image'); drawnow;
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/Example_TVDNoA.html000066400000000000000000000125111277570055300213520ustar00rootroot00000000000000 Description of Example_TVDNoA
Home > matlab > Example_TVDNoA.m

Example_TVDNoA

PURPOSE ^

% Example TVDNoA

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Example TVDNoA
 Example to demonstrate TVoA_B2 solver, where an additional operator is
 included in the TV norm of the TVDN problem.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Example TVDNoA
0002 % Example to demonstrate TVoA_B2 solver, where an additional operator is
0003 % included in the TV norm of the TVDN problem.
0004 
0005 
0006 %% Clear workspace
0007 clc;
0008 clear;
0009 
0010 %% Define paths
0011 addpath misc/
0012 addpath prox_operators/
0013 
0014 %% Parameters
0015 N = 32;
0016 input_snr = 1; % Noise level (on the measurements)
0017 randn('seed', 1);
0018 
0019 %% Load an image
0020 im_ref = phantom(N);
0021 %
0022 figure(1);
0023 imagesc(im_ref); axis image; axis off;
0024 colormap gray; title('Original image'); drawnow;
0025 
0026 %% Create an artifial operator to test prox_TVoS
0027 S = randn(N^2, N^2)/N^2;
0028 im = reshape(S\im_ref(:), N, N); % S*im is thus sparse in TV
0029 
0030 %% Add Gaussian i.i.d. noise to im
0031 y = im;
0032 sigma_noise = 10^(-input_snr/20)*std(im_ref(:));
0033 y = y + randn(size(y))*sigma_noise;
0034 
0035 %% Solving modified ROF
0036 % Parameters for TVDN
0037 param.verbose = 2; % Print log or not
0038 param.gamma = 1; % Converge parameter
0039 param.rel_obj = 1e-4; % Stopping criterion for the TVDN problem
0040 param.max_iter = 100; % Max. number of iterations for the TVDN problem
0041 param.max_iter_TV = 100; % Max. nb. of iter. for the sub-problem (proximal TV operator)
0042 param.nu_B2 = 1; % Bound on the norm of the measurement operator (Id here)
0043 param.tight_B2 = 1; % Indicate that A is a tight frame (1) or not (0)
0044 param.nu_TV = norm(S)^2; % Bound on the norm of the operator S
0045 func_S = @(x) reshape(S*x(:), N, N);
0046 func_St = @(x) reshape(S'*x(:), N, N);
0047 % Tolerance on noise
0048 epsilon = sqrt(chi2inv(0.99, numel(im)))*sigma_noise;
0049 % Identity
0050 A = @(x) x;
0051 % Solve
0052 sol = sopt_mltb_solve_TVDNoA(y, epsilon, A, A, func_S, func_St, param);
0053 % Show reconstructed image
0054 figure(2);
0055 imagesc(func_S(sol)); axis image; axis off;
0056 colormap gray; title('Reconstructed image'); drawnow;
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/Example_weighted_BPDN.html000066400000000000000000000160221277570055300227230ustar00rootroot00000000000000 Description of Example_weighted_BPDN
Home > matlab > Example_weighted_BPDN.m

Example_weighted_BPDN

PURPOSE ^

% Exampled_weighted_L1

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Exampled_weighted_L1
 Example to demonstrate use of BPDN solver when incorporating weights
 (performs one re-weighting of previous solution).

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Exampled_weighted_L1
0002 % Example to demonstrate use of BPDN solver when incorporating weights
0003 % (performs one re-weighting of previous solution).
0004 
0005 
0006 %% Clear workspace
0007 clc;
0008 clear;
0009 
0010 %% Define paths
0011 addpath misc/
0012 addpath prox_operators/
0013 
0014 %% Parameters
0015 N = 64;
0016 input_snr = 30; % Noise level (on the measurements)
0017 randn('seed', 1); rand('seed', 1);
0018 
0019 %% Create an image with few spikes
0020 im = zeros(N); ind = randperm(N^2); im(ind(1:100)) = 1;
0021 %
0022 figure(1);
0023 subplot(221), imagesc(im); axis image; axis off;
0024 colormap gray; title('Original image'); drawnow;
0025 
0026 %% Create a mask
0027 % Mask
0028 mask = rand(size(im)) < 0.095; ind = find(mask==1);
0029 % Masking matrix (sparse matrix in matlab)
0030 Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im));
0031 % Masking operator
0032 A = @(x) Ma*x(:);
0033 At = @(x) reshape(Ma'*x(:), size(im));
0034 
0035 %% Reconstruct from a few Fourier measurements
0036 
0037 % Composition (Masking o Fourier)
0038 A = @(x) Ma*reshape(fft2(x)/sqrt(numel(im)), numel(x), 1);
0039 At = @(x) ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(im)));
0040 
0041 % Sparsity operator
0042 Psit = @(x) x; Psi = Psit;
0043 
0044 % Select 33% of Fourier coefficients
0045 y = A(im);
0046 
0047 % Add Gaussian i.i.d. noise
0048 sigma_noise = 10^(-input_snr/20)*std(im(:));
0049 y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0050 
0051 % Display the downsampled image
0052 figure(1);
0053 subplot(222); imagesc(real(At(y))); axis image; axis off;
0054 colormap gray; title('Measured image'); drawnow;
0055 
0056 % Tolerance on noise
0057 epsilon = sqrt(chi2inv(0.99, 2*numel(ind))/2)*sigma_noise;
0058 
0059 % Parameters for BPDN
0060 param.verbose = 1; % Print log or not
0061 param.gamma = 1e-1; % Converge parameter
0062 param.rel_obj = 1e-4; % Stopping criterion for the TVDN problem
0063 param.max_iter = 300; % Max. number of iterations for the TVDN problem
0064 param.nu_B2 = 1; % Bound on the norm of the operator A
0065 param.tol_B2 = 1e-4; % Tolerance for the projection onto the L2-ball
0066 param.tight_B2 = 1; % Indicate if A is a tight frame (1) or not (0)
0067 param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0068 param.pos_l1 = 1; %
0069 
0070 % Solve BPDN problem (without weights)
0071 sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0072 
0073 % Show reconstructed image
0074 figure(1);
0075 subplot(223); imagesc(real(sol)); axis image; axis off;
0076 colormap gray; title(['First estimate - ', ...
0077     num2str(sopt_mltb_SNR(im, real(sol))), 'dB']); drawnow;
0078 
0079 % Refine the estimate
0080 param.weights = 1./(abs(sol)+1e-5);
0081 sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0082 
0083 % Show reconstructed image
0084 figure(1);
0085 subplot(224); imagesc(real(sol)); axis image; axis off;
0086 colormap gray; title(['Second estimate - ', ...
0087     num2str(sopt_mltb_SNR(im, real(sol))), 'dB']); drawnow;
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/Example_weighted_TVDN.html000066400000000000000000000166741277570055300227700ustar00rootroot00000000000000 Description of Example_weighted_TVDN
Home > matlab > Example_weighted_TVDN.m

Example_weighted_TVDN

PURPOSE ^

% Exampled_weighted_TVDN

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Exampled_weighted_TVDN
 Example to demonstrate use of TVDN solver when incorporating weights
 (performs one re-weighting of previous solution).

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Exampled_weighted_TVDN
0002 % Example to demonstrate use of TVDN solver when incorporating weights
0003 % (performs one re-weighting of previous solution).
0004 
0005 
0006 %% Clear workspace
0007 clc;
0008 clear;
0009 
0010 %% Define paths
0011 addpath misc/
0012 addpath prox_operators/
0013 
0014 %% Parameters
0015 N = 64;
0016 input_snr = 30; % Noise level (on the measurements)
0017 randn('seed', 1); rand('seed', 1);
0018 
0019 %% Load image
0020 im = phantom(N);
0021 %
0022 figure(1); clf;
0023 subplot(141), imagesc(im); axis image; axis off;
0024 colormap gray; title('Original image'); drawnow;
0025 
0026 %% Create a mask
0027 % Mask
0028 mask = rand(size(im)) < 0.33; ind = find(mask==1);
0029 % Masking matrix (sparse matrix in matlab)
0030 Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im));
0031 
0032 %% Measure a few Fourier measurements
0033 
0034 % Composition (Masking o Fourier)
0035 A = @(x) Ma*reshape(fft2(x)/sqrt(numel(im)), numel(x), 1);
0036 At = @(x) ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(im)));
0037 
0038 % TV sparsity operator
0039 Psit = @(x) x; Psi = Psit;
0040 
0041 % Select 33% of Fourier coefficients
0042 y = A(im);
0043 
0044 % Add Gaussian i.i.d. noise
0045 sigma_noise = 10^(-input_snr/20)*std(im(:));
0046 y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0047 
0048 % Display the downsampled image
0049 figure(1);
0050 subplot(142); imagesc(real(At(y))); axis image; axis off;
0051 colormap gray; title('Measured image'); drawnow;
0052 
0053 %% Reconstruct with TV
0054 
0055 % Tolerance on noise
0056 epsilon = sqrt(chi2inv(0.99, 2*numel(ind))/2)*sigma_noise;
0057 
0058 % Parameters for TVDN
0059 param.verbose = 1; % Print log or not
0060 param.rel_obj = 1e-4; % Stopping criterion for the TVDN problem
0061 param.max_iter = 200; % Max. nb. of iterations for the TVDN problem
0062 param.gamma = 1e-1; % Converge parameter
0063 param.nu_B2 = 1; % Bound on the norm of the operator A
0064 param.tol_B2 = 1e-4; % Tolerance for the projection onto the L2-ball
0065 param.tight_B2 = 1; % Indicate if A is a tight frame (1) or not (0)
0066 param.max_iter_TV = 500; %
0067 param.zero_weights_flag_TV = 0; %
0068 param.identical_weights_flag_TV = 1; %
0069 
0070 % Solve TVDN problem (without weights)
0071 sol_1 = sopt_mltb_solve_TVDNoA(y, epsilon, A, At, Psi, Psit, param);
0072 
0073 % Show first reconstructed image
0074 figure(1);
0075 subplot(143); imagesc(real(sol_1)); axis image; axis off;
0076 colormap gray; 
0077 title(['First estimate: ', ...
0078     num2str(sopt_mltb_SNR(im, real(sol_1))), 'dB']);
0079 drawnow;
0080 clc;
0081 
0082 %% Re-fine the estimate with weighted TV
0083 % Weights
0084 [param.weights_dx_TV param.weights_dy_TV] = sopt_mltb_gradient_op(real(sol_1));
0085 param.weights_dx_TV = 1./(abs(param.weights_dx_TV)+1e-3);
0086 param.weights_dy_TV = 1./(abs(param.weights_dy_TV)+1e-3);
0087 param.identical_weights_flag_TV = 0;
0088 param.gamma = 1e-3;
0089 
0090 % First reconstruction with weights in the gradient
0091 param.zero_weights_flag_TV = 1;
0092 sol_2 = sopt_mltb_solve_TVDNoA(y, epsilon, A, At, Psi, Psit, param);
0093 % Show second reconstructed image
0094 figure(1);
0095 subplot(144); imagesc(real(sol_2)); axis image; axis off;
0096 colormap gray; 
0097 title(['Second estimate: ', ...
0098     num2str(sopt_mltb_SNR(im, real(sol_2))), 'dB']); 
0099 drawnow;
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/Experiment1.html000066400000000000000000000426341277570055300210560ustar00rootroot00000000000000 Description of Experiment1
Home > matlab > Experiment1.m

Experiment1

PURPOSE ^

% Experiment1

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Experiment1
 In this experiment we evaluate the performance of SARA for spread 
 spectrum acquisition. We use a 256x256 version of Lena as a test image. 
 Number of measurements is M = 0.2N and input SNR is set to 30 dB. These
 parameters can be changed by modifying the variables p (for the
 undersampling ratio) and input_snr (for the input SNR).

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Experiment1
0002 % In this experiment we evaluate the performance of SARA for spread
0003 % spectrum acquisition. We use a 256x256 version of Lena as a test image.
0004 % Number of measurements is M = 0.2N and input SNR is set to 30 dB. These
0005 % parameters can be changed by modifying the variables p (for the
0006 % undersampling ratio) and input_snr (for the input SNR).
0007 
0008 
0009 %% Clear workspace
0010 
0011 clc
0012 clear;
0013 
0014 
0015 %% Define paths
0016 
0017 addpath misc/
0018 addpath prox_operators/
0019 addpath test_images/
0020 
0021 
0022 
0023 %% Read image
0024 
0025 imagename = 'lena_256.tiff';
0026 
0027 % Load image
0028 im = im2double(imread(imagename));
0029 
0030 % Normalise
0031 im = im/max(max(im));
0032 
0033 % Enforce positivity
0034 im(im<0) = 0;
0035 
0036 %% Parameters
0037 
0038 input_snr = 30; % Noise level (on the measurements)
0039 
0040 %Undersampling ratio M/N
0041 p=0.2;
0042 
0043 
0044 %% Sparsity operators
0045 
0046 %Wavelet decomposition depth
0047 nlevel=4;
0048 
0049 dwtmode('per');
0050 [C,S]=wavedec2(im,nlevel,'db8'); 
0051 ncoef=length(C);
0052 [C1,S1]=wavedec2(im,nlevel,'db1'); 
0053 ncoef1=length(C1);
0054 [C2,S2]=wavedec2(im,nlevel,'db2'); 
0055 ncoef2=length(C2);
0056 [C3,S3]=wavedec2(im,nlevel,'db3'); 
0057 ncoef3=length(C3);
0058 [C4,S4]=wavedec2(im,nlevel,'db4'); 
0059 ncoef4=length(C4);
0060 [C5,S5]=wavedec2(im,nlevel,'db5'); 
0061 ncoef5=length(C5);
0062 [C6,S6]=wavedec2(im,nlevel,'db6'); 
0063 ncoef6=length(C6);
0064 [C7,S7]=wavedec2(im,nlevel,'db7'); 
0065 ncoef7=length(C7);
0066 
0067 %SARA
0068 
0069 Psit = @(x) [wavedec2(x,nlevel,'db1')'; wavedec2(x,nlevel,'db2')';wavedec2(x,nlevel,'db3')';...
0070     wavedec2(x,nlevel,'db4')'; wavedec2(x,nlevel,'db5')'; wavedec2(x,nlevel,'db6')';...
0071     wavedec2(x,nlevel,'db7')';wavedec2(x,nlevel,'db8')']/sqrt(8); 
0072 
0073 Psi = @(x) (waverec2(x(1:ncoef1),S1,'db1')+waverec2(x(ncoef1+1:ncoef1+ncoef2),S2,'db2')+...
0074     waverec2(x(ncoef1+ncoef2+1:ncoef1+ncoef2+ncoef3),S3,'db3')+...
0075     waverec2(x(ncoef1+ncoef2+ncoef3+1:ncoef1+ncoef2+ncoef3+ncoef4),S4,'db4')+...
0076     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5),S5,'db5')+...
0077     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6),S6,'db6')+...
0078     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7),S7,'db7')+...
0079     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+ncoef),S,'db8'))/sqrt(8);
0080 
0081 %Db8 wavelet basis
0082 Psit2 = @(x) wavedec2(x, nlevel,'db8'); 
0083 Psi2 = @(x) waverec2(x,S,'db8');
0084 
0085 %Curvelet
0086 %CurveLab needs to be installed to run Curvelet simulations
0087 realv = 1;
0088 Cv = fdct_usfft(im,realv);
0089 Mod = sopt_mltb_struct2size(Cv);
0090 
0091 Psit3 = @(x) sopt_mltb_fwdcurvelet(x,realv); 
0092 Psi3 = @(x) sopt_mltb_adjcurvelet(x,Mod,realv);
0093 
0094 %% Spread spectrum operator
0095 % Mask
0096 mask = rand(size(im)) < p; 
0097 ind = find(mask==1);
0098 % Masking matrix (sparse matrix in matlab)
0099 Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im));
0100     
0101 %Spread spectrum sequence
0102     
0103 ss=rand(size(im));
0104 C=(2*(ss<0.5)-1);
0105 
0106 A = @(x) Ma*reshape(fft2(C.*x)/sqrt(numel(ind)), numel(x), 1);
0107 At = @(x) C.*(ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(ind))));
0108     
0109 % Sampling
0110 y = A(im);
0111 % Add Gaussian i.i.d. noise
0112 sigma_noise = 10^(-input_snr/20)*std(im(:));
0113 y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0114     
0115     
0116 % Tolerance on noise
0117 epsilon = sqrt(numel(y)+2*sqrt(numel(y)))*sigma_noise;
0118 epsilon_up = sqrt(numel(y)+2.1*sqrt(numel(y)))*sigma_noise;
0119     
0120     
0121 % Parameters for BPDN
0122 param.verbose = 1; % Print log or not
0123 param.gamma = 1e-1; % Converge parameter
0124 param.rel_obj = 5e-4; % Stopping criterion for the L1 problem
0125 param.max_iter = 200; % Max. number of iterations for the L1 problem
0126 param.nu_B2 = 1; % Bound on the norm of the operator A
0127 param.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0128 param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0129 param.pos_B2 = 1; %Positivity constraint: (1) active, (0) not active
0130 param.max_iter_B2=300;
0131 param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0132 param.nu_L1 = 1;
0133 param.max_iter_L1 = 20;
0134 param.rel_obj_L1 = 1e-2;
0135     
0136     
0137 % Solve BPSA problem
0138     
0139 sol1 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0140     
0141 RSNR1=20*log10(norm(im,'fro')/norm(im-sol1,'fro'));
0142     
0143 % SARA
0144 % It uses the solution to BPSA as a warm start
0145 maxiter=10;
0146 sigma=sigma_noise*sqrt(numel(y)/(numel(im)*8));
0147 tol=1e-3;
0148   
0149 sol2 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi, Psit, param, sigma, tol, maxiter, sol1);
0150 
0151 RSNR2=20*log10(norm(im,'fro')/norm(im-sol2,'fro'));
0152 
0153 % Solve BPBb8 problem
0154     
0155 sol3 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi2, Psit2, param);
0156     
0157 RSNR3=20*log10(norm(im,'fro')/norm(im-sol3,'fro'));
0158     
0159 % RWBPDb8
0160 % It uses the solution to BPDBb8 as a warm start
0161 maxiter=10;
0162 sigma=sigma_noise*sqrt(numel(y)/(numel(im)));
0163 tol=1e-3;
0164   
0165 sol4 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi2, Psit2, param, sigma, tol, maxiter, sol3);
0166       
0167 RSNR4=20*log10(norm(im,'fro')/norm(im-sol4,'fro'));
0168 
0169 % Parameters for Curvelet
0170 
0171 % Parameters for BPDN
0172 param3.verbose = 1; % Print log or not
0173 param3.gamma = 1e-1; % Converge parameter
0174 param3.rel_obj = 5e-4; % Stopping criterion for the L1 problem
0175 param3.max_iter = 200; % Max. number of iterations for the L1 problem
0176 param3.nu_B2 = 1; % Bound on the norm of the operator A
0177 param3.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0178 param3.tight_B2 = 1; % Indicate if A is a tight frame (1) or not (0)
0179 param3.pos_B2 = 1; % Positivity constraint flag. (1) active (0) otherwise
0180 param3.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0181 
0182     
0183 
0184 
0185 % Solve BP Curvelet problem
0186     
0187 sol5 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi3, Psit3, param3);
0188     
0189 RSNR5=20*log10(norm(im,'fro')/norm(im-sol5,'fro'));
0190     
0191 % RW-Curvelet
0192 % It uses the solution to BPDBb8 as a warm start
0193 maxiter=10;
0194 sigma=sigma_noise*sqrt(numel(y)/(numel(im)));
0195 tol=1e-3;
0196   
0197 sol6 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi3, Psit3, param3, sigma, tol, maxiter, sol5);
0198      
0199 RSNR6=20*log10(norm(im,'fro')/norm(im-sol6,'fro'));
0200 
0201     
0202 % Parameters for TVDN
0203 param1.verbose = 1; % Print log or not
0204 param1.gamma = 1e-1; % Converge parameter
0205 param1.rel_obj = 5e-4; % Stopping criterion for the TVDN problem
0206 param1.max_iter = 200; % Max. number of iterations for the TVDN problem
0207 param1.max_iter_TV = 200; % Max. nb. of iter. for the sub-problem (proximal TV operator)
0208 param1.nu_B2 = 1; % Bound on the norm of the operator A
0209 param1.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0210 param1.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0211 param1.max_iter_B2 = 300;
0212 param1.pos_B2 = 1; % Positivity constraint flag. (1) active (0) otherwise
0213     
0214 % Solve TV problem
0215     
0216 sol7 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param1);
0217     
0218 RSNR7=20*log10(norm(im,'fro')/norm(im-sol7,'fro'));
0219     
0220 % RWTV
0221 % It uses the solution to TV as a warm start
0222 maxiter=10;
0223 sigma=sigma_noise*sqrt(numel(y)/(numel(im)));
0224 tol=1e-3;
0225   
0226 sol8 = sopt_mltb_solve_rwTVDN(y, epsilon, A, At, param1,sigma, tol, maxiter, sol7);
0227     
0228 RSNR8=20*log10(norm(im,'fro')/norm(im-sol8,'fro'));
0229 
0230 
0231 %Show reconstructed images
0232 
0233 figure, imagesc(sol1,[0 1]); axis image; axis off; colormap gray;
0234 title(['BPSA, SNR=',num2str(RSNR1), 'dB'])
0235 figure, imagesc(sol2,[0 1]); axis image; axis off; colormap gray;
0236 title(['SARA, SNR=',num2str(RSNR2), 'dB'])
0237 
0238 figure, imagesc(sol3,[0 1]); axis image; axis off; colormap gray;
0239 title(['BPDb8, SNR=',num2str(RSNR3), 'dB'])
0240 figure, imagesc(sol4,[0 1]); axis image; axis off; colormap gray;
0241 title(['RW- BPDb8, SNR=',num2str(RSNR4), 'dB'])
0242 
0243 figure, imagesc(sol5,[0 1]); axis image; axis off; colormap gray;
0244 title(['Curvelet, SNR=',num2str(RSNR5), 'dB'])
0245 figure, imagesc(sol6,[0 1]); axis image; axis off; colormap gray;
0246 title(['RW-Curvelet, SNR=',num2str(RSNR6), 'dB'])
0247 
0248 figure, imagesc(sol7,[0 1]); axis image; axis off; colormap gray;
0249 title(['TV, SNR=',num2str(RSNR7), 'dB'])
0250 figure, imagesc(sol8,[0 1]); axis image; axis off; colormap gray;
0251 title(['RW-TV, SNR=',num2str(RSNR8), 'dB'])
0252 
0253 
0254 
0255 
0256 
0257 
0258 
0259 
0260 
0261 
0262
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sopt-2.0.0/matlab/doc/matlab/Experiment2.html000066400000000000000000000426531277570055300210600ustar00rootroot00000000000000 Description of Experiment2
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Experiment2

PURPOSE ^

% Experiment2

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Experiment2
 In this experiment we study the performance of SARA with Gaussian random 
 matrices as measurements operators. Due to computational limitations for 
 the use of a dense sensing matrix, for this experiment we use a cropped 
 version of Lena, around the head, of dimension 128x128 as a test image. 
 Number of measurements is M = 0.3N and input SNR is set to 30 dB. These
 parameters can be changed by modifying the variables p (for the
 undersampling ratio) and input_snr (for the input SNR).

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Experiment2
0002 % In this experiment we study the performance of SARA with Gaussian random
0003 % matrices as measurements operators. Due to computational limitations for
0004 % the use of a dense sensing matrix, for this experiment we use a cropped
0005 % version of Lena, around the head, of dimension 128x128 as a test image.
0006 % Number of measurements is M = 0.3N and input SNR is set to 30 dB. These
0007 % parameters can be changed by modifying the variables p (for the
0008 % undersampling ratio) and input_snr (for the input SNR).
0009 
0010 
0011 %% Clear workspace
0012 
0013 clc
0014 clear;
0015 
0016 
0017 %% Define paths
0018 
0019 addpath misc/
0020 addpath prox_operators/
0021 addpath test_images/
0022 
0023 
0024 %% Read image
0025 
0026 imagename = 'lena_256.tiff';
0027 
0028 % Load image
0029 im1 = im2double(imread(imagename));
0030 
0031 %Crope image
0032 a=70;
0033 b=96;
0034 im=im1(a+1:a+128,b+1:b+128);
0035 
0036 % Enforce positivity
0037 im(im<0) = 0;
0038 
0039 %% Parameters
0040 
0041 input_snr = 30; % Noise level (on the measurements)
0042 
0043 %Undersampling ratio M/N
0044 p=0.3;
0045 
0046 %% Sparsity operators
0047 
0048 %Wavelet decomposition depth
0049 nlevel=4;
0050 
0051 dwtmode('per');
0052 [C,S]=wavedec2(im,nlevel,'db8'); 
0053 ncoef=length(C);
0054 [C1,S1]=wavedec2(im,nlevel,'db1'); 
0055 ncoef1=length(C1);
0056 [C2,S2]=wavedec2(im,nlevel,'db2'); 
0057 ncoef2=length(C2);
0058 [C3,S3]=wavedec2(im,nlevel,'db3'); 
0059 ncoef3=length(C3);
0060 [C4,S4]=wavedec2(im,nlevel,'db4'); 
0061 ncoef4=length(C4);
0062 [C5,S5]=wavedec2(im,nlevel,'db5'); 
0063 ncoef5=length(C5);
0064 [C6,S6]=wavedec2(im,nlevel,'db6'); 
0065 ncoef6=length(C6);
0066 [C7,S7]=wavedec2(im,nlevel,'db7'); 
0067 ncoef7=length(C7);
0068 
0069 %Sparsity averaging operator
0070 
0071 Psit = @(x) [wavedec2(x,nlevel,'db1')'; wavedec2(x,nlevel,'db2')';wavedec2(x,nlevel,'db3')';...
0072     wavedec2(x,nlevel,'db4')'; wavedec2(x,nlevel,'db5')'; wavedec2(x,nlevel,'db6')';...
0073     wavedec2(x,nlevel,'db7')';wavedec2(x,nlevel,'db8')']/sqrt(8); 
0074 
0075 Psi = @(x) (waverec2(x(1:ncoef1),S1,'db1')+waverec2(x(ncoef1+1:ncoef1+ncoef2),S2,'db2')+...
0076     waverec2(x(ncoef1+ncoef2+1:ncoef1+ncoef2+ncoef3),S3,'db3')+...
0077     waverec2(x(ncoef1+ncoef2+ncoef3+1:ncoef1+ncoef2+ncoef3+ncoef4),S4,'db4')+...
0078     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5),S5,'db5')+...
0079     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6),S6,'db6')+...
0080     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7),S7,'db7')+...
0081     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+ncoef),S,'db8'))/sqrt(8);
0082 
0083 %Db8 wavelet basis
0084 Psit2 = @(x) wavedec2(x, nlevel,'db8'); 
0085 Psi2 = @(x) waverec2(x,S,'db8');
0086 
0087 %Curvelet
0088 %CurveLab needs to be installed to run Curvelet simulations
0089 realv = 1;
0090 Cv = fdct_usfft(im,realv);
0091 Mod = sopt_mltb_struct2size(Cv);
0092 
0093 Psit3 = @(x) sopt_mltb_fwdcurvelet(x,realv); 
0094 Psi3 = @(x) sopt_mltb_adjcurvelet(x,Mod,realv);
0095 
0096 %% Gaussian matrix operator
0097 
0098 num_meas=floor(p*numel(im));
0099 
0100 Phi = randn(num_meas,numel(im))/sqrt(num_meas);
0101         
0102 bound=svds(Phi,1)^2;
0103      
0104 A = @(x) Phi*x(:);
0105     
0106 At = @(x) reshape(Phi'*x(:),size(im));
0107     
0108    
0109 % Sampling
0110 y = A(im);
0111 % Add Gaussian i.i.d. noise
0112 sigma_noise = 10^(-input_snr/20)*std(im(:));
0113 y = y + (randn(size(y)))*sigma_noise;
0114     
0115     
0116 % Tolerance on noise
0117 epsilon = sqrt(numel(y)+2*sqrt(numel(y)))*sigma_noise;
0118 epsilon_up = sqrt(numel(y)+2.1*sqrt(numel(y)))*sigma_noise;
0119     
0120     
0121 % Parameters for BPDN
0122 param.verbose = 1; % Print log or not
0123 param.gamma = 1e-1; % Converge parameter
0124 param.rel_obj = 5e-4; % Stopping criterion for the L1 problem
0125 param.max_iter = 200; % Max. number of iterations for the L1 problem
0126 param.nu_B2 = bound; % Bound on the norm of the operator A
0127 param.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0128 param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0129 param.pos_B2 = 1; %Positivity constraint: (1) active, (0) not active
0130 param.max_iter_B2=300;
0131 param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0132 param.nu_L1 = 1;
0133 param.max_iter_L1 = 20;
0134 param.rel_obj_L1 = 1e-2;
0135     
0136     
0137 % Solve BPSA problem
0138     
0139 sol1 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0140     
0141 RSNR1=20*log10(norm(im,'fro')/norm(im-sol1,'fro'));
0142     
0143 % SARA
0144 % It uses the solution to BPSA as a warm start
0145 maxiter=10;
0146 sigma=sigma_noise*sqrt(numel(y)/(numel(im)*8));
0147 tol=1e-3;
0148   
0149 sol2 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi, Psit, param, sigma, tol, maxiter, sol1);
0150 
0151 RSNR2=20*log10(norm(im,'fro')/norm(im-sol2,'fro'));
0152 
0153 % Solve BPBb8 problem
0154     
0155 sol3 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi2, Psit2, param);
0156     
0157 RSNR3=20*log10(norm(im,'fro')/norm(im-sol3,'fro'));
0158     
0159 % RWBPDb8
0160 % It uses the solution to BPDBb8 as a warm start
0161 maxiter=10;
0162 sigma=sigma_noise*sqrt(numel(y)/(numel(im)));
0163 tol=1e-3;
0164   
0165 sol4 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi2, Psit2, param, sigma, tol, maxiter, sol3);
0166       
0167 RSNR4=20*log10(norm(im,'fro')/norm(im-sol4,'fro'));
0168 
0169 % Parameters for Curvelet
0170 
0171 % Parameters for BPDN
0172 param3.verbose = 1; % Print log or not
0173 param3.gamma = 1e-1; % Converge parameter
0174 param3.rel_obj = 5e-4; % Stopping criterion for the L1 problem
0175 param3.max_iter = 200; % Max. number of iterations for the L1 problem
0176 param3.nu_B2 = bound; % Bound on the norm of the operator A
0177 param3.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0178 param3.tight_B2 = 1; % Indicate if A is a tight frame (1) or not (0)
0179 param3.pos_B2 = 1; % Positivity constraint flag. (1) active (0) otherwise
0180 param3.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0181 
0182     
0183 
0184 
0185 % Solve BP Curvelet problem
0186     
0187 sol5 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi3, Psit3, param3);
0188     
0189 RSNR5=20*log10(norm(im,'fro')/norm(im-sol5,'fro'));
0190     
0191 % RW-Curvelet
0192 % It uses the solution to BPDBb8 as a warm start
0193 maxiter=10;
0194 sigma=sigma_noise*sqrt(numel(y)/(numel(im)));
0195 tol=1e-3;
0196   
0197 sol6 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi3, Psit3, param3, sigma, tol, maxiter, sol5);
0198      
0199 RSNR6=20*log10(norm(im,'fro')/norm(im-sol6,'fro'));
0200 
0201     
0202 % Parameters for TVDN
0203 param1.verbose = 1; % Print log or not
0204 param1.gamma = 1e-1; % Converge parameter
0205 param1.rel_obj = 5e-4; % Stopping criterion for the TVDN problem
0206 param1.max_iter = 200; % Max. number of iterations for the TVDN problem
0207 param1.max_iter_TV = 200; % Max. nb. of iter. for the sub-problem (proximal TV operator)
0208 param1.nu_B2 = bound; % Bound on the norm of the operator A
0209 param1.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0210 param1.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0211 param1.max_iter_B2 = 300;
0212 param1.pos_B2 = 1;
0213     
0214 % Solve TV problem
0215     
0216 sol7 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param1);
0217     
0218 RSNR7=20*log10(norm(im,'fro')/norm(im-sol7,'fro'));
0219     
0220 % RWTV
0221 % It uses the solution to TV as a warm start
0222 maxiter=10;
0223 sigma=sigma_noise*sqrt(numel(y)/(numel(im)));
0224 tol=1e-3;
0225   
0226 sol8 = sopt_mltb_solve_rwTVDN(y, epsilon, A, At, param1,sigma, tol, maxiter, sol7);
0227     
0228 RSNR8=20*log10(norm(im,'fro')/norm(im-sol8,'fro'));
0229 
0230 %Show reconstructed images
0231 
0232 figure, imagesc(sol1,[0 1]); axis image; axis off; colormap gray;
0233 title(['BPSA, SNR=',num2str(RSNR1), 'dB'])
0234 figure, imagesc(sol2,[0 1]); axis image; axis off; colormap gray;
0235 title(['SARA, SNR=',num2str(RSNR2), 'dB'])
0236 
0237 figure, imagesc(sol3,[0 1]); axis image; axis off; colormap gray;
0238 title(['BPDb8, SNR=',num2str(RSNR3), 'dB'])
0239 figure, imagesc(sol4,[0 1]); axis image; axis off; colormap gray;
0240 title(['RW- BPDb8, SNR=',num2str(RSNR4), 'dB'])
0241 
0242 figure, imagesc(sol5,[0 1]); axis image; axis off; colormap gray;
0243 title(['Curvelet, SNR=',num2str(RSNR5), 'dB'])
0244 figure, imagesc(sol6,[0 1]); axis image; axis off; colormap gray;
0245 title(['RW-Curvelet, SNR=',num2str(RSNR6), 'dB'])
0246 
0247 figure, imagesc(sol7,[0 1]); axis image; axis off; colormap gray;
0248 title(['TV, SNR=',num2str(RSNR7), 'dB'])
0249 figure, imagesc(sol8,[0 1]); axis image; axis off; colormap gray;
0250 title(['RW-TV, SNR=',num2str(RSNR8), 'dB'])
0251 
0252 
0253 
0254 
0255 
0256 
0257 
0258 
0259 
0260 
0261
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sopt-2.0.0/matlab/doc/matlab/Experiment3.html000066400000000000000000000304561277570055300210570ustar00rootroot00000000000000 Description of Experiment3
Home > matlab > Experiment3.m

Experiment3

PURPOSE ^

% Experiement3

SYNOPSIS ^

This is a script file.

DESCRIPTION ^

% Experiement3
 In this experiment we illustrate the application of SARA to Fourier 
 imaging. We reconstruct a 224x168 positive brain image from standard 
 variable density sampling. Fourier measurements, for an undersampling 
 ratio of M = 0.05N and input SNR set to 30 dB.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 %% Experiement3
0002 % In this experiment we illustrate the application of SARA to Fourier
0003 % imaging. We reconstruct a 224x168 positive brain image from standard
0004 % variable density sampling. Fourier measurements, for an undersampling
0005 % ratio of M = 0.05N and input SNR set to 30 dB.
0006 
0007 
0008 %% Clear workspace
0009 
0010 clc
0011 clear;
0012 
0013 
0014 %% Define paths
0015 
0016 addpath misc/
0017 addpath prox_operators/
0018 addpath test_images/
0019 
0020 
0021 %% Read image
0022 
0023 % Load image
0024 load Brain_low_res
0025 im=abs(map_ref);
0026 
0027 % Normalise
0028 im = im/max(max(im));
0029 
0030 % Enforce positivity
0031 im(im<0) = 0;
0032 
0033 %% Parameters
0034 
0035 input_snr = 30; % Noise level (on the measurements)
0036 
0037 %Undersampling ratio M/N
0038 %The range of p is 0 < p <= 0.5 since the vdsmask
0039 %function samples only half plane. Therefore, full
0040 %sampling is p = 0.5
0041 p=0.05;
0042 
0043 
0044 %% Sparsity operators
0045 
0046 %Wavelet decomposition depth
0047 nlevel=4;
0048 
0049 dwtmode('per');
0050 [C,S]=wavedec2(im,nlevel,'db8'); 
0051 ncoef=length(C);
0052 [C1,S1]=wavedec2(im,nlevel,'db1'); 
0053 ncoef1=length(C1);
0054 [C2,S2]=wavedec2(im,nlevel,'db2'); 
0055 ncoef2=length(C2);
0056 [C3,S3]=wavedec2(im,nlevel,'db3'); 
0057 ncoef3=length(C3);
0058 [C4,S4]=wavedec2(im,nlevel,'db4'); 
0059 ncoef4=length(C4);
0060 [C5,S5]=wavedec2(im,nlevel,'db5'); 
0061 ncoef5=length(C5);
0062 [C6,S6]=wavedec2(im,nlevel,'db6'); 
0063 ncoef6=length(C6);
0064 [C7,S7]=wavedec2(im,nlevel,'db7'); 
0065 ncoef7=length(C7);
0066 
0067 %Sparsity averaging operator
0068 
0069 Psit = @(x) [wavedec2(x,nlevel,'db1')'; wavedec2(x,nlevel,'db2')';wavedec2(x,nlevel,'db3')';...
0070     wavedec2(x,nlevel,'db4')'; wavedec2(x,nlevel,'db5')'; wavedec2(x,nlevel,'db6')';...
0071     wavedec2(x,nlevel,'db7')';wavedec2(x,nlevel,'db8')']/sqrt(8); 
0072 
0073 Psi = @(x) (waverec2(x(1:ncoef1),S1,'db1')+waverec2(x(ncoef1+1:ncoef1+ncoef2),S2,'db2')+...
0074     waverec2(x(ncoef1+ncoef2+1:ncoef1+ncoef2+ncoef3),S3,'db3')+...
0075     waverec2(x(ncoef1+ncoef2+ncoef3+1:ncoef1+ncoef2+ncoef3+ncoef4),S4,'db4')+...
0076     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5),S5,'db5')+...
0077     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6),S6,'db6')+...
0078     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7),S7,'db7')+...
0079     waverec2(x(ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+1:ncoef1+ncoef2+ncoef3+ncoef4+ncoef5+ncoef6+ncoef7+ncoef),S,'db8'))/sqrt(8);
0080 
0081 %Db8 wavelet basis
0082 Psit2 = @(x) wavedec2(x, nlevel,'db8'); 
0083 Psi2 = @(x) waverec2(x,S,'db8');
0084 
0085 
0086 % Variable density mask
0087 mask = sopt_mltb_vdsmask(size(im,1),size(im,2),p);
0088 ind = find(mask==1);
0089 % Masking matrix (sparse matrix in matlab)
0090 Ma = sparse(1:numel(ind), ind, ones(numel(ind), 1), numel(ind), numel(im));
0091 
0092   
0093 % Composition (Masking o Fourier)
0094     
0095 A = @(x) Ma*reshape(fft2(x)/sqrt(numel(ind)), numel(x), 1);
0096 At = @(x) (ifft2(reshape(Ma'*x(:), size(im))*sqrt(numel(ind))));
0097       
0098 % Select p% of Fourier coefficients
0099 y = A(im);
0100 % Add Gaussian i.i.d. noise
0101 sigma_noise = 10^(-input_snr/20)*std(im(:));
0102 y = y + (randn(size(y)) + 1i*randn(size(y)))*sigma_noise/sqrt(2);
0103 
0104 
0105 % Tolerance on noise
0106 epsilon = sqrt(numel(y)+2*sqrt(numel(y)))*sigma_noise;
0107 epsilon_up = sqrt(numel(y)+2.1*sqrt(numel(y)))*sigma_noise;
0108 
0109 
0110 %% Parameters for BPDN
0111 param.verbose = 1; % Print log or not
0112 param.gamma = 1e-1; % Converge parameter
0113 param.rel_obj = 4e-4; % Stopping criterion for the L1 problem
0114 param.max_iter = 200; % Max. number of iterations for the L1 problem
0115 param.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0116 param.nu_B2 = 1; % Bound on the norm of the operator A
0117 param.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0118 param.pos_B2 = 1;  % Positivity constraint flag. (1) active (0) otherwise
0119 param.tight_L1 = 1; % Indicate if Psit is a tight frame (1) or not (0)
0120 param.nu_L1 = 1;
0121 param.max_iter_L1 = 20;
0122 param.rel_obj_L1 = 1e-2;
0123     
0124 % Solve SARA
0125 maxiter=10;
0126 sigma=sigma_noise*sqrt(numel(y)/(numel(im)*9));
0127 tol=1e-3;
0128 sol1 = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi, Psit, param, sigma, tol, maxiter);
0129     
0130 RSNR1=20*log10(norm(im,'fro')/norm(im-sol1,'fro'));
0131     
0132     
0133 %% Parameters for TVDN
0134 param1.verbose = 1; % Print log or not
0135 param1.gamma = 3e-1; % Converge parameter
0136 param1.rel_obj = 5e-4; % Stopping criterion for the TVDN problem
0137 param1.max_iter = 200; % Max. number of iterations for the TVDN problem
0138 param1.max_iter_TV = 200; % Max. nb. of iter. for the sub-problem (proximal TV operator)
0139 param1.nu_B2 = 1; % Bound on the norm of the operator A
0140 param1.tol_B2 = 1-(epsilon/epsilon_up); % Tolerance for the projection onto the L2-ball
0141 param1.tight_B2 = 0; % Indicate if A is a tight frame (1) or not (0)
0142 param1.max_iter_B2 = 200;
0143 param1.pos_B2 = 1; % Positivity constraint flag. (1) active (0) otherwise
0144 
0145 % Solve TVDN problem
0146 sol2 = sopt_mltb_solve_TVDN(y, epsilon, A, At, param1);
0147 
0148 RSNR2=20*log10(norm(im,'fro')/norm(im-sol2,'fro'));
0149    
0150     
0151 % Solve BPDN problem
0152 sol3 = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi2, Psit2, param);
0153 
0154 
0155 RSNR3=20*log10(norm(im,'fro')/norm(im-sol3,'fro'));
0156 
0157 figure, imagesc(sol1,[0 1]);axis image;axis off; colormap gray;
0158 title(['SARA, SNR=',num2str(RSNR1), 'dB'])
0159 figure, imagesc(sol2,[0 1]);axis image;axis off; colormap gray;
0160 title(['TV, SNR=',num2str(RSNR2), 'dB'])
0161 figure, imagesc(sol3,[0 1]);axis image;axis off; colormap gray;
0162 title(['BPDb8, SNR=',num2str(RSNR3), 'dB'])
0163 figure, imagesc(im, [0 1]); axis image; axis off; colormap gray;
0164 title('Original image')
0165     
0166     
0167     
0168     
0169     
0170     
0171     
0172 
0173 
0174 
0175 
0176 
0177 
0178 
0179 
0180 
0181 
0182 
0183
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/index.html000066400000000000000000000073541277570055300177640ustar00rootroot00000000000000 Index for Directory matlab
< Master index Index for matlab >

Index for matlab

Matlab files in this directory:

 Example_BPDN% Example_BPDN
 Example_TVDN% Example_TVDN
 Example_TVDN2% Example_TVDN2
 Example_TVDNoA% Example TVDNoA
 Example_weighted_BPDN% Exampled_weighted_L1
 Example_weighted_TVDN% Exampled_weighted_TVDN
 Experiment1% Experiment1
 Experiment2% Experiment2
 Experiment3% Experiement3
 sopt_mltb_solve_BPDNsopt_mltb_solve_BPDN - Solve BPDN problem
 sopt_mltb_solve_L2DNsopt_mltb_solve_L2DN - Solve L2DN problem.
 sopt_mltb_solve_TVDNsopt_mltb_solve_TVDN - Solve TVDN problem
 sopt_mltb_solve_TVDNoAsopt_mltb_solve_TVDNoA - Solve augmented TVDN problem
 sopt_mltb_solve_rwBPDNsopt_mltb_solve_rwBPDN - Solve reweighted BPDN problem
 sopt_mltb_solve_rwTVDNsopt_mltb_solve_rwTVDN - Solve reweighted TVDN problem

Subsequent directories:

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sopt-2.0.0/matlab/doc/matlab/misc/000077500000000000000000000000001277570055300167115ustar00rootroot00000000000000sopt-2.0.0/matlab/doc/matlab/misc/.svn/000077500000000000000000000000001277570055300175755ustar00rootroot00000000000000sopt-2.0.0/matlab/doc/matlab/misc/.svn/all-wcprops000066400000000000000000000036021277570055300217640ustar00rootroot00000000000000K 25 svn:wc:ra_dav:version-url V 51 /svn/sopt/!svn/ver/231/trunk/matlab/doc/matlab/misc END sopt_mltb_fwdcurvelet.html K 25 svn:wc:ra_dav:version-url V 78 /svn/sopt/!svn/ver/231/trunk/matlab/doc/matlab/misc/sopt_mltb_fwdcurvelet.html END sopt_mltb_gradient_op.html K 25 svn:wc:ra_dav:version-url V 78 /svn/sopt/!svn/ver/231/trunk/matlab/doc/matlab/misc/sopt_mltb_gradient_op.html END sopt_mltb_div_op.html K 25 svn:wc:ra_dav:version-url V 73 /svn/sopt/!svn/ver/231/trunk/matlab/doc/matlab/misc/sopt_mltb_div_op.html END sopt_mltb_gradient_op_sphere.html K 25 svn:wc:ra_dav:version-url V 85 /svn/sopt/!svn/ver/231/trunk/matlab/doc/matlab/misc/sopt_mltb_gradient_op_sphere.html END sopt_mltb_modifypdf.html K 25 svn:wc:ra_dav:version-url V 76 /svn/sopt/!svn/ver/231/trunk/matlab/doc/matlab/misc/sopt_mltb_modifypdf.html END sopt_mltb_struct2size.html K 25 svn:wc:ra_dav:version-url V 78 /svn/sopt/!svn/ver/231/trunk/matlab/doc/matlab/misc/sopt_mltb_struct2size.html END sopt_mltb_SNR.html K 25 svn:wc:ra_dav:version-url V 70 /svn/sopt/!svn/ver/231/trunk/matlab/doc/matlab/misc/sopt_mltb_SNR.html END index.html K 25 svn:wc:ra_dav:version-url V 62 /svn/sopt/!svn/ver/231/trunk/matlab/doc/matlab/misc/index.html END sopt_mltb_TV_norm.html K 25 svn:wc:ra_dav:version-url V 74 /svn/sopt/!svn/ver/231/trunk/matlab/doc/matlab/misc/sopt_mltb_TV_norm.html END sopt_mltb_genmask.html K 25 svn:wc:ra_dav:version-url V 74 /svn/sopt/!svn/ver/231/trunk/matlab/doc/matlab/misc/sopt_mltb_genmask.html END sopt_mltb_div_op_sphere.html K 25 svn:wc:ra_dav:version-url V 80 /svn/sopt/!svn/ver/231/trunk/matlab/doc/matlab/misc/sopt_mltb_div_op_sphere.html END sopt_mltb_adjcurvelet.html K 25 svn:wc:ra_dav:version-url V 78 /svn/sopt/!svn/ver/231/trunk/matlab/doc/matlab/misc/sopt_mltb_adjcurvelet.html END sopt_mltb_vdsmask.html K 25 svn:wc:ra_dav:version-url V 74 /svn/sopt/!svn/ver/231/trunk/matlab/doc/matlab/misc/sopt_mltb_vdsmask.html END sopt-2.0.0/matlab/doc/matlab/misc/.svn/entries000066400000000000000000000047241277570055300212000ustar00rootroot0000000000000010 dir 238 https://svn.epfl.ch/svn/sopt/trunk/matlab/doc/matlab/misc https://svn.epfl.ch/svn/sopt 2013-02-22T15:55:11.317825Z 231 jason.mcewen@ucl.ac.uk 970ed402-9328-4b70-8940-a90ce832aee1 sopt_mltb_fwdcurvelet.html file 2013-02-22T16:01:56.000000Z 61a0afe7e1051dc663bd813e1946c991 2013-02-22T15:55:11.317825Z 231 jason.mcewen@ucl.ac.uk 4018 sopt_mltb_gradient_op.html file 2013-02-22T16:01:56.000000Z e9e53c757aa0220be1de8391a45e1c8e 2013-02-22T15:55:11.317825Z 231 jason.mcewen@ucl.ac.uk 4290 sopt_mltb_div_op.html file 2013-02-22T16:01:56.000000Z f767067fa21bf8a7dd2648939365f06d 2013-02-22T15:55:11.317825Z 231 jason.mcewen@ucl.ac.uk 4153 sopt_mltb_gradient_op_sphere.html file 2013-02-22T16:01:56.000000Z b0006e097fcc454e00f339ca786d607b 2013-02-22T15:55:11.317825Z 231 jason.mcewen@ucl.ac.uk 5896 sopt_mltb_modifypdf.html file 2013-02-22T16:01:56.000000Z e697c0b0875c1de406230cb54cc36181 2013-02-22T15:55:11.317825Z 231 jason.mcewen@ucl.ac.uk 4983 sopt_mltb_struct2size.html file 2013-02-22T16:01:56.000000Z 60316959c86e3778d036f3a6df6363da 2013-02-22T15:55:11.317825Z 231 jason.mcewen@ucl.ac.uk 3818 sopt_mltb_SNR.html file 2013-02-22T16:01:56.000000Z 86b350223ad23e2d92d3fe209cc3e133 2013-02-22T15:55:11.317825Z 231 jason.mcewen@ucl.ac.uk 3975 index.html file 2013-02-22T16:01:56.000000Z b0f89bce4130abc4f87f642aa8b78fe0 2013-02-22T15:55:11.317825Z 231 jason.mcewen@ucl.ac.uk 3461 sopt_mltb_TV_norm.html file 2013-02-22T16:01:56.000000Z d14c505596d34092576f2f6ee3239888 2013-02-22T15:55:11.317825Z 231 jason.mcewen@ucl.ac.uk 6268 sopt_mltb_genmask.html file 2013-02-22T16:01:56.000000Z 93760df61086aca94f64462a975b7618 2013-02-22T15:55:11.317825Z 231 jason.mcewen@ucl.ac.uk 4002 sopt_mltb_div_op_sphere.html file 2013-02-22T16:01:56.000000Z f5bb5a2ef42036cc1fa9f6303946661e 2013-02-22T15:55:11.317825Z 231 jason.mcewen@ucl.ac.uk 5252 sopt_mltb_adjcurvelet.html file 2013-02-22T16:01:56.000000Z 7b04f5f1d1c1efc204e0be9b6f9f351a 2013-02-22T15:55:11.317825Z 231 jason.mcewen@ucl.ac.uk 4614 sopt_mltb_vdsmask.html file 2013-02-22T16:01:56.000000Z 31d41f1ec21b4ed34cf5908ed92f5048 2013-02-22T15:55:11.317825Z 231 jason.mcewen@ucl.ac.uk 6317 sopt-2.0.0/matlab/doc/matlab/misc/.svn/text-base/000077500000000000000000000000001277570055300214715ustar00rootroot00000000000000sopt-2.0.0/matlab/doc/matlab/misc/.svn/text-base/index.html.svn-base000066400000000000000000000066051277570055300252120ustar00rootroot00000000000000 Index for Directory matlab/misc
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Index for matlab/misc

Matlab files in this directory:

 sopt_mltb_SNRSNR - Compute the SNR between two images
 sopt_mltb_TV_normsopt_mltb_TV_norm - Compute TV norm
 sopt_mltb_adjcurveletsopt_mltb_adjcurvelet - Adjoint curvelet transform
 sopt_mltb_div_opsopt_mltb_div_op - Compute divergence
 sopt_mltb_div_op_spheresopt_mltb_div_op_sphere - Compute divergence on sphere
 sopt_mltb_fwdcurveletsopt_mltb_fwdcurvelet - Forward curvelet transform
 sopt_mltb_genmasksopt_mltb_genmask - Generate mask
 sopt_mltb_gradient_opsopt_mltb_gradient_op - Compute gradient
 sopt_mltb_gradient_op_spheresopt_mltb_gradient_op_sphere - Compute gradient on sphere
 sopt_mltb_modifypdfsopt_mltb_modifypdf - Modify PDF of sampling profile
 sopt_mltb_struct2sizesopt_mltb_struct2size - Compute curvelet data structure
 sopt_mltb_vdsmasksopt_mltb_vdsmask - Create variable density sampling profile
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sopt-2.0.0/matlab/doc/matlab/misc/.svn/text-base/sopt_mltb_SNR.html.svn-base000066400000000000000000000076071277570055300266330ustar00rootroot00000000000000 Description of sopt_mltb_SNR
Home > matlab > misc > sopt_mltb_SNR.m

sopt_mltb_SNR

PURPOSE ^

SNR - Compute the SNR between two images

SYNOPSIS ^

function snr = sopt_mltb_SNR(map_init, map_recon)

DESCRIPTION ^

 SNR - Compute the SNR between two images
 
 C omputes the SNR between the maps map_init and map_recon.  The SNR is 
 computed by
    10 * log10( var(MAP_INIT) / var(MAP_INIT-MAP_NOISY) )
  where var stands for the matlab built-in function that computes the
  variance.

 Inputs:

   - map_init: Initial image.

   - map_recon: Reconstructed image.

 Outputs:

   - snr: SNR.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function snr = sopt_mltb_SNR(map_init, map_recon)
0002 % SNR - Compute the SNR between two images
0003 %
0004 % C omputes the SNR between the maps map_init and map_recon.  The SNR is
0005 % computed by
0006 %    10 * log10( var(MAP_INIT) / var(MAP_INIT-MAP_NOISY) )
0007 %  where var stands for the matlab built-in function that computes the
0008 %  variance.
0009 %
0010 % Inputs:
0011 %
0012 %   - map_init: Initial image.
0013 %
0014 %   - map_recon: Reconstructed image.
0015 %
0016 % Outputs:
0017 %
0018 %   - snr: SNR.
0019 
0020 noise = map_init(:)-map_recon(:);
0021 var_init = var(map_init(:));
0022 var_den = var(noise(:));
0023 snr = 10 * log10(var_init/var_den);
0024 
0025 end
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sopt-2.0.0/matlab/doc/matlab/misc/.svn/text-base/sopt_mltb_TV_norm.html.svn-base000066400000000000000000000141741277570055300275520ustar00rootroot00000000000000 Description of sopt_mltb_TV_norm
Home > matlab > misc > sopt_mltb_TV_norm.m

sopt_mltb_TV_norm

PURPOSE ^

sopt_mltb_TV_norm - Compute TV norm

SYNOPSIS ^

function y = sopt_mltb_TV_norm(u, sphere_flag,incNP, weights_dx, weights_dy)

DESCRIPTION ^

 sopt_mltb_TV_norm - Compute TV norm

 Compute the TV of an image on the plane or sphere.

 Inputs:

   - u: Image to compute TV norm of.

   - sphere_flag: Flag indicating whether to compute the TV norm on the
       sphere (1 = on sphere, 2 = on plane).

   - includeNP: Flag indicating whether the North pole is included
       in the sampling grid (1 = North pole included, 0 = North pole not
       included).

   - weights_dx: Weights in the x (phi) direction.

   - weights_dy: Weights in the y (theta) direction.

 Outputs:

   - y: TV norm of image.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function y = sopt_mltb_TV_norm(u, sphere_flag, ...
0002     incNP, weights_dx, weights_dy)
0003 % sopt_mltb_TV_norm - Compute TV norm
0004 %
0005 % Compute the TV of an image on the plane or sphere.
0006 %
0007 % Inputs:
0008 %
0009 %   - u: Image to compute TV norm of.
0010 %
0011 %   - sphere_flag: Flag indicating whether to compute the TV norm on the
0012 %       sphere (1 = on sphere, 2 = on plane).
0013 %
0014 %   - includeNP: Flag indicating whether the North pole is included
0015 %       in the sampling grid (1 = North pole included, 0 = North pole not
0016 %       included).
0017 %
0018 %   - weights_dx: Weights in the x (phi) direction.
0019 %
0020 %   - weights_dy: Weights in the y (theta) direction.
0021 %
0022 % Outputs:
0023 %
0024 %   - y: TV norm of image.
0025 
0026 if sphere_flag
0027     if nargin>3 
0028         [dx, dy] = sopt_mltb_gradient_op_sphere(u, incNP, weights_dx, weights_dy);
0029     else
0030         [dx, dy] = sopt_mltb_gradient_op_sphere(u, incNP);
0031     end
0032 else
0033     if nargin>3 
0034         [dx, dy] = sopt_mltb_gradient_op(u, weights_dx, weights_dy);
0035     else
0036         [dx, dy] = sopt_mltb_gradient_op(u);
0037     end
0038 end
0039 temp = sqrt(abs(dx).^2 + abs(dy).^2);
0040 y = sum(temp(:));
0041 
0042 end
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sopt-2.0.0/matlab/doc/matlab/misc/.svn/text-base/sopt_mltb_adjcurvelet.html.svn-base000066400000000000000000000110061277570055300304650ustar00rootroot00000000000000 Description of sopt_mltb_adjcurvelet
Home > matlab > misc > sopt_mltb_adjcurvelet.m

sopt_mltb_adjcurvelet

PURPOSE ^

sopt_mltb_adjcurvelet - Adjoint curvelet transform

SYNOPSIS ^

function restim = sopt_mltb_adjcurvelet(coef, Mod, real)

DESCRIPTION ^

 sopt_mltb_adjcurvelet - Adjoint curvelet transform

 Compute the adjoint curvelet transform from the curvelet
 coefficient vector.

 Inputs:

   - coef: Input curvelet coefficient vector.

   - Mod: Data structure that stores sizes of the curvelet transform 
       generated from the CurveLab toolbox.

   - real: Flag indicating if the transform is real or complex (1 = real,  
       0 = complex).

 Outputs:

   - restim: Estimated image from the Curvelet coefficients.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function restim = sopt_mltb_adjcurvelet(coef, Mod, real)
0002 % sopt_mltb_adjcurvelet - Adjoint curvelet transform
0003 %
0004 % Compute the adjoint curvelet transform from the curvelet
0005 % coefficient vector.
0006 %
0007 % Inputs:
0008 %
0009 %   - coef: Input curvelet coefficient vector.
0010 %
0011 %   - Mod: Data structure that stores sizes of the curvelet transform
0012 %       generated from the CurveLab toolbox.
0013 %
0014 %   - real: Flag indicating if the transform is real or complex (1 = real,
0015 %       0 = complex).
0016 %
0017 % Outputs:
0018 %
0019 %   - restim: Estimated image from the Curvelet coefficients.
0020 
0021 CR=cell(size(Mod));
0022 marker=0;
0023 
0024 for s=1:length(Mod)
0025    CR{s} = cell(size(Mod{s}));
0026    for w=1:length(Mod{s})
0027      m=Mod{s}{w}(1);
0028      n=Mod{s}{w}(2);
0029      CR{s}{w}=reshape(coef(marker+1:marker+(m*n)),m,n);
0030      marker=marker+m*n;
0031    end
0032 end
0033  
0034 % Apply adjoint curvelet transform
0035 restim = ifdct_usfft(CR,real);
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sopt-2.0.0/matlab/doc/matlab/misc/.svn/text-base/sopt_mltb_div_op.html.svn-base000066400000000000000000000100711277570055300274360ustar00rootroot00000000000000 Description of sopt_mltb_div_op
Home > matlab > misc > sopt_mltb_div_op.m

sopt_mltb_div_op

PURPOSE ^

sopt_mltb_div_op - Compute divergence

SYNOPSIS ^

function I = sopt_mltb_div_op(dx, dy, weights_dx, weights_dy)

DESCRIPTION ^

 sopt_mltb_div_op - Compute divergence

 Compute the divergence (adjoint of the gradient) of a two dimensional
 signal from the horizontal and vertical gradients.

 Inputs:

   - dx: Gradient in x.

   - dy: Gradient in y.

   - weights_dx: Weights in the x direction.

   - weights_dy: Weights in the y direction.

 Outputs:

   - I: Divergence.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function I = sopt_mltb_div_op(dx, dy, weights_dx, weights_dy)
0002 % sopt_mltb_div_op - Compute divergence
0003 %
0004 % Compute the divergence (adjoint of the gradient) of a two dimensional
0005 % signal from the horizontal and vertical gradients.
0006 %
0007 % Inputs:
0008 %
0009 %   - dx: Gradient in x.
0010 %
0011 %   - dy: Gradient in y.
0012 %
0013 %   - weights_dx: Weights in the x direction.
0014 %
0015 %   - weights_dy: Weights in the y direction.
0016 %
0017 % Outputs:
0018 %
0019 %   - I: Divergence.
0020 
0021 if nargin > 2
0022     dx = dx .* conj(weights_dx);
0023     dy = dy .* conj(weights_dy);
0024 end
0025 
0026 I = [dx(1, :) ; dx(2:end-1, :)-dx(1:end-2, :) ; -dx(end-1, :)];
0027 I = I + [dy(:, 1) , dy(:, 2:end-1)-dy(:, 1:end-2) , -dy(:, end-1)];
0028 
0029 end
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sopt-2.0.0/matlab/doc/matlab/misc/.svn/text-base/sopt_mltb_div_op_sphere.html.svn-base000066400000000000000000000122041277570055300310040ustar00rootroot00000000000000 Description of sopt_mltb_div_op_sphere
Home > matlab > misc > sopt_mltb_div_op_sphere.m

sopt_mltb_div_op_sphere

PURPOSE ^

sopt_mltb_div_op_sphere - Compute divergence on sphere

SYNOPSIS ^

function I = sopt_mltb_div_op_sphere(dx, dy, includeNorthpole,weights_dx, weights_dy)

DESCRIPTION ^

 sopt_mltb_div_op_sphere - Compute divergence on sphere

 Compute the divergence (adjoint of the gradient) of a two dimensional
 signal on the sphere.  The phi direction (x) is periodic, while the theta
 direction (y) is not.

 Inputs:

   - dx: Gradient in x.

   - dy: Gradient in y.

   - includeNorthPole: Flag indicating whether the North pole is included
       in the sampling grid (1 = North pole included, 0 = North pole not
       included).

   - weights_dx: Weights in the x (phi) direction.

   - weights_dy: Weights in the y (theta) direction.

 Outputs:

   - I: Divergence.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function I = sopt_mltb_div_op_sphere(dx, dy, includeNorthpole, ...
0002   weights_dx, weights_dy)
0003 % sopt_mltb_div_op_sphere - Compute divergence on sphere
0004 %
0005 % Compute the divergence (adjoint of the gradient) of a two dimensional
0006 % signal on the sphere.  The phi direction (x) is periodic, while the theta
0007 % direction (y) is not.
0008 %
0009 % Inputs:
0010 %
0011 %   - dx: Gradient in x.
0012 %
0013 %   - dy: Gradient in y.
0014 %
0015 %   - includeNorthPole: Flag indicating whether the North pole is included
0016 %       in the sampling grid (1 = North pole included, 0 = North pole not
0017 %       included).
0018 %
0019 %   - weights_dx: Weights in the x (phi) direction.
0020 %
0021 %   - weights_dy: Weights in the y (theta) direction.
0022 %
0023 % Outputs:
0024 %
0025 %   - I: Divergence.
0026 
0027 if nargin > 3
0028     dx = dx .* conj(weights_dx);
0029     dy = dy .* conj(weights_dy);
0030 end
0031 if(includeNorthpole)
0032     I = [zeros(1, size(dx,2)) ; dx(2, :); dx(3:end-1, :)-dx(2:end-2, :); -dx(end-1, :)];
0033     I = I + [dy(:, 1) - dy(:,end) , dy(:, 2:end)-dy(:, 1:end-1)];
0034 else
0035     I = [dx(1, :) ; dx(2:end-1, :)-dx(1:end-2, :) ; -dx(end-1, :)];
0036     I = I + [dy(:, 1) - dy(:,end) , dy(:, 2:end)-dy(:, 1:end-1)];
0037 end
0038 end
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sopt-2.0.0/matlab/doc/matlab/misc/.svn/text-base/sopt_mltb_fwdcurvelet.html.svn-base000066400000000000000000000076621277570055300305240ustar00rootroot00000000000000 Description of sopt_mltb_fwdcurvelet
Home > matlab > misc > sopt_mltb_fwdcurvelet.m

sopt_mltb_fwdcurvelet

PURPOSE ^

sopt_mltb_fwdcurvelet - Forward curvelet transform

SYNOPSIS ^

function coef = sopt_mltb_fwdcurvelet(im,real)

DESCRIPTION ^

 sopt_mltb_fwdcurvelet - Forward curvelet transform

 Compute the forward curvelet transform of an image and stores it in
 vector coef.

 Inputs:

   - im: Input image.

   - real: Flag indicating if the transform is real or complex (1 = real,  
       0 = complex).

 Outputs:

   - coef: Curvelet coefficients.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function coef = sopt_mltb_fwdcurvelet(im,real)
0002 % sopt_mltb_fwdcurvelet - Forward curvelet transform
0003 %
0004 % Compute the forward curvelet transform of an image and stores it in
0005 % vector coef.
0006 %
0007 % Inputs:
0008 %
0009 %   - im: Input image.
0010 %
0011 %   - real: Flag indicating if the transform is real or complex (1 = real,
0012 %       0 = complex).
0013 %
0014 % Outputs:
0015 %
0016 %   - coef: Curvelet coefficients.
0017 
0018 C = fdct_usfft(im,real); 
0019 
0020 % Compute vector version of curvelets coeffcients
0021 coef = [];
0022 
0023 for s=1:length(C)
0024   for w=1:length(C{s})
0025     A = C{s}{w};
0026     coef= [coef; A(:)];
0027   end
0028 end
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sopt-2.0.0/matlab/doc/matlab/misc/.svn/text-base/sopt_mltb_genmask.html.svn-base000066400000000000000000000076421277570055300276150ustar00rootroot00000000000000 Description of sopt_mltb_genmask
Home > matlab > misc > sopt_mltb_genmask.m

sopt_mltb_genmask

PURPOSE ^

sopt_mltb_genmask - Generate mask

SYNOPSIS ^

function mask = sopt_mltb_genmask(pdf, seed)

DESCRIPTION ^

 sopt_mltb_genmask - Generate mask

 Generate a binary mask with variable density sampling. It is used
 in sopt_mltb_vdsmask in the generation of variable density sampling
 profiles.

 Inputs:

   - pdf: Sampling profile function.

   - seed: Seed for the random number generator. Optional.

 Outputs:

   - mask: Binary mask.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function mask = sopt_mltb_genmask(pdf, seed)
0002 % sopt_mltb_genmask - Generate mask
0003 %
0004 % Generate a binary mask with variable density sampling. It is used
0005 % in sopt_mltb_vdsmask in the generation of variable density sampling
0006 % profiles.
0007 %
0008 % Inputs:
0009 %
0010 %   - pdf: Sampling profile function.
0011 %
0012 %   - seed: Seed for the random number generator. Optional.
0013 %
0014 % Outputs:
0015 %
0016 %   - mask: Binary mask.
0017 
0018 if nargin==2
0019     rand('seed', seed);
0020 end
0021 
0022 mask = rand(size(pdf))<pdf;
0023
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Home > matlab > misc > sopt_mltb_gradient_op.m

sopt_mltb_gradient_op

PURPOSE ^

sopt_mltb_gradient_op - Compute gradient

SYNOPSIS ^

function [dx, dy] = sopt_mltb_gradient_op(I, weights_dx, weights_dy)

DESCRIPTION ^

 sopt_mltb_gradient_op - Compute gradient

 Compute the gradient of a two dimensional signal.

 Inputs:

   - I: Input two dimesional signal.

   - weights_dx: Weights in the x direction.

   - weights_dy: Weights in the y direction.

 Outputs:
 
   - dx: Gradient in x direction.
 
   - dy: Gradient in y direction.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function [dx, dy] = sopt_mltb_gradient_op(I, weights_dx, weights_dy)
0002 % sopt_mltb_gradient_op - Compute gradient
0003 %
0004 % Compute the gradient of a two dimensional signal.
0005 %
0006 % Inputs:
0007 %
0008 %   - I: Input two dimesional signal.
0009 %
0010 %   - weights_dx: Weights in the x direction.
0011 %
0012 %   - weights_dy: Weights in the y direction.
0013 %
0014 % Outputs:
0015 %
0016 %   - dx: Gradient in x direction.
0017 %
0018 %   - dy: Gradient in y direction.
0019 
0020 dx = [I(2:end, :)-I(1:end-1, :) ; zeros(1, size(I, 2))];
0021 dy = [I(:, 2:end)-I(:, 1:end-1) , zeros(size(I, 1), 1)];
0022 
0023 if nargin>1
0024     dx = dx .* weights_dx;
0025     dy = dy .* weights_dy;
0026 end
0027 
0028 end
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sopt-2.0.0/matlab/doc/matlab/misc/.svn/text-base/sopt_mltb_gradient_op_sphere.html.svn-base000066400000000000000000000134101277570055300320170ustar00rootroot00000000000000 Description of sopt_mltb_gradient_op_sphere
Home > matlab > misc > sopt_mltb_gradient_op_sphere.m

sopt_mltb_gradient_op_sphere

PURPOSE ^

sopt_mltb_gradient_op_sphere - Compute gradient on sphere

SYNOPSIS ^

function [dx, dy] = sopt_mltb_gradient_op_sphere(I, includeNorthpole,weights_dx, weights_dy)

DESCRIPTION ^

 sopt_mltb_gradient_op_sphere - Compute gradient on sphere

 Compute the gradientof a signal on the sphere.  The phi direction (x) is
 periodic, while the theta direction (y) is not.

 Inputs:

   - I: Input two dimesional signal on the sphere.

   - includeNorthPole: Flag indicating whether the North pole is included
       in the sampling grid (1 = North pole included, 0 = North pole not
       included).

   - weights_dx: Weights in the x (phi) direction.

   - weights_dy: Weights in the y (theta) direction.

 Outputs:
 
   - dx: Gradient in x (phi) direction.
 
   - dy: Gradient in y (theta) direction.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function [dx, dy] = sopt_mltb_gradient_op_sphere(I, includeNorthpole, ...
0002   weights_dx, weights_dy)
0003 % sopt_mltb_gradient_op_sphere - Compute gradient on sphere
0004 %
0005 % Compute the gradientof a signal on the sphere.  The phi direction (x) is
0006 % periodic, while the theta direction (y) is not.
0007 %
0008 % Inputs:
0009 %
0010 %   - I: Input two dimesional signal on the sphere.
0011 %
0012 %   - includeNorthPole: Flag indicating whether the North pole is included
0013 %       in the sampling grid (1 = North pole included, 0 = North pole not
0014 %       included).
0015 %
0016 %   - weights_dx: Weights in the x (phi) direction.
0017 %
0018 %   - weights_dy: Weights in the y (theta) direction.
0019 %
0020 % Outputs:
0021 %
0022 %   - dx: Gradient in x (phi) direction.
0023 %
0024 %   - dy: Gradient in y (theta) direction.
0025 
0026 dx = zeros(size(I, 1),size(I, 2));
0027 I_big = zeros(size(I, 1)+2,size(I, 2));
0028 I_big(2:end-1, 1:end) = I;
0029 
0030 % Theta direction
0031 if(includeNorthpole)
0032     dx = [I(2:end, :)-I(1:end-1, :) ; zeros(1, size(I, 2))];
0033     dx(1,:) = zeros();
0034 else
0035     dx = [I(2:end, :)-I(1:end-1, :) ; zeros(1, size(I, 2))];
0036 end
0037 
0038 % Phi direction
0039 if(includeNorthpole)
0040     dy = [I(:, 2:end)-I(:, 1:end-1) , I(:, 1)-I(:, end)];
0041 else
0042     dy = [I(:, 2:end)-I(:, 1:end-1) , I(:, 1)-I(:, end)];
0043 end
0044 
0045 if nargin>2
0046     dx = dx .* weights_dx;
0047     dy = dy .* weights_dy;
0048 end
0049 
0050 end
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sopt-2.0.0/matlab/doc/matlab/misc/.svn/text-base/sopt_mltb_modifypdf.html.svn-base000066400000000000000000000115671277570055300301520ustar00rootroot00000000000000 Description of sopt_mltb_modifypdf
Home > matlab > misc > sopt_mltb_modifypdf.m

sopt_mltb_modifypdf

PURPOSE ^

sopt_mltb_modifypdf - Modify PDF of sampling profile

SYNOPSIS ^

function [new_pdf, alpha] = sopt_mltb_modifypdf(pdf, nb_meas)

DESCRIPTION ^

 sopt_mltb_modifypdf - Modify PDF of sampling profile
 
 Checks PDF of the sampling profile and normalizes it. It is used
 in sopt_mltb_vdsmask in the generation of variable density sampling
 profiles.

 Inputs:

   - pdf: Sampling profile function.

   - nb_meas: Number of measurements.

 Outputs:

   - new_pdf: New sampling profile function.

   - alpha: DC level for the required number of samples.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function [new_pdf, alpha] = sopt_mltb_modifypdf(pdf, nb_meas)
0002 % sopt_mltb_modifypdf - Modify PDF of sampling profile
0003 %
0004 % Checks PDF of the sampling profile and normalizes it. It is used
0005 % in sopt_mltb_vdsmask in the generation of variable density sampling
0006 % profiles.
0007 %
0008 % Inputs:
0009 %
0010 %   - pdf: Sampling profile function.
0011 %
0012 %   - nb_meas: Number of measurements.
0013 %
0014 % Outputs:
0015 %
0016 %   - new_pdf: New sampling profile function.
0017 %
0018 %   - alpha: DC level for the required number of samples.
0019 
0020 if sum(pdf(:)<0)>0
0021     error('PDF contains negative values');
0022 end
0023 pdf = pdf/max(pdf(:));
0024 
0025 % Find alpha
0026 alpha_min = -1; alpha_max = 1;
0027 while 1
0028     alpha = (alpha_min + alpha_max)/2;
0029     new_pdf = pdf + alpha;
0030     new_pdf(new_pdf>1) = 1; new_pdf(new_pdf<0) = 0;
0031     M = round(sum(new_pdf(:)));
0032     if M > nb_meas
0033         alpha_max = alpha;
0034     elseif M < nb_meas
0035         alpha_min = alpha;
0036     else
0037         break;
0038     end
0039 end
0040
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sopt-2.0.0/matlab/doc/matlab/misc/.svn/text-base/sopt_mltb_struct2size.html.svn-base000066400000000000000000000073521277570055300304670ustar00rootroot00000000000000 Description of sopt_mltb_struct2size
Home > matlab > misc > sopt_mltb_struct2size.m

sopt_mltb_struct2size

PURPOSE ^

sopt_mltb_struct2size - Compute curvelet data structure

SYNOPSIS ^

function Mod = sopt_mltb_struct2size(C)

DESCRIPTION ^

sopt_mltb_struct2size - Compute curvelet data structure

 Compute data structure to store the sizes of the curvelet transform 
 generated from the CurveLab toolbox.

 Inputs:

   - C: Curvelet coefficients arranged in the original format.

 Outputs:

   - Mod: Data structure with the appropiate sizes.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function Mod = sopt_mltb_struct2size(C)
0002 %sopt_mltb_struct2size - Compute curvelet data structure
0003 %
0004 % Compute data structure to store the sizes of the curvelet transform
0005 % generated from the CurveLab toolbox.
0006 %
0007 % Inputs:
0008 %
0009 %   - C: Curvelet coefficients arranged in the original format.
0010 %
0011 % Outputs:
0012 %
0013 %   - Mod: Data structure with the appropiate sizes.
0014 
0015  Mod=cell(size(C));
0016  for s=1:length(C)
0017    Mod{s} = cell(size(C{s}));
0018    for w=1:length(C{s})
0019      [m,n]=size(C{s}{w});
0020      Mod{s}{w}=[m,n];
0021    end
0022  end
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sopt-2.0.0/matlab/doc/matlab/misc/.svn/text-base/sopt_mltb_vdsmask.html.svn-base000066400000000000000000000142551277570055300276360ustar00rootroot00000000000000 Description of sopt_mltb_vdsmask
Home > matlab > misc > sopt_mltb_vdsmask.m

sopt_mltb_vdsmask

PURPOSE ^

sopt_mltb_vdsmask - Create variable density sampling profile

SYNOPSIS ^

function mask = sopt_mltb_vdsmask(N,M,p)

DESCRIPTION ^

 sopt_mltb_vdsmask - Create variable density sampling profile

 Creates a binary mask generated from a variable density sampling 
 profile for two dimensional images in the frequency domain.   The mask 
 contains on average p*N*M ones's.

 Inputs:

   - N and M: Size of mask.

   - p: Coverage percentage.

 Note: The undersampling ratio is passed as 2*p and then the function
 only takes samples in half plane to account for signal reality. The range
 of p is 0 < p <= 0.5.

 Outputs:

   - mask: Binary mask with the sampling profile in the frequency domain.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function mask = sopt_mltb_vdsmask(N,M,p)
0002 % sopt_mltb_vdsmask - Create variable density sampling profile
0003 %
0004 % Creates a binary mask generated from a variable density sampling
0005 % profile for two dimensional images in the frequency domain.   The mask
0006 % contains on average p*N*M ones's.
0007 %
0008 % Inputs:
0009 %
0010 %   - N and M: Size of mask.
0011 %
0012 %   - p: Coverage percentage.
0013 %
0014 % Note: The undersampling ratio is passed as 2*p and then the function
0015 % only takes samples in half plane to account for signal reality. The range
0016 % of p is 0 < p <= 0.5.
0017 %
0018 % Outputs:
0019 %
0020 %   - mask: Binary mask with the sampling profile in the frequency domain.
0021 
0022 p = 2*p;
0023 
0024 if p==1
0025     mask=ones(N,M);
0026 else
0027     
0028     nb_meas=round(p*N*M);
0029     tol=ceil(p*N*M)-floor(p*N*M);
0030     d=1;
0031 
0032     [x,y] = meshgrid(linspace(-1, 1, M), linspace(-1, 1, N)); % Cartesian grid
0033     r = sqrt(x.^2+y.^2); r = r/max(r(:)); % Polar grid
0034 
0035     alpha=-1;
0036     it=0;
0037     maxit=20;
0038     while (alpha<-0.01 || alpha>0.01) && it<maxit
0039         pdf = (1-r).^d;
0040         [new_pdf,alpha] = sopt_mltb_modifypdf(pdf, nb_meas);
0041         if alpha<0
0042             d=d+0.1;
0043         else
0044             d=d-0.1;
0045         end
0046         it=it+1;
0047     end
0048 
0049     mask = zeros(size(new_pdf));
0050     while sum(mask(:))>nb_meas+tol || sum(mask(:))<nb_meas-tol
0051         mask = sopt_mltb_genmask(new_pdf);
0052     end
0053 
0054 end
0055 
0056 %Samples in half plane to account for signal reality.
0057 M1=floor(M/2);
0058 mask(:,1:M1)=0;
0059 mask = ifftshift(mask);
0060 
0061 N1=floor(N/2);
0062 mask(N1+1:N,1)=0;
0063
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sopt-2.0.0/matlab/doc/matlab/misc/index.html000066400000000000000000000066051277570055300207150ustar00rootroot00000000000000 Index for Directory matlab/misc
< Master index Index for matlab/misc >

Index for matlab/misc

Matlab files in this directory:

 sopt_mltb_SNRSNR - Compute the SNR between two images
 sopt_mltb_TV_normsopt_mltb_TV_norm - Compute TV norm
 sopt_mltb_adjcurveletsopt_mltb_adjcurvelet - Adjoint curvelet transform
 sopt_mltb_div_opsopt_mltb_div_op - Compute divergence
 sopt_mltb_div_op_spheresopt_mltb_div_op_sphere - Compute divergence on sphere
 sopt_mltb_fwdcurveletsopt_mltb_fwdcurvelet - Forward curvelet transform
 sopt_mltb_genmasksopt_mltb_genmask - Generate mask
 sopt_mltb_gradient_opsopt_mltb_gradient_op - Compute gradient
 sopt_mltb_gradient_op_spheresopt_mltb_gradient_op_sphere - Compute gradient on sphere
 sopt_mltb_modifypdfsopt_mltb_modifypdf - Modify PDF of sampling profile
 sopt_mltb_struct2sizesopt_mltb_struct2size - Compute curvelet data structure
 sopt_mltb_vdsmasksopt_mltb_vdsmask - Create variable density sampling profile
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sopt-2.0.0/matlab/doc/matlab/misc/sopt_mltb_SNR.html000066400000000000000000000076071277570055300223360ustar00rootroot00000000000000 Description of sopt_mltb_SNR
Home > matlab > misc > sopt_mltb_SNR.m

sopt_mltb_SNR

PURPOSE ^

SNR - Compute the SNR between two images

SYNOPSIS ^

function snr = sopt_mltb_SNR(map_init, map_recon)

DESCRIPTION ^

 SNR - Compute the SNR between two images
 
 C omputes the SNR between the maps map_init and map_recon.  The SNR is 
 computed by
    10 * log10( var(MAP_INIT) / var(MAP_INIT-MAP_NOISY) )
  where var stands for the matlab built-in function that computes the
  variance.

 Inputs:

   - map_init: Initial image.

   - map_recon: Reconstructed image.

 Outputs:

   - snr: SNR.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function snr = sopt_mltb_SNR(map_init, map_recon)
0002 % SNR - Compute the SNR between two images
0003 %
0004 % C omputes the SNR between the maps map_init and map_recon.  The SNR is
0005 % computed by
0006 %    10 * log10( var(MAP_INIT) / var(MAP_INIT-MAP_NOISY) )
0007 %  where var stands for the matlab built-in function that computes the
0008 %  variance.
0009 %
0010 % Inputs:
0011 %
0012 %   - map_init: Initial image.
0013 %
0014 %   - map_recon: Reconstructed image.
0015 %
0016 % Outputs:
0017 %
0018 %   - snr: SNR.
0019 
0020 noise = map_init(:)-map_recon(:);
0021 var_init = var(map_init(:));
0022 var_den = var(noise(:));
0023 snr = 10 * log10(var_init/var_den);
0024 
0025 end
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sopt-2.0.0/matlab/doc/matlab/misc/sopt_mltb_TV_norm.html000066400000000000000000000141741277570055300232550ustar00rootroot00000000000000 Description of sopt_mltb_TV_norm
Home > matlab > misc > sopt_mltb_TV_norm.m

sopt_mltb_TV_norm

PURPOSE ^

sopt_mltb_TV_norm - Compute TV norm

SYNOPSIS ^

function y = sopt_mltb_TV_norm(u, sphere_flag,incNP, weights_dx, weights_dy)

DESCRIPTION ^

 sopt_mltb_TV_norm - Compute TV norm

 Compute the TV of an image on the plane or sphere.

 Inputs:

   - u: Image to compute TV norm of.

   - sphere_flag: Flag indicating whether to compute the TV norm on the
       sphere (1 = on sphere, 2 = on plane).

   - includeNP: Flag indicating whether the North pole is included
       in the sampling grid (1 = North pole included, 0 = North pole not
       included).

   - weights_dx: Weights in the x (phi) direction.

   - weights_dy: Weights in the y (theta) direction.

 Outputs:

   - y: TV norm of image.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function y = sopt_mltb_TV_norm(u, sphere_flag, ...
0002     incNP, weights_dx, weights_dy)
0003 % sopt_mltb_TV_norm - Compute TV norm
0004 %
0005 % Compute the TV of an image on the plane or sphere.
0006 %
0007 % Inputs:
0008 %
0009 %   - u: Image to compute TV norm of.
0010 %
0011 %   - sphere_flag: Flag indicating whether to compute the TV norm on the
0012 %       sphere (1 = on sphere, 2 = on plane).
0013 %
0014 %   - includeNP: Flag indicating whether the North pole is included
0015 %       in the sampling grid (1 = North pole included, 0 = North pole not
0016 %       included).
0017 %
0018 %   - weights_dx: Weights in the x (phi) direction.
0019 %
0020 %   - weights_dy: Weights in the y (theta) direction.
0021 %
0022 % Outputs:
0023 %
0024 %   - y: TV norm of image.
0025 
0026 if sphere_flag
0027     if nargin>3 
0028         [dx, dy] = sopt_mltb_gradient_op_sphere(u, incNP, weights_dx, weights_dy);
0029     else
0030         [dx, dy] = sopt_mltb_gradient_op_sphere(u, incNP);
0031     end
0032 else
0033     if nargin>3 
0034         [dx, dy] = sopt_mltb_gradient_op(u, weights_dx, weights_dy);
0035     else
0036         [dx, dy] = sopt_mltb_gradient_op(u);
0037     end
0038 end
0039 temp = sqrt(abs(dx).^2 + abs(dy).^2);
0040 y = sum(temp(:));
0041 
0042 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/misc/sopt_mltb_adjcurvelet.html000066400000000000000000000110061277570055300241700ustar00rootroot00000000000000 Description of sopt_mltb_adjcurvelet
Home > matlab > misc > sopt_mltb_adjcurvelet.m

sopt_mltb_adjcurvelet

PURPOSE ^

sopt_mltb_adjcurvelet - Adjoint curvelet transform

SYNOPSIS ^

function restim = sopt_mltb_adjcurvelet(coef, Mod, real)

DESCRIPTION ^

 sopt_mltb_adjcurvelet - Adjoint curvelet transform

 Compute the adjoint curvelet transform from the curvelet
 coefficient vector.

 Inputs:

   - coef: Input curvelet coefficient vector.

   - Mod: Data structure that stores sizes of the curvelet transform 
       generated from the CurveLab toolbox.

   - real: Flag indicating if the transform is real or complex (1 = real,  
       0 = complex).

 Outputs:

   - restim: Estimated image from the Curvelet coefficients.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function restim = sopt_mltb_adjcurvelet(coef, Mod, real)
0002 % sopt_mltb_adjcurvelet - Adjoint curvelet transform
0003 %
0004 % Compute the adjoint curvelet transform from the curvelet
0005 % coefficient vector.
0006 %
0007 % Inputs:
0008 %
0009 %   - coef: Input curvelet coefficient vector.
0010 %
0011 %   - Mod: Data structure that stores sizes of the curvelet transform
0012 %       generated from the CurveLab toolbox.
0013 %
0014 %   - real: Flag indicating if the transform is real or complex (1 = real,
0015 %       0 = complex).
0016 %
0017 % Outputs:
0018 %
0019 %   - restim: Estimated image from the Curvelet coefficients.
0020 
0021 CR=cell(size(Mod));
0022 marker=0;
0023 
0024 for s=1:length(Mod)
0025    CR{s} = cell(size(Mod{s}));
0026    for w=1:length(Mod{s})
0027      m=Mod{s}{w}(1);
0028      n=Mod{s}{w}(2);
0029      CR{s}{w}=reshape(coef(marker+1:marker+(m*n)),m,n);
0030      marker=marker+m*n;
0031    end
0032 end
0033  
0034 % Apply adjoint curvelet transform
0035 restim = ifdct_usfft(CR,real);
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sopt-2.0.0/matlab/doc/matlab/misc/sopt_mltb_div_op.html000066400000000000000000000100711277570055300231410ustar00rootroot00000000000000 Description of sopt_mltb_div_op
Home > matlab > misc > sopt_mltb_div_op.m

sopt_mltb_div_op

PURPOSE ^

sopt_mltb_div_op - Compute divergence

SYNOPSIS ^

function I = sopt_mltb_div_op(dx, dy, weights_dx, weights_dy)

DESCRIPTION ^

 sopt_mltb_div_op - Compute divergence

 Compute the divergence (adjoint of the gradient) of a two dimensional
 signal from the horizontal and vertical gradients.

 Inputs:

   - dx: Gradient in x.

   - dy: Gradient in y.

   - weights_dx: Weights in the x direction.

   - weights_dy: Weights in the y direction.

 Outputs:

   - I: Divergence.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function I = sopt_mltb_div_op(dx, dy, weights_dx, weights_dy)
0002 % sopt_mltb_div_op - Compute divergence
0003 %
0004 % Compute the divergence (adjoint of the gradient) of a two dimensional
0005 % signal from the horizontal and vertical gradients.
0006 %
0007 % Inputs:
0008 %
0009 %   - dx: Gradient in x.
0010 %
0011 %   - dy: Gradient in y.
0012 %
0013 %   - weights_dx: Weights in the x direction.
0014 %
0015 %   - weights_dy: Weights in the y direction.
0016 %
0017 % Outputs:
0018 %
0019 %   - I: Divergence.
0020 
0021 if nargin > 2
0022     dx = dx .* conj(weights_dx);
0023     dy = dy .* conj(weights_dy);
0024 end
0025 
0026 I = [dx(1, :) ; dx(2:end-1, :)-dx(1:end-2, :) ; -dx(end-1, :)];
0027 I = I + [dy(:, 1) , dy(:, 2:end-1)-dy(:, 1:end-2) , -dy(:, end-1)];
0028 
0029 end
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sopt-2.0.0/matlab/doc/matlab/misc/sopt_mltb_div_op_sphere.html000066400000000000000000000122041277570055300245070ustar00rootroot00000000000000 Description of sopt_mltb_div_op_sphere
Home > matlab > misc > sopt_mltb_div_op_sphere.m

sopt_mltb_div_op_sphere

PURPOSE ^

sopt_mltb_div_op_sphere - Compute divergence on sphere

SYNOPSIS ^

function I = sopt_mltb_div_op_sphere(dx, dy, includeNorthpole,weights_dx, weights_dy)

DESCRIPTION ^

 sopt_mltb_div_op_sphere - Compute divergence on sphere

 Compute the divergence (adjoint of the gradient) of a two dimensional
 signal on the sphere.  The phi direction (x) is periodic, while the theta
 direction (y) is not.

 Inputs:

   - dx: Gradient in x.

   - dy: Gradient in y.

   - includeNorthPole: Flag indicating whether the North pole is included
       in the sampling grid (1 = North pole included, 0 = North pole not
       included).

   - weights_dx: Weights in the x (phi) direction.

   - weights_dy: Weights in the y (theta) direction.

 Outputs:

   - I: Divergence.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function I = sopt_mltb_div_op_sphere(dx, dy, includeNorthpole, ...
0002   weights_dx, weights_dy)
0003 % sopt_mltb_div_op_sphere - Compute divergence on sphere
0004 %
0005 % Compute the divergence (adjoint of the gradient) of a two dimensional
0006 % signal on the sphere.  The phi direction (x) is periodic, while the theta
0007 % direction (y) is not.
0008 %
0009 % Inputs:
0010 %
0011 %   - dx: Gradient in x.
0012 %
0013 %   - dy: Gradient in y.
0014 %
0015 %   - includeNorthPole: Flag indicating whether the North pole is included
0016 %       in the sampling grid (1 = North pole included, 0 = North pole not
0017 %       included).
0018 %
0019 %   - weights_dx: Weights in the x (phi) direction.
0020 %
0021 %   - weights_dy: Weights in the y (theta) direction.
0022 %
0023 % Outputs:
0024 %
0025 %   - I: Divergence.
0026 
0027 if nargin > 3
0028     dx = dx .* conj(weights_dx);
0029     dy = dy .* conj(weights_dy);
0030 end
0031 if(includeNorthpole)
0032     I = [zeros(1, size(dx,2)) ; dx(2, :); dx(3:end-1, :)-dx(2:end-2, :); -dx(end-1, :)];
0033     I = I + [dy(:, 1) - dy(:,end) , dy(:, 2:end)-dy(:, 1:end-1)];
0034 else
0035     I = [dx(1, :) ; dx(2:end-1, :)-dx(1:end-2, :) ; -dx(end-1, :)];
0036     I = I + [dy(:, 1) - dy(:,end) , dy(:, 2:end)-dy(:, 1:end-1)];
0037 end
0038 end
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sopt-2.0.0/matlab/doc/matlab/misc/sopt_mltb_fwdcurvelet.html000066400000000000000000000076621277570055300242270ustar00rootroot00000000000000 Description of sopt_mltb_fwdcurvelet
Home > matlab > misc > sopt_mltb_fwdcurvelet.m

sopt_mltb_fwdcurvelet

PURPOSE ^

sopt_mltb_fwdcurvelet - Forward curvelet transform

SYNOPSIS ^

function coef = sopt_mltb_fwdcurvelet(im,real)

DESCRIPTION ^

 sopt_mltb_fwdcurvelet - Forward curvelet transform

 Compute the forward curvelet transform of an image and stores it in
 vector coef.

 Inputs:

   - im: Input image.

   - real: Flag indicating if the transform is real or complex (1 = real,  
       0 = complex).

 Outputs:

   - coef: Curvelet coefficients.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function coef = sopt_mltb_fwdcurvelet(im,real)
0002 % sopt_mltb_fwdcurvelet - Forward curvelet transform
0003 %
0004 % Compute the forward curvelet transform of an image and stores it in
0005 % vector coef.
0006 %
0007 % Inputs:
0008 %
0009 %   - im: Input image.
0010 %
0011 %   - real: Flag indicating if the transform is real or complex (1 = real,
0012 %       0 = complex).
0013 %
0014 % Outputs:
0015 %
0016 %   - coef: Curvelet coefficients.
0017 
0018 C = fdct_usfft(im,real); 
0019 
0020 % Compute vector version of curvelets coeffcients
0021 coef = [];
0022 
0023 for s=1:length(C)
0024   for w=1:length(C{s})
0025     A = C{s}{w};
0026     coef= [coef; A(:)];
0027   end
0028 end
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sopt-2.0.0/matlab/doc/matlab/misc/sopt_mltb_genmask.html000066400000000000000000000076421277570055300233200ustar00rootroot00000000000000 Description of sopt_mltb_genmask
Home > matlab > misc > sopt_mltb_genmask.m

sopt_mltb_genmask

PURPOSE ^

sopt_mltb_genmask - Generate mask

SYNOPSIS ^

function mask = sopt_mltb_genmask(pdf, seed)

DESCRIPTION ^

 sopt_mltb_genmask - Generate mask

 Generate a binary mask with variable density sampling. It is used
 in sopt_mltb_vdsmask in the generation of variable density sampling
 profiles.

 Inputs:

   - pdf: Sampling profile function.

   - seed: Seed for the random number generator. Optional.

 Outputs:

   - mask: Binary mask.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function mask = sopt_mltb_genmask(pdf, seed)
0002 % sopt_mltb_genmask - Generate mask
0003 %
0004 % Generate a binary mask with variable density sampling. It is used
0005 % in sopt_mltb_vdsmask in the generation of variable density sampling
0006 % profiles.
0007 %
0008 % Inputs:
0009 %
0010 %   - pdf: Sampling profile function.
0011 %
0012 %   - seed: Seed for the random number generator. Optional.
0013 %
0014 % Outputs:
0015 %
0016 %   - mask: Binary mask.
0017 
0018 if nargin==2
0019     rand('seed', seed);
0020 end
0021 
0022 mask = rand(size(pdf))<pdf;
0023
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sopt-2.0.0/matlab/doc/matlab/misc/sopt_mltb_gradient_op.html000066400000000000000000000103021277570055300241510ustar00rootroot00000000000000 Description of sopt_mltb_gradient_op
Home > matlab > misc > sopt_mltb_gradient_op.m

sopt_mltb_gradient_op

PURPOSE ^

sopt_mltb_gradient_op - Compute gradient

SYNOPSIS ^

function [dx, dy] = sopt_mltb_gradient_op(I, weights_dx, weights_dy)

DESCRIPTION ^

 sopt_mltb_gradient_op - Compute gradient

 Compute the gradient of a two dimensional signal.

 Inputs:

   - I: Input two dimesional signal.

   - weights_dx: Weights in the x direction.

   - weights_dy: Weights in the y direction.

 Outputs:
 
   - dx: Gradient in x direction.
 
   - dy: Gradient in y direction.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function [dx, dy] = sopt_mltb_gradient_op(I, weights_dx, weights_dy)
0002 % sopt_mltb_gradient_op - Compute gradient
0003 %
0004 % Compute the gradient of a two dimensional signal.
0005 %
0006 % Inputs:
0007 %
0008 %   - I: Input two dimesional signal.
0009 %
0010 %   - weights_dx: Weights in the x direction.
0011 %
0012 %   - weights_dy: Weights in the y direction.
0013 %
0014 % Outputs:
0015 %
0016 %   - dx: Gradient in x direction.
0017 %
0018 %   - dy: Gradient in y direction.
0019 
0020 dx = [I(2:end, :)-I(1:end-1, :) ; zeros(1, size(I, 2))];
0021 dy = [I(:, 2:end)-I(:, 1:end-1) , zeros(size(I, 1), 1)];
0022 
0023 if nargin>1
0024     dx = dx .* weights_dx;
0025     dy = dy .* weights_dy;
0026 end
0027 
0028 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/misc/sopt_mltb_gradient_op_sphere.html000066400000000000000000000134101277570055300255220ustar00rootroot00000000000000 Description of sopt_mltb_gradient_op_sphere
Home > matlab > misc > sopt_mltb_gradient_op_sphere.m

sopt_mltb_gradient_op_sphere

PURPOSE ^

sopt_mltb_gradient_op_sphere - Compute gradient on sphere

SYNOPSIS ^

function [dx, dy] = sopt_mltb_gradient_op_sphere(I, includeNorthpole,weights_dx, weights_dy)

DESCRIPTION ^

 sopt_mltb_gradient_op_sphere - Compute gradient on sphere

 Compute the gradientof a signal on the sphere.  The phi direction (x) is
 periodic, while the theta direction (y) is not.

 Inputs:

   - I: Input two dimesional signal on the sphere.

   - includeNorthPole: Flag indicating whether the North pole is included
       in the sampling grid (1 = North pole included, 0 = North pole not
       included).

   - weights_dx: Weights in the x (phi) direction.

   - weights_dy: Weights in the y (theta) direction.

 Outputs:
 
   - dx: Gradient in x (phi) direction.
 
   - dy: Gradient in y (theta) direction.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function [dx, dy] = sopt_mltb_gradient_op_sphere(I, includeNorthpole, ...
0002   weights_dx, weights_dy)
0003 % sopt_mltb_gradient_op_sphere - Compute gradient on sphere
0004 %
0005 % Compute the gradientof a signal on the sphere.  The phi direction (x) is
0006 % periodic, while the theta direction (y) is not.
0007 %
0008 % Inputs:
0009 %
0010 %   - I: Input two dimesional signal on the sphere.
0011 %
0012 %   - includeNorthPole: Flag indicating whether the North pole is included
0013 %       in the sampling grid (1 = North pole included, 0 = North pole not
0014 %       included).
0015 %
0016 %   - weights_dx: Weights in the x (phi) direction.
0017 %
0018 %   - weights_dy: Weights in the y (theta) direction.
0019 %
0020 % Outputs:
0021 %
0022 %   - dx: Gradient in x (phi) direction.
0023 %
0024 %   - dy: Gradient in y (theta) direction.
0025 
0026 dx = zeros(size(I, 1),size(I, 2));
0027 I_big = zeros(size(I, 1)+2,size(I, 2));
0028 I_big(2:end-1, 1:end) = I;
0029 
0030 % Theta direction
0031 if(includeNorthpole)
0032     dx = [I(2:end, :)-I(1:end-1, :) ; zeros(1, size(I, 2))];
0033     dx(1,:) = zeros();
0034 else
0035     dx = [I(2:end, :)-I(1:end-1, :) ; zeros(1, size(I, 2))];
0036 end
0037 
0038 % Phi direction
0039 if(includeNorthpole)
0040     dy = [I(:, 2:end)-I(:, 1:end-1) , I(:, 1)-I(:, end)];
0041 else
0042     dy = [I(:, 2:end)-I(:, 1:end-1) , I(:, 1)-I(:, end)];
0043 end
0044 
0045 if nargin>2
0046     dx = dx .* weights_dx;
0047     dy = dy .* weights_dy;
0048 end
0049 
0050 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/misc/sopt_mltb_modifypdf.html000066400000000000000000000115671277570055300236550ustar00rootroot00000000000000 Description of sopt_mltb_modifypdf
Home > matlab > misc > sopt_mltb_modifypdf.m

sopt_mltb_modifypdf

PURPOSE ^

sopt_mltb_modifypdf - Modify PDF of sampling profile

SYNOPSIS ^

function [new_pdf, alpha] = sopt_mltb_modifypdf(pdf, nb_meas)

DESCRIPTION ^

 sopt_mltb_modifypdf - Modify PDF of sampling profile
 
 Checks PDF of the sampling profile and normalizes it. It is used
 in sopt_mltb_vdsmask in the generation of variable density sampling
 profiles.

 Inputs:

   - pdf: Sampling profile function.

   - nb_meas: Number of measurements.

 Outputs:

   - new_pdf: New sampling profile function.

   - alpha: DC level for the required number of samples.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function [new_pdf, alpha] = sopt_mltb_modifypdf(pdf, nb_meas)
0002 % sopt_mltb_modifypdf - Modify PDF of sampling profile
0003 %
0004 % Checks PDF of the sampling profile and normalizes it. It is used
0005 % in sopt_mltb_vdsmask in the generation of variable density sampling
0006 % profiles.
0007 %
0008 % Inputs:
0009 %
0010 %   - pdf: Sampling profile function.
0011 %
0012 %   - nb_meas: Number of measurements.
0013 %
0014 % Outputs:
0015 %
0016 %   - new_pdf: New sampling profile function.
0017 %
0018 %   - alpha: DC level for the required number of samples.
0019 
0020 if sum(pdf(:)<0)>0
0021     error('PDF contains negative values');
0022 end
0023 pdf = pdf/max(pdf(:));
0024 
0025 % Find alpha
0026 alpha_min = -1; alpha_max = 1;
0027 while 1
0028     alpha = (alpha_min + alpha_max)/2;
0029     new_pdf = pdf + alpha;
0030     new_pdf(new_pdf>1) = 1; new_pdf(new_pdf<0) = 0;
0031     M = round(sum(new_pdf(:)));
0032     if M > nb_meas
0033         alpha_max = alpha;
0034     elseif M < nb_meas
0035         alpha_min = alpha;
0036     else
0037         break;
0038     end
0039 end
0040
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sopt-2.0.0/matlab/doc/matlab/misc/sopt_mltb_struct2size.html000066400000000000000000000073521277570055300241720ustar00rootroot00000000000000 Description of sopt_mltb_struct2size
Home > matlab > misc > sopt_mltb_struct2size.m

sopt_mltb_struct2size

PURPOSE ^

sopt_mltb_struct2size - Compute curvelet data structure

SYNOPSIS ^

function Mod = sopt_mltb_struct2size(C)

DESCRIPTION ^

sopt_mltb_struct2size - Compute curvelet data structure

 Compute data structure to store the sizes of the curvelet transform 
 generated from the CurveLab toolbox.

 Inputs:

   - C: Curvelet coefficients arranged in the original format.

 Outputs:

   - Mod: Data structure with the appropiate sizes.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function Mod = sopt_mltb_struct2size(C)
0002 %sopt_mltb_struct2size - Compute curvelet data structure
0003 %
0004 % Compute data structure to store the sizes of the curvelet transform
0005 % generated from the CurveLab toolbox.
0006 %
0007 % Inputs:
0008 %
0009 %   - C: Curvelet coefficients arranged in the original format.
0010 %
0011 % Outputs:
0012 %
0013 %   - Mod: Data structure with the appropiate sizes.
0014 
0015  Mod=cell(size(C));
0016  for s=1:length(C)
0017    Mod{s} = cell(size(C{s}));
0018    for w=1:length(C{s})
0019      [m,n]=size(C{s}{w});
0020      Mod{s}{w}=[m,n];
0021    end
0022  end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/misc/sopt_mltb_vdsmask.html000066400000000000000000000142551277570055300233410ustar00rootroot00000000000000 Description of sopt_mltb_vdsmask
Home > matlab > misc > sopt_mltb_vdsmask.m

sopt_mltb_vdsmask

PURPOSE ^

sopt_mltb_vdsmask - Create variable density sampling profile

SYNOPSIS ^

function mask = sopt_mltb_vdsmask(N,M,p)

DESCRIPTION ^

 sopt_mltb_vdsmask - Create variable density sampling profile

 Creates a binary mask generated from a variable density sampling 
 profile for two dimensional images in the frequency domain.   The mask 
 contains on average p*N*M ones's.

 Inputs:

   - N and M: Size of mask.

   - p: Coverage percentage.

 Note: The undersampling ratio is passed as 2*p and then the function
 only takes samples in half plane to account for signal reality. The range
 of p is 0 < p <= 0.5.

 Outputs:

   - mask: Binary mask with the sampling profile in the frequency domain.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function mask = sopt_mltb_vdsmask(N,M,p)
0002 % sopt_mltb_vdsmask - Create variable density sampling profile
0003 %
0004 % Creates a binary mask generated from a variable density sampling
0005 % profile for two dimensional images in the frequency domain.   The mask
0006 % contains on average p*N*M ones's.
0007 %
0008 % Inputs:
0009 %
0010 %   - N and M: Size of mask.
0011 %
0012 %   - p: Coverage percentage.
0013 %
0014 % Note: The undersampling ratio is passed as 2*p and then the function
0015 % only takes samples in half plane to account for signal reality. The range
0016 % of p is 0 < p <= 0.5.
0017 %
0018 % Outputs:
0019 %
0020 %   - mask: Binary mask with the sampling profile in the frequency domain.
0021 
0022 p = 2*p;
0023 
0024 if p==1
0025     mask=ones(N,M);
0026 else
0027     
0028     nb_meas=round(p*N*M);
0029     tol=ceil(p*N*M)-floor(p*N*M);
0030     d=1;
0031 
0032     [x,y] = meshgrid(linspace(-1, 1, M), linspace(-1, 1, N)); % Cartesian grid
0033     r = sqrt(x.^2+y.^2); r = r/max(r(:)); % Polar grid
0034 
0035     alpha=-1;
0036     it=0;
0037     maxit=20;
0038     while (alpha<-0.01 || alpha>0.01) && it<maxit
0039         pdf = (1-r).^d;
0040         [new_pdf,alpha] = sopt_mltb_modifypdf(pdf, nb_meas);
0041         if alpha<0
0042             d=d+0.1;
0043         else
0044             d=d-0.1;
0045         end
0046         it=it+1;
0047     end
0048 
0049     mask = zeros(size(new_pdf));
0050     while sum(mask(:))>nb_meas+tol || sum(mask(:))<nb_meas-tol
0051         mask = sopt_mltb_genmask(new_pdf);
0052     end
0053 
0054 end
0055 
0056 %Samples in half plane to account for signal reality.
0057 M1=floor(M/2);
0058 mask(:,1:M1)=0;
0059 mask = ifftshift(mask);
0060 
0061 N1=floor(N/2);
0062 mask(N1+1:N,1)=0;
0063
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Index for matlab/prox_operators

Matlab files in this directory:

 sopt_mltb_fast_proj_B2sopt_mltb_fast_proj_B2 - Fast projection onto L2-ball
 sopt_mltb_proj_B2sopt_mltb_proj_B2 - Projection onto L2-ball
 sopt_mltb_prox_L1sopt_mltb_prox_L1 - Proximal operator with L1 norm
 sopt_mltb_prox_TVsopt_mltb_prox_TV - Total variation proximal operator
 sopt_mltb_prox_TVoAsopt_mltb_prox_TVoA - Agumented total variation proximal operator
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sopt-2.0.0/matlab/doc/matlab/prox_operators/.svn/text-base/sopt_mltb_fast_proj_B2.html.svn-base000066400000000000000000000337071277570055300326360ustar00rootroot00000000000000 Description of sopt_mltb_fast_proj_B2
Home > matlab > prox_operators > sopt_mltb_fast_proj_B2.m

sopt_mltb_fast_proj_B2

PURPOSE ^

sopt_mltb_fast_proj_B2 - Fast projection onto L2-ball

SYNOPSIS ^

function [sol, u] = sopt_mltb_fast_proj_B2(x, param)

DESCRIPTION ^

 sopt_mltb_fast_proj_B2 - Fast projection onto L2-ball

 Compute the projection onto the L2 ball, i.e. solve

   min_{z} ||x - z||_2^2   s.t.  ||y - A z||_2 < epsilon

 where x is the input vector and the solution z* is returned as sol.
 The structure param should contain the following fields:

   - y: Measurements (default = 0).

   - A: Forward operator (default = Identity).

   - At: Adjoint operator (default = Identity).

   - epsilon: Radius of the L2 ball (default = 1e-3).

   - tight: 1 if A is a tight frame or 0 otherwise (default = 1).

   - nu: Bound on the norm^2 of the operator A, i.e.
       ||A x||^2 <= nu * ||x||^2 (default = 1).

   - tol: Tolerance for the projection onto the L2 ball. The algorithms
       stops if
         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
       (default = 1e-3).

   - max_iter: Maximum number of iterations (default: 200).

   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - u: Initial vector for the dual problem, same dimension as y
       (default = 0).
   
   - pos: Positivity flag (1 = positive solution,
       0 general complex case; default = 0).

   - real: Reality flag (1 = real solution,
       0 = general complex case; default = 0).

 Outputs:

   - sol: Final solution.

   - u: Final dual vector.

 References:
 [1] M.J. Fadili and J-L. Starck, "Monotone operator splitting for
 optimization problems in sparse recovery" , IEEE ICIP, Cairo,
 Egypt, 2009.
 [2] Amir Beck and Marc Teboulle, "A Fast Iterative Shrinkage-Thresholding
 Algorithm for Linear Inverse Problems",  SIAM Journal on Imaging Sciences
 2 (2009), no. 1, 183--202.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function [sol, u] = sopt_mltb_fast_proj_B2(x, param)
0002 % sopt_mltb_fast_proj_B2 - Fast projection onto L2-ball
0003 %
0004 % Compute the projection onto the L2 ball, i.e. solve
0005 %
0006 %   min_{z} ||x - z||_2^2   s.t.  ||y - A z||_2 < epsilon
0007 %
0008 % where x is the input vector and the solution z* is returned as sol.
0009 % The structure param should contain the following fields:
0010 %
0011 %   - y: Measurements (default = 0).
0012 %
0013 %   - A: Forward operator (default = Identity).
0014 %
0015 %   - At: Adjoint operator (default = Identity).
0016 %
0017 %   - epsilon: Radius of the L2 ball (default = 1e-3).
0018 %
0019 %   - tight: 1 if A is a tight frame or 0 otherwise (default = 1).
0020 %
0021 %   - nu: Bound on the norm^2 of the operator A, i.e.
0022 %       ||A x||^2 <= nu * ||x||^2 (default = 1).
0023 %
0024 %   - tol: Tolerance for the projection onto the L2 ball. The algorithms
0025 %       stops if
0026 %         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
0027 %       (default = 1e-3).
0028 %
0029 %   - max_iter: Maximum number of iterations (default: 200).
0030 %
0031 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0032 %       2 = print main steps; default = 1).
0033 %
0034 %   - u: Initial vector for the dual problem, same dimension as y
0035 %       (default = 0).
0036 %
0037 %   - pos: Positivity flag (1 = positive solution,
0038 %       0 general complex case; default = 0).
0039 %
0040 %   - real: Reality flag (1 = real solution,
0041 %       0 = general complex case; default = 0).
0042 %
0043 % Outputs:
0044 %
0045 %   - sol: Final solution.
0046 %
0047 %   - u: Final dual vector.
0048 %
0049 % References:
0050 % [1] M.J. Fadili and J-L. Starck, "Monotone operator splitting for
0051 % optimization problems in sparse recovery" , IEEE ICIP, Cairo,
0052 % Egypt, 2009.
0053 % [2] Amir Beck and Marc Teboulle, "A Fast Iterative Shrinkage-Thresholding
0054 % Algorithm for Linear Inverse Problems",  SIAM Journal on Imaging Sciences
0055 % 2 (2009), no. 1, 183--202.
0056 
0057 % Optional input arguments
0058 if ~isfield(param, 'y'), param.y = 0; end
0059 if ~isfield(param, 'A'), param.A = @(x) x; end
0060 if ~isfield(param, 'At'), param.At = @(x) x; end
0061 if ~isfield(param, 'epsilon'), param.epsilon = 1e-3; end
0062 if ~isfield(param, 'tight'), param.tight = 1; end
0063 if ~isfield(param, 'tol'), param.tol = 1e-3; end
0064 if ~isfield(param, 'verbose'), param.verbose = 1; end
0065 if ~isfield(param, 'nu'), param.nu = 1; end
0066 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0067 if ~isfield(param, 'u'), param.u = zeros(size(param.y)); end
0068 if ~isfield(param, 'pos'), param.pos = 0; end
0069 if ~isfield(param, 'real'), param.real = 0; end
0070 
0071 % Useful functions for the projection
0072 sc = @(z) z*min(param.epsilon/norm(z(:)), 1); % scaling
0073 
0074 % Projection
0075 if (param.tight && ~(param.pos||param.real)) % TIGHT FRAME CASE
0076     
0077     temp = param.A(x) - param.y;
0078     sol = x + 1/param.nu * param.At(sc(temp)-temp);
0079     crit_B2 = 'TOL_EPS'; iter = 0;
0080     u = 0;
0081     
0082 else % NON TIGHT FRAME CASE
0083     
0084     % Initializations
0085     sol = x; u = param.u; v = u;
0086     iter = 1; true = 1; told = 1;
0087     
0088     % Tolerance onto the L2 ball
0089     epsilon_low = param.epsilon/(1+param.tol);
0090     epsilon_up = param.epsilon/(1-param.tol);
0091     
0092     % Check if we are in the L2 ball
0093     dummy = param.A(sol);
0094     norm_res = norm(param.y(:)-dummy(:), 2);
0095     if norm_res <= epsilon_up
0096         crit_B2 = 'TOL_EPS'; true = 0;
0097     end
0098     
0099     % Projection onto the L2-ball
0100     % Init
0101     if param.verbose > 1
0102         fprintf('  Proj. B2:\n');
0103     end
0104     while true
0105         
0106         % Residual
0107         res = param.A(sol) - param.y; norm_res = norm(res(:), 2);
0108         
0109         % Scaling for the projection
0110         res = u*param.nu + res; norm_proj = norm(res(:), 2);
0111         
0112         % Log
0113         if param.verbose>1
0114             fprintf('   Iter %i, epsilon = %e, ||y - Ax||_2 = %e\n', ...
0115                 iter, param.epsilon, norm_res);
0116         end
0117         
0118         % Stopping criterion
0119         if (norm_res>=epsilon_low && norm_res<=epsilon_up)
0120             crit_B2 = 'TOL_EPS'; break;
0121         elseif iter >= param.max_iter
0122             crit_B2 = 'MAX_IT'; break;
0123         end
0124         
0125         % Projection onto the L2 ball
0126         t = (1+sqrt(1+4*told^2))/2;
0127         ratio = min(1, param.epsilon/norm_proj);
0128         u = v;
0129         v = 1/param.nu * (res - res*ratio);
0130         u = v + (told-1)/t * (v - u);
0131         
0132         % Current estimate
0133         sol = x - param.At(u);
0134         
0135         % Projection onto the non-negative orthant (positivity constraint)
0136         if (param.pos)
0137             sol = real(sol);
0138             sol(sol<0) = 0;
0139             
0140         end
0141         
0142         % Projection onto the real orthant (reality constraint)
0143         if (param.real)
0144             sol = real(sol);            
0145         end        
0146         
0147         % Update number of iteration
0148         told = t;
0149         
0150         % Update number of iterations
0151         iter = iter + 1;
0152         
0153     end
0154 end
0155 
0156 % Log after the projection onto the L2-ball
0157 if param.verbose >= 1
0158     temp = param.A(sol);
0159     fprintf(['  Proj. B2: epsilon = %e, ||y-Ax||_2 = %e,', ...
0160         ' %s, iter = %i\n'], param.epsilon, norm(param.y(:)-temp(:)), ...
0161         crit_B2, iter);
0162 end
0163 
0164 
0165 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/prox_operators/.svn/text-base/sopt_mltb_proj_B2.html.svn-base000066400000000000000000000273051277570055300316160ustar00rootroot00000000000000 Description of sopt_mltb_proj_B2
Home > matlab > prox_operators > sopt_mltb_proj_B2.m

sopt_mltb_proj_B2

PURPOSE ^

sopt_mltb_proj_B2 - Projection onto L2-ball

SYNOPSIS ^

function [sol, u] = sopt_mltb_proj_B2(x, param)

DESCRIPTION ^

 sopt_mltb_proj_B2 - Projection onto L2-ball

 Compute the projection onto the L2 ball, i.e. solve

   min_{z} ||x - z||_2^2   s.t.  ||y - A z||_2 < epsilon

 where x is the input vector and the solution z* is returned as sol.
 The structure param should contain the following fields:

   - y: Measurements (default = 0).

   - A: Forward operator (default = Identity).

   - At: Adjoint operator (default = Identity).

   - epsilon: Radius of the L2 ball (default = 1e-3).

   - tight: 1 if A is a tight frame or 0 otherwise (default = 1).

   - nu: Bound on the norm^2 of the operator A, i.e.
       ||A x||^2 <= nu * ||x||^2 (default = 1).

   - tol: Tolerance for the projection onto the L2 ball. The algorithms
       stops if
         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
       (default = 1e-3).

   - max_iter: Maximum number of iterations (default: 200).

   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - u: Initial vector for the dual problem, same dimension as y
       (default = 0).

 Outputs:

   - sol: Final solution.

   - u: Final dual vector.

 References:
 [1] M.J. Fadili and J-L. Starck, "Monotone operator splitting for 
 optimization problems in sparse recovery" , IEEE ICIP, Cairo, 
 Egypt, 2009.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function [sol, u] = sopt_mltb_proj_B2(x, param)
0002 % sopt_mltb_proj_B2 - Projection onto L2-ball
0003 %
0004 % Compute the projection onto the L2 ball, i.e. solve
0005 %
0006 %   min_{z} ||x - z||_2^2   s.t.  ||y - A z||_2 < epsilon
0007 %
0008 % where x is the input vector and the solution z* is returned as sol.
0009 % The structure param should contain the following fields:
0010 %
0011 %   - y: Measurements (default = 0).
0012 %
0013 %   - A: Forward operator (default = Identity).
0014 %
0015 %   - At: Adjoint operator (default = Identity).
0016 %
0017 %   - epsilon: Radius of the L2 ball (default = 1e-3).
0018 %
0019 %   - tight: 1 if A is a tight frame or 0 otherwise (default = 1).
0020 %
0021 %   - nu: Bound on the norm^2 of the operator A, i.e.
0022 %       ||A x||^2 <= nu * ||x||^2 (default = 1).
0023 %
0024 %   - tol: Tolerance for the projection onto the L2 ball. The algorithms
0025 %       stops if
0026 %         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
0027 %       (default = 1e-3).
0028 %
0029 %   - max_iter: Maximum number of iterations (default: 200).
0030 %
0031 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0032 %       2 = print main steps; default = 1).
0033 %
0034 %   - u: Initial vector for the dual problem, same dimension as y
0035 %       (default = 0).
0036 %
0037 % Outputs:
0038 %
0039 %   - sol: Final solution.
0040 %
0041 %   - u: Final dual vector.
0042 %
0043 % References:
0044 % [1] M.J. Fadili and J-L. Starck, "Monotone operator splitting for
0045 % optimization problems in sparse recovery" , IEEE ICIP, Cairo,
0046 % Egypt, 2009.
0047 
0048 % Optional input arguments
0049 if ~isfield(param, 'y'), param.y = 0; end
0050 if ~isfield(param, 'A'), param.A = @(x) x; end
0051 if ~isfield(param, 'At'), param.At = @(x) x; end
0052 if ~isfield(param, 'epsilon'), param.epsilon = 1e-3; end
0053 if ~isfield(param, 'tight'), param.tight = 1; end
0054 if ~isfield(param, 'tol'), param.tol = 1e-3; end
0055 if ~isfield(param, 'verbose'), param.verbose = 1; end
0056 if ~isfield(param, 'nu'), param.nu = 1; end
0057 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0058 if ~isfield(param, 'u'), param.u = zeros(size(param.y)); end
0059 
0060 % Useful functions for the projection
0061 sc = @(z) z*min(param.epsilon/norm(z(:)), 1); % scaling
0062 
0063 % Projection
0064 if param.tight % TIGHT FRAME CASE
0065     
0066     temp = param.A(x) - param.y;
0067     sol = x + 1/param.nu * param.At(sc(temp)-temp);
0068     crit_B2 = 'TOL_EPS'; iter = 0; u = NaN;
0069     
0070 else % NON TIGHT FRAME CASE
0071     
0072     % Initializations
0073     sol = x; u = param.u;
0074     iter = 1; true = 1;
0075     
0076     % Tolerance onto the L2 ball
0077     epsilon_low = param.epsilon/(1+param.tol);
0078     epsilon_up = param.epsilon/(1-param.tol);
0079     
0080     % Check if we are in the L2 ball
0081     norm_res = norm(param.y(:)-param.A(sol), 2);
0082     if norm_res <= epsilon_up
0083         crit_B2 = 'TOL_EPS'; true = 0;
0084     end
0085     
0086     % Projection onto the L2-ball
0087     % Init
0088     if param.verbose > 1
0089         fprintf('  Proj. B2:\n');
0090     end
0091     while true
0092         
0093         % Residual
0094         res = param.A(sol) - param.y; norm_res = norm(res(:), 2);
0095         
0096         % Scaling for the projection
0097         res = u*param.nu + res; norm_proj = norm(res(:), 2);
0098         
0099         % Log
0100         if param.verbose>1
0101             fprintf('   Iter %i, epsilon = %e, ||y - Ax||_2 = %e\n', ...
0102                 iter, param.epsilon, norm_res);
0103         end
0104         
0105         % Stopping criterion
0106         if (norm_res>=epsilon_low && norm_res<=epsilon_up)
0107             crit_B2 = 'TOL_EPS'; break;
0108         elseif iter >= param.max_iter
0109             crit_B2 = 'MAX_IT'; break;
0110         end
0111         
0112         % Projection onto the L2 ball
0113         ratio = min(1, param.epsilon/norm_proj);
0114         u = 1/param.nu * (res - res*ratio);
0115         
0116         % Current estimate
0117         sol = x - param.At(u);
0118         
0119         % Update number of iteration
0120         iter = iter + 1;
0121         
0122     end
0123 end
0124 
0125 % Log after the projection onto the L2-ball
0126 if param.verbose >= 1
0127     temp = param.A(sol);
0128     fprintf(['  Proj. B2: epsilon = %e, ||y-Ax||_2 = %e,', ...
0129         ' %s, iter = %i\n'], param.epsilon, norm(param.y(:)-temp(:)), ...
0130         crit_B2, iter);
0131 end
0132 
0133 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/prox_operators/.svn/text-base/sopt_mltb_prox_L1.html.svn-base000066400000000000000000000272541277570055300316500ustar00rootroot00000000000000 Description of sopt_mltb_prox_L1
Home > matlab > prox_operators > sopt_mltb_prox_L1.m

sopt_mltb_prox_L1

PURPOSE ^

sopt_mltb_prox_L1 - Proximal operator with L1 norm

SYNOPSIS ^

function sol = sopt_mltb_prox_L1(x, lambda, param)

DESCRIPTION ^

 sopt_mltb_prox_L1 - Proximal operator with L1 norm

 Compute the L1 proximal operator, i.e. solve

   min_{z} 0.5*||x - z||_2^2 + lambda * ||Psit x||_1 ,

 where x is the input vector and the solution z* is returned as sol.  
 The structure param should contain the following fields:

   - Psit: Sparsifying transform (default = Identity).

   - Psi: Adjoint of Psit (default = Identity).

   - tight: 1 if Psit is a tight frame or 0 otherwise (default = 1).

   - nu: Bound on the norm^2 of the operator Psi, i.e.
       ||Psi x||^2 <= nu * ||x||^2 (default = 1).

   - max_iter: Maximum number of iterations (default = 200).

   - rel_obj: Minimum relative change of the objective value 
       (default = 1e-4).  The algorithm stops if
           | ||x(t)||_1 - ||x(t-1)||_1 | / ||x(t)||_1 < rel_obj,
       where x(t) is the estimate of the solution at iteration t.

   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - weights: Weights for a weighted L1-norm (default = 1).

   - pos: Positivity flag (1 = positive solution,
       0 = general complex case; default = 0).

 References:
 [1] M.J. Fadili and J-L. Starck, "Monotone operator splitting for
 optimization problems in sparse recovery" , IEEE ICIP, Cairo,
 Egypt, 2009.
 [2] Amir Beck and Marc Teboulle, "A Fast Iterative Shrinkage-Thresholding
 Algorithm for Linear Inverse Problems",  SIAM Journal on Imaging Sciences
 2 (2009), no. 1, 183--202.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_prox_L1(x, lambda, param)
0002 % sopt_mltb_prox_L1 - Proximal operator with L1 norm
0003 %
0004 % Compute the L1 proximal operator, i.e. solve
0005 %
0006 %   min_{z} 0.5*||x - z||_2^2 + lambda * ||Psit x||_1 ,
0007 %
0008 % where x is the input vector and the solution z* is returned as sol.
0009 % The structure param should contain the following fields:
0010 %
0011 %   - Psit: Sparsifying transform (default = Identity).
0012 %
0013 %   - Psi: Adjoint of Psit (default = Identity).
0014 %
0015 %   - tight: 1 if Psit is a tight frame or 0 otherwise (default = 1).
0016 %
0017 %   - nu: Bound on the norm^2 of the operator Psi, i.e.
0018 %       ||Psi x||^2 <= nu * ||x||^2 (default = 1).
0019 %
0020 %   - max_iter: Maximum number of iterations (default = 200).
0021 %
0022 %   - rel_obj: Minimum relative change of the objective value
0023 %       (default = 1e-4).  The algorithm stops if
0024 %           | ||x(t)||_1 - ||x(t-1)||_1 | / ||x(t)||_1 < rel_obj,
0025 %       where x(t) is the estimate of the solution at iteration t.
0026 %
0027 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0028 %       2 = print main steps; default = 1).
0029 %
0030 %   - weights: Weights for a weighted L1-norm (default = 1).
0031 %
0032 %   - pos: Positivity flag (1 = positive solution,
0033 %       0 = general complex case; default = 0).
0034 %
0035 % References:
0036 % [1] M.J. Fadili and J-L. Starck, "Monotone operator splitting for
0037 % optimization problems in sparse recovery" , IEEE ICIP, Cairo,
0038 % Egypt, 2009.
0039 % [2] Amir Beck and Marc Teboulle, "A Fast Iterative Shrinkage-Thresholding
0040 % Algorithm for Linear Inverse Problems",  SIAM Journal on Imaging Sciences
0041 % 2 (2009), no. 1, 183--202.
0042 
0043 % Optional input arguments
0044 if ~isfield(param, 'verbose'), param.verbose = 1; end
0045 if ~isfield(param, 'Psit'), param.Psi = @(x) x; param.Psit = @(x) x; end
0046 if ~isfield(param, 'tight'), param.tight = 1; end
0047 if ~isfield(param, 'nu'), param.nu = 1; end
0048 if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end
0049 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0050 if ~isfield(param, 'Psit'), param.Psit = @(x) x; end
0051 if ~isfield(param, 'Psi'), param.Psi = @(x) x; end
0052 if ~isfield(param, 'weights'), param.weights = 1; end
0053 if ~isfield(param, 'pos'), param.pos = 0; end
0054 
0055 % Useful functions
0056 soft = @(z, T) sign(z).*max(abs(z)-T, 0);
0057 
0058 % Projection
0059 if param.tight && ~param.pos % TIGHT FRAME CASE
0060     
0061     temp = param.Psit(x);
0062     sol = x + 1/param.nu * param.Psi(soft(temp, ...
0063         lambda*param.nu*param.weights)-temp);
0064     crit_L1 = 'REL_OBJ'; iter_L1 = 1;
0065     dummy = param.Psit(sol);
0066     norm_l1 = sum(param.weights(:).*abs(dummy(:)));
0067     
0068 else % NON TIGHT FRAME CASE OR CONSTRAINT INVOLVED
0069     
0070     % Initializations
0071     u_l1 = zeros(size(param.Psit(x)));
0072     sol = x - param.Psi(u_l1);
0073     prev_l1 = 0; iter_L1 = 0;
0074     
0075     % Soft-thresholding
0076     % Init
0077     if param.verbose > 1
0078         fprintf('  Proximal l1 operator:\n');
0079     end
0080     while 1
0081         
0082         % L1 norm of the estimate
0083         dummy = param.Psit(sol);
0084         
0085         norm_l1 = .5*norm(x(:) - sol(:), 2)^2 + lambda * ...
0086             sum(param.weights(:).*abs(dummy(:)));
0087         rel_l1 = abs(norm_l1-prev_l1)/norm_l1;
0088         
0089         % Log
0090         if param.verbose>1
0091             fprintf('   Iter %i, ||Psit x||_1 = %e, rel_l1 = %e\n', ...
0092                 iter_L1, norm_l1, rel_l1);
0093         end
0094         
0095         % Stopping criterion
0096         if (rel_l1 < param.rel_obj)
0097             crit_L1 = 'REL_OB'; break;
0098         elseif iter_L1 >= param.max_iter
0099             crit_L1 = 'MAX_IT'; break;
0100         end
0101         
0102         % Soft-thresholding
0103         res = u_l1*param.nu + param.Psit(sol);
0104         dummy = soft(res, lambda*param.nu*param.weights);
0105         if param.pos
0106             dummy = real(dummy); dummy(dummy<0) = 0;
0107         end
0108         u_l1 = 1/param.nu * (res - dummy);
0109         sol = x - param.Psi(u_l1);
0110         
0111         % Update
0112         prev_l1 = norm_l1;
0113         iter_L1 = iter_L1 + 1;
0114         
0115     end
0116 end
0117 
0118 % Log after the projection onto the L2-ball
0119 if param.verbose >= 1
0120     fprintf(['  prox_L1: ||Psi x||_1 = %e,', ...
0121         ' %s, iter = %i\n'], norm_l1, crit_L1, iter_L1);
0122 end
0123 
0124 end
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sopt-2.0.0/matlab/doc/matlab/prox_operators/.svn/text-base/sopt_mltb_prox_TV.html.svn-base000066400000000000000000000200761277570055300317200ustar00rootroot00000000000000 Description of sopt_mltb_prox_TV
Home > matlab > prox_operators > sopt_mltb_prox_TV.m

sopt_mltb_prox_TV

PURPOSE ^

sopt_mltb_prox_TV - Total variation proximal operator

SYNOPSIS ^

function sol = sopt_mltb_prox_TV(x, lambda, param)

DESCRIPTION ^

 sopt_mltb_prox_TV - Total variation proximal operator

 Compute the TV proximal operator, i.e. solve

   min_{z} ||x - z||_2^2 + lambda * ||z||_{TV}

 where x is the input vector and the solution z* is returned as sol.  
 The structure param should contain the following fields:

   - max_iter: Maximum number of iterations (default = 200).

   - rel_obj: Minimum relative change of the objective value 
       (default = 1e-4).  The algorithm stops if
           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj,
       where x(t) is the estimate of the solution at iteration t.

   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

 Reference:
 [1] A. Beck and  M. Teboulle, "Fast gradient-based algorithms for
 constrained Total Variation Image Denoising and Deblurring Problems", 
 IEEE Transactions on Image Processing, VOL. 18, NO. 11, 2419-2434, 
 November 2009.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_prox_TV(x, lambda, param)
0002 % sopt_mltb_prox_TV - Total variation proximal operator
0003 %
0004 % Compute the TV proximal operator, i.e. solve
0005 %
0006 %   min_{z} ||x - z||_2^2 + lambda * ||z||_{TV}
0007 %
0008 % where x is the input vector and the solution z* is returned as sol.
0009 % The structure param should contain the following fields:
0010 %
0011 %   - max_iter: Maximum number of iterations (default = 200).
0012 %
0013 %   - rel_obj: Minimum relative change of the objective value
0014 %       (default = 1e-4).  The algorithm stops if
0015 %           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj,
0016 %       where x(t) is the estimate of the solution at iteration t.
0017 %
0018 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0019 %       2 = print main steps; default = 1).
0020 %
0021 % Reference:
0022 % [1] A. Beck and  M. Teboulle, "Fast gradient-based algorithms for
0023 % constrained Total Variation Image Denoising and Deblurring Problems",
0024 % IEEE Transactions on Image Processing, VOL. 18, NO. 11, 2419-2434,
0025 % November 2009.
0026 
0027 % Optional input arguments
0028 if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end
0029 if ~isfield(param, 'verbose'), param.verbose = 1; end
0030 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0031 
0032 % Initializations
0033 [r, s] = sopt_mltb_gradient_op(x*0);
0034 pold = r; qold = s;
0035 told = 1; prev_obj = 0;
0036 
0037 % Main iterations
0038 if param.verbose > 1
0039     fprintf('  Proximal TV operator:\n');
0040 end
0041 for iter = 1:param.max_iter
0042     
0043     % Current solution
0044     sol = x - lambda*sopt_mltb_div_op(r, s);
0045     
0046     % Objective function value
0047     obj = .5*norm(x(:)-sol(:), 2)^2 + lambda * sopt_mltb_TV_norm(sol, 0);
0048     rel_obj = abs(obj-prev_obj)/obj;
0049     prev_obj = obj;
0050     
0051     % Stopping criterion
0052     if param.verbose>1
0053         fprintf('   Iter %i, obj = %e, rel_obj = %e\n', ...
0054             iter, obj, rel_obj);
0055     end
0056     if rel_obj < param.rel_obj
0057         crit_TV = 'TOL_EPS'; break;
0058     end
0059     
0060     % Udpate divergence vectors and project
0061     [dx, dy] = sopt_mltb_gradient_op(sol);
0062     r = r - 1/(8*lambda) * dx; s = s - 1/(8*lambda) * dy;
0063     weights = max(1, sqrt(abs(r).^2+abs(s).^2));
0064     p = r./weights; q = s./weights;
0065     
0066     % FISTA update
0067     t = (1+sqrt(4*told^2))/2;
0068     r = p + (told-1)/t * (p - pold); pold = p;
0069     s = q + (told-1)/t * (q - qold); qold = q;
0070     told = t;
0071     
0072 end
0073 
0074 % Log after the minimization
0075 if ~exist('crit_TV', 'var'), crit_TV = 'MAX_IT'; end
0076 if param.verbose >= 1
0077     fprintf(['  Prox_TV: obj = %e, rel_obj = %e,' ...
0078         ' %s, iter = %i\n'], obj, rel_obj, crit_TV, iter);
0079 end
0080 
0081 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/prox_operators/.svn/text-base/sopt_mltb_prox_TVoA.html.svn-base000066400000000000000000000306471277570055300322050ustar00rootroot00000000000000 Description of sopt_mltb_prox_TVoA
Home > matlab > prox_operators > sopt_mltb_prox_TVoA.m

sopt_mltb_prox_TVoA

PURPOSE ^

sopt_mltb_prox_TVoA - Agumented total variation proximal operator

SYNOPSIS ^

function sol = sopt_mltb_prox_TVoA(b, lambda, param)

DESCRIPTION ^

 sopt_mltb_prox_TVoA - Agumented total variation proximal operator

 Compute the TV proximal operator when an additional linear operator A is
 incorporated in the TV norm, i.e. solve

   min_{x} ||y - x||_2^2 + lambda * ||A x||_{TV}

 where x is the input vector and the solution z* is returned as sol.  
 The structure param should contain the following fields:

   - max_iter: Maximum number of iterations (default = 200).

   - rel_obj: Minimum relative change of the objective value 
       (default = 1e-4).  The algorithm stops if
           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_1 < rel_obj,
       where x(t) is the estimate of the solution at iteration t.

   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - A: Forward transform (default = Identity).

   - At: Adjoint of At (default = Identity).

   - nu: Bound on the norm^2 of the operator A, i.e.
       ||A x||^2 <= nu * ||x||^2 (default = 1)

 Reference:
 [1] A. Beck and  M. Teboulle, "Fast gradient-based algorithms for
 constrained Total Variation Image Denoising and Deblurring Problems",
 IEEE Transactions on Image Processing, VOL. 18, NO. 11, 2419-2434,
 November 2009.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_prox_TVoA(b, lambda, param)
0002 % sopt_mltb_prox_TVoA - Agumented total variation proximal operator
0003 %
0004 % Compute the TV proximal operator when an additional linear operator A is
0005 % incorporated in the TV norm, i.e. solve
0006 %
0007 %   min_{x} ||y - x||_2^2 + lambda * ||A x||_{TV}
0008 %
0009 % where x is the input vector and the solution z* is returned as sol.
0010 % The structure param should contain the following fields:
0011 %
0012 %   - max_iter: Maximum number of iterations (default = 200).
0013 %
0014 %   - rel_obj: Minimum relative change of the objective value
0015 %       (default = 1e-4).  The algorithm stops if
0016 %           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_1 < rel_obj,
0017 %       where x(t) is the estimate of the solution at iteration t.
0018 %
0019 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0020 %       2 = print main steps; default = 1).
0021 %
0022 %   - A: Forward transform (default = Identity).
0023 %
0024 %   - At: Adjoint of At (default = Identity).
0025 %
0026 %   - nu: Bound on the norm^2 of the operator A, i.e.
0027 %       ||A x||^2 <= nu * ||x||^2 (default = 1)
0028 %
0029 % Reference:
0030 % [1] A. Beck and  M. Teboulle, "Fast gradient-based algorithms for
0031 % constrained Total Variation Image Denoising and Deblurring Problems",
0032 % IEEE Transactions on Image Processing, VOL. 18, NO. 11, 2419-2434,
0033 % November 2009.
0034 
0035 % Optional input arguments
0036 if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end
0037 if ~isfield(param, 'verbose'), param.verbose = 1; end
0038 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0039 if ~isfield(param, 'At'), param.At = @(x) x; end
0040 if ~isfield(param, 'A'), param.A = @(x) x; end
0041 if ~isfield(param, 'nu'), param.nu = 1; end
0042 
0043 % Advanced input arguments (not exposed in documentation)
0044 if ~isfield(param, 'weights_dx'), param.weights_dx = 1; end
0045 if ~isfield(param, 'weights_dy'), param.weights_dy = 1; end
0046 if ~isfield(param, 'zero_weights_flag'), param.zero_weights_flag = 1; end
0047 if ~isfield(param, 'identical_weights_flag')
0048     param.identical_weights_flag = 0; 
0049 end
0050 if ~isfield(param, 'sphere_flag'), param.sphere_flag = 0; end
0051 if ~isfield(param, 'incNP'), param.incNP = 0; end
0052 
0053 % Set grad and div operators to planar or spherical case and also
0054 % include weights or not (depending on parameter flags).
0055 if (param.sphere_flag)
0056    G = @sopt_mltb_gradient_op_sphere;
0057    D = @sopt_mltb_div_op_sphere;
0058 else
0059    G = @sopt_mltb_gradient_op;
0060    D = @sopt_mltb_div_op;
0061 end
0062 
0063 if (~param.identical_weights_flag && param.zero_weights_flag)
0064     grad = @(x) G(x, param.weights_dx, param.weights_dy);
0065     div = @(r, s) D(r, s, param.weights_dx, param.weights_dy);
0066     max_weights = max([abs(param.weights_dx(:)); ...
0067         abs(param.weights_dy(:))])^2;
0068 else
0069     grad = @(x) G(x);
0070     div = @(r, s) D(r, s);
0071 end
0072 
0073 % Initializations
0074 [r, s] = grad(param.A(b*0));
0075 pold = r; qold = s;
0076 told = 1; prev_obj = 0;
0077 
0078 % Main iterations
0079 if param.verbose > 1
0080     fprintf('  Proximal TV operator:\n');
0081 end
0082 for iter = 1:param.max_iter
0083     
0084     % Current solution
0085     sol = b - lambda*param.At(div(r, s));
0086     
0087     % Objective function value
0088     obj = .5*norm(b(:)-sol(:), 2) + lambda * ...
0089         sopt_mltb_TV_norm(param.A(sol), param.weights_dx, param.weights_dy);
0090     rel_obj = abs(obj-prev_obj)/obj;
0091     prev_obj = obj;
0092     
0093     % Stopping criterion
0094     if param.verbose>1
0095         fprintf('   Iter %i, obj = %e, rel_obj = %e\n', ...
0096             iter, obj, rel_obj);
0097     end
0098     if rel_obj < param.rel_obj
0099         crit_TV = 'TOL_EPS'; break;
0100     end
0101     
0102     % Udpate divergence vectors and project
0103     [dx, dy] = grad(param.A(sol));
0104     if (param.identical_weights_flag)
0105         r = r - 1/(8*lambda*param.nu) * dx;
0106         s = s - 1/(8*lambda*param.nu) * dy;
0107         weights = max(param.weights_dx, sqrt(abs(r).^2+abs(s).^2));
0108         p = r./weights.*param.weights_dx; q = s./weights.*param.weights_dx;
0109     else
0110         if (~param.zero_weights_flag)
0111             r = r - 1/(8*lambda*param.nu) * dx;
0112             s = s - 1/(8*lambda*param.nu) * dy;
0113             weights = max(1, sqrt(abs(r./param.weights_dx).^2+...
0114                 abs(s./param.weights_dy).^2));
0115             p = r./weights; q = s./weights;
0116         else
0117             % Weights go into grad and div operators so usual update
0118             r = r - 1/(8*lambda*param.nu*max_weights) * dx;
0119             s = s - 1/(8*lambda*param.nu*max_weights) * dy;
0120             weights = max(1, sqrt(abs(r).^2+abs(s).^2));
0121             p = r./weights; q = s./weights;
0122         end
0123     end
0124     
0125     % FISTA update
0126     t = (1+sqrt(4*told^2))/2;
0127     r = p + (told-1)/t * (p - pold); pold = p;
0128     s = q + (told-1)/t * (q - qold); qold = q;
0129     told = t;
0130     
0131 end
0132 
0133 % Log after the minimization
0134 if ~exist('crit_TV', 'var'), crit_TV = 'MAX_IT'; end
0135 if param.verbose >= 1
0136     fprintf(['  Prox_TV: obj = %e, rel_obj = %e,' ...
0137         ' %s, iter = %i\n'], obj, rel_obj, crit_TV, iter);
0138 end
0139 
0140 end
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sopt-2.0.0/matlab/doc/matlab/prox_operators/index.html000066400000000000000000000042371277570055300230470ustar00rootroot00000000000000 Index for Directory matlab/prox_operators
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Index for matlab/prox_operators

Matlab files in this directory:

 sopt_mltb_fast_proj_B2sopt_mltb_fast_proj_B2 - Fast projection onto L2-ball
 sopt_mltb_proj_B2sopt_mltb_proj_B2 - Projection onto L2-ball
 sopt_mltb_prox_L1sopt_mltb_prox_L1 - Proximal operator with L1 norm
 sopt_mltb_prox_TVsopt_mltb_prox_TV - Total variation proximal operator
 sopt_mltb_prox_TVoAsopt_mltb_prox_TVoA - Agumented total variation proximal operator
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/prox_operators/sopt_mltb_fast_proj_B2.html000066400000000000000000000337071277570055300263410ustar00rootroot00000000000000 Description of sopt_mltb_fast_proj_B2
Home > matlab > prox_operators > sopt_mltb_fast_proj_B2.m

sopt_mltb_fast_proj_B2

PURPOSE ^

sopt_mltb_fast_proj_B2 - Fast projection onto L2-ball

SYNOPSIS ^

function [sol, u] = sopt_mltb_fast_proj_B2(x, param)

DESCRIPTION ^

 sopt_mltb_fast_proj_B2 - Fast projection onto L2-ball

 Compute the projection onto the L2 ball, i.e. solve

   min_{z} ||x - z||_2^2   s.t.  ||y - A z||_2 < epsilon

 where x is the input vector and the solution z* is returned as sol.
 The structure param should contain the following fields:

   - y: Measurements (default = 0).

   - A: Forward operator (default = Identity).

   - At: Adjoint operator (default = Identity).

   - epsilon: Radius of the L2 ball (default = 1e-3).

   - tight: 1 if A is a tight frame or 0 otherwise (default = 1).

   - nu: Bound on the norm^2 of the operator A, i.e.
       ||A x||^2 <= nu * ||x||^2 (default = 1).

   - tol: Tolerance for the projection onto the L2 ball. The algorithms
       stops if
         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
       (default = 1e-3).

   - max_iter: Maximum number of iterations (default: 200).

   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - u: Initial vector for the dual problem, same dimension as y
       (default = 0).
   
   - pos: Positivity flag (1 = positive solution,
       0 general complex case; default = 0).

   - real: Reality flag (1 = real solution,
       0 = general complex case; default = 0).

 Outputs:

   - sol: Final solution.

   - u: Final dual vector.

 References:
 [1] M.J. Fadili and J-L. Starck, "Monotone operator splitting for
 optimization problems in sparse recovery" , IEEE ICIP, Cairo,
 Egypt, 2009.
 [2] Amir Beck and Marc Teboulle, "A Fast Iterative Shrinkage-Thresholding
 Algorithm for Linear Inverse Problems",  SIAM Journal on Imaging Sciences
 2 (2009), no. 1, 183--202.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function [sol, u] = sopt_mltb_fast_proj_B2(x, param)
0002 % sopt_mltb_fast_proj_B2 - Fast projection onto L2-ball
0003 %
0004 % Compute the projection onto the L2 ball, i.e. solve
0005 %
0006 %   min_{z} ||x - z||_2^2   s.t.  ||y - A z||_2 < epsilon
0007 %
0008 % where x is the input vector and the solution z* is returned as sol.
0009 % The structure param should contain the following fields:
0010 %
0011 %   - y: Measurements (default = 0).
0012 %
0013 %   - A: Forward operator (default = Identity).
0014 %
0015 %   - At: Adjoint operator (default = Identity).
0016 %
0017 %   - epsilon: Radius of the L2 ball (default = 1e-3).
0018 %
0019 %   - tight: 1 if A is a tight frame or 0 otherwise (default = 1).
0020 %
0021 %   - nu: Bound on the norm^2 of the operator A, i.e.
0022 %       ||A x||^2 <= nu * ||x||^2 (default = 1).
0023 %
0024 %   - tol: Tolerance for the projection onto the L2 ball. The algorithms
0025 %       stops if
0026 %         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
0027 %       (default = 1e-3).
0028 %
0029 %   - max_iter: Maximum number of iterations (default: 200).
0030 %
0031 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0032 %       2 = print main steps; default = 1).
0033 %
0034 %   - u: Initial vector for the dual problem, same dimension as y
0035 %       (default = 0).
0036 %
0037 %   - pos: Positivity flag (1 = positive solution,
0038 %       0 general complex case; default = 0).
0039 %
0040 %   - real: Reality flag (1 = real solution,
0041 %       0 = general complex case; default = 0).
0042 %
0043 % Outputs:
0044 %
0045 %   - sol: Final solution.
0046 %
0047 %   - u: Final dual vector.
0048 %
0049 % References:
0050 % [1] M.J. Fadili and J-L. Starck, "Monotone operator splitting for
0051 % optimization problems in sparse recovery" , IEEE ICIP, Cairo,
0052 % Egypt, 2009.
0053 % [2] Amir Beck and Marc Teboulle, "A Fast Iterative Shrinkage-Thresholding
0054 % Algorithm for Linear Inverse Problems",  SIAM Journal on Imaging Sciences
0055 % 2 (2009), no. 1, 183--202.
0056 
0057 % Optional input arguments
0058 if ~isfield(param, 'y'), param.y = 0; end
0059 if ~isfield(param, 'A'), param.A = @(x) x; end
0060 if ~isfield(param, 'At'), param.At = @(x) x; end
0061 if ~isfield(param, 'epsilon'), param.epsilon = 1e-3; end
0062 if ~isfield(param, 'tight'), param.tight = 1; end
0063 if ~isfield(param, 'tol'), param.tol = 1e-3; end
0064 if ~isfield(param, 'verbose'), param.verbose = 1; end
0065 if ~isfield(param, 'nu'), param.nu = 1; end
0066 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0067 if ~isfield(param, 'u'), param.u = zeros(size(param.y)); end
0068 if ~isfield(param, 'pos'), param.pos = 0; end
0069 if ~isfield(param, 'real'), param.real = 0; end
0070 
0071 % Useful functions for the projection
0072 sc = @(z) z*min(param.epsilon/norm(z(:)), 1); % scaling
0073 
0074 % Projection
0075 if (param.tight && ~(param.pos||param.real)) % TIGHT FRAME CASE
0076     
0077     temp = param.A(x) - param.y;
0078     sol = x + 1/param.nu * param.At(sc(temp)-temp);
0079     crit_B2 = 'TOL_EPS'; iter = 0;
0080     u = 0;
0081     
0082 else % NON TIGHT FRAME CASE
0083     
0084     % Initializations
0085     sol = x; u = param.u; v = u;
0086     iter = 1; true = 1; told = 1;
0087     
0088     % Tolerance onto the L2 ball
0089     epsilon_low = param.epsilon/(1+param.tol);
0090     epsilon_up = param.epsilon/(1-param.tol);
0091     
0092     % Check if we are in the L2 ball
0093     dummy = param.A(sol);
0094     norm_res = norm(param.y(:)-dummy(:), 2);
0095     if norm_res <= epsilon_up
0096         crit_B2 = 'TOL_EPS'; true = 0;
0097     end
0098     
0099     % Projection onto the L2-ball
0100     % Init
0101     if param.verbose > 1
0102         fprintf('  Proj. B2:\n');
0103     end
0104     while true
0105         
0106         % Residual
0107         res = param.A(sol) - param.y; norm_res = norm(res(:), 2);
0108         
0109         % Scaling for the projection
0110         res = u*param.nu + res; norm_proj = norm(res(:), 2);
0111         
0112         % Log
0113         if param.verbose>1
0114             fprintf('   Iter %i, epsilon = %e, ||y - Ax||_2 = %e\n', ...
0115                 iter, param.epsilon, norm_res);
0116         end
0117         
0118         % Stopping criterion
0119         if (norm_res>=epsilon_low && norm_res<=epsilon_up)
0120             crit_B2 = 'TOL_EPS'; break;
0121         elseif iter >= param.max_iter
0122             crit_B2 = 'MAX_IT'; break;
0123         end
0124         
0125         % Projection onto the L2 ball
0126         t = (1+sqrt(1+4*told^2))/2;
0127         ratio = min(1, param.epsilon/norm_proj);
0128         u = v;
0129         v = 1/param.nu * (res - res*ratio);
0130         u = v + (told-1)/t * (v - u);
0131         
0132         % Current estimate
0133         sol = x - param.At(u);
0134         
0135         % Projection onto the non-negative orthant (positivity constraint)
0136         if (param.pos)
0137             sol = real(sol);
0138             sol(sol<0) = 0;
0139             
0140         end
0141         
0142         % Projection onto the real orthant (reality constraint)
0143         if (param.real)
0144             sol = real(sol);            
0145         end        
0146         
0147         % Update number of iteration
0148         told = t;
0149         
0150         % Update number of iterations
0151         iter = iter + 1;
0152         
0153     end
0154 end
0155 
0156 % Log after the projection onto the L2-ball
0157 if param.verbose >= 1
0158     temp = param.A(sol);
0159     fprintf(['  Proj. B2: epsilon = %e, ||y-Ax||_2 = %e,', ...
0160         ' %s, iter = %i\n'], param.epsilon, norm(param.y(:)-temp(:)), ...
0161         crit_B2, iter);
0162 end
0163 
0164 
0165 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/prox_operators/sopt_mltb_proj_B2.html000066400000000000000000000273051277570055300253210ustar00rootroot00000000000000 Description of sopt_mltb_proj_B2
Home > matlab > prox_operators > sopt_mltb_proj_B2.m

sopt_mltb_proj_B2

PURPOSE ^

sopt_mltb_proj_B2 - Projection onto L2-ball

SYNOPSIS ^

function [sol, u] = sopt_mltb_proj_B2(x, param)

DESCRIPTION ^

 sopt_mltb_proj_B2 - Projection onto L2-ball

 Compute the projection onto the L2 ball, i.e. solve

   min_{z} ||x - z||_2^2   s.t.  ||y - A z||_2 < epsilon

 where x is the input vector and the solution z* is returned as sol.
 The structure param should contain the following fields:

   - y: Measurements (default = 0).

   - A: Forward operator (default = Identity).

   - At: Adjoint operator (default = Identity).

   - epsilon: Radius of the L2 ball (default = 1e-3).

   - tight: 1 if A is a tight frame or 0 otherwise (default = 1).

   - nu: Bound on the norm^2 of the operator A, i.e.
       ||A x||^2 <= nu * ||x||^2 (default = 1).

   - tol: Tolerance for the projection onto the L2 ball. The algorithms
       stops if
         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
       (default = 1e-3).

   - max_iter: Maximum number of iterations (default: 200).

   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - u: Initial vector for the dual problem, same dimension as y
       (default = 0).

 Outputs:

   - sol: Final solution.

   - u: Final dual vector.

 References:
 [1] M.J. Fadili and J-L. Starck, "Monotone operator splitting for 
 optimization problems in sparse recovery" , IEEE ICIP, Cairo, 
 Egypt, 2009.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function [sol, u] = sopt_mltb_proj_B2(x, param)
0002 % sopt_mltb_proj_B2 - Projection onto L2-ball
0003 %
0004 % Compute the projection onto the L2 ball, i.e. solve
0005 %
0006 %   min_{z} ||x - z||_2^2   s.t.  ||y - A z||_2 < epsilon
0007 %
0008 % where x is the input vector and the solution z* is returned as sol.
0009 % The structure param should contain the following fields:
0010 %
0011 %   - y: Measurements (default = 0).
0012 %
0013 %   - A: Forward operator (default = Identity).
0014 %
0015 %   - At: Adjoint operator (default = Identity).
0016 %
0017 %   - epsilon: Radius of the L2 ball (default = 1e-3).
0018 %
0019 %   - tight: 1 if A is a tight frame or 0 otherwise (default = 1).
0020 %
0021 %   - nu: Bound on the norm^2 of the operator A, i.e.
0022 %       ||A x||^2 <= nu * ||x||^2 (default = 1).
0023 %
0024 %   - tol: Tolerance for the projection onto the L2 ball. The algorithms
0025 %       stops if
0026 %         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
0027 %       (default = 1e-3).
0028 %
0029 %   - max_iter: Maximum number of iterations (default: 200).
0030 %
0031 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0032 %       2 = print main steps; default = 1).
0033 %
0034 %   - u: Initial vector for the dual problem, same dimension as y
0035 %       (default = 0).
0036 %
0037 % Outputs:
0038 %
0039 %   - sol: Final solution.
0040 %
0041 %   - u: Final dual vector.
0042 %
0043 % References:
0044 % [1] M.J. Fadili and J-L. Starck, "Monotone operator splitting for
0045 % optimization problems in sparse recovery" , IEEE ICIP, Cairo,
0046 % Egypt, 2009.
0047 
0048 % Optional input arguments
0049 if ~isfield(param, 'y'), param.y = 0; end
0050 if ~isfield(param, 'A'), param.A = @(x) x; end
0051 if ~isfield(param, 'At'), param.At = @(x) x; end
0052 if ~isfield(param, 'epsilon'), param.epsilon = 1e-3; end
0053 if ~isfield(param, 'tight'), param.tight = 1; end
0054 if ~isfield(param, 'tol'), param.tol = 1e-3; end
0055 if ~isfield(param, 'verbose'), param.verbose = 1; end
0056 if ~isfield(param, 'nu'), param.nu = 1; end
0057 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0058 if ~isfield(param, 'u'), param.u = zeros(size(param.y)); end
0059 
0060 % Useful functions for the projection
0061 sc = @(z) z*min(param.epsilon/norm(z(:)), 1); % scaling
0062 
0063 % Projection
0064 if param.tight % TIGHT FRAME CASE
0065     
0066     temp = param.A(x) - param.y;
0067     sol = x + 1/param.nu * param.At(sc(temp)-temp);
0068     crit_B2 = 'TOL_EPS'; iter = 0; u = NaN;
0069     
0070 else % NON TIGHT FRAME CASE
0071     
0072     % Initializations
0073     sol = x; u = param.u;
0074     iter = 1; true = 1;
0075     
0076     % Tolerance onto the L2 ball
0077     epsilon_low = param.epsilon/(1+param.tol);
0078     epsilon_up = param.epsilon/(1-param.tol);
0079     
0080     % Check if we are in the L2 ball
0081     norm_res = norm(param.y(:)-param.A(sol), 2);
0082     if norm_res <= epsilon_up
0083         crit_B2 = 'TOL_EPS'; true = 0;
0084     end
0085     
0086     % Projection onto the L2-ball
0087     % Init
0088     if param.verbose > 1
0089         fprintf('  Proj. B2:\n');
0090     end
0091     while true
0092         
0093         % Residual
0094         res = param.A(sol) - param.y; norm_res = norm(res(:), 2);
0095         
0096         % Scaling for the projection
0097         res = u*param.nu + res; norm_proj = norm(res(:), 2);
0098         
0099         % Log
0100         if param.verbose>1
0101             fprintf('   Iter %i, epsilon = %e, ||y - Ax||_2 = %e\n', ...
0102                 iter, param.epsilon, norm_res);
0103         end
0104         
0105         % Stopping criterion
0106         if (norm_res>=epsilon_low && norm_res<=epsilon_up)
0107             crit_B2 = 'TOL_EPS'; break;
0108         elseif iter >= param.max_iter
0109             crit_B2 = 'MAX_IT'; break;
0110         end
0111         
0112         % Projection onto the L2 ball
0113         ratio = min(1, param.epsilon/norm_proj);
0114         u = 1/param.nu * (res - res*ratio);
0115         
0116         % Current estimate
0117         sol = x - param.At(u);
0118         
0119         % Update number of iteration
0120         iter = iter + 1;
0121         
0122     end
0123 end
0124 
0125 % Log after the projection onto the L2-ball
0126 if param.verbose >= 1
0127     temp = param.A(sol);
0128     fprintf(['  Proj. B2: epsilon = %e, ||y-Ax||_2 = %e,', ...
0129         ' %s, iter = %i\n'], param.epsilon, norm(param.y(:)-temp(:)), ...
0130         crit_B2, iter);
0131 end
0132 
0133 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/prox_operators/sopt_mltb_prox_L1.html000066400000000000000000000272541277570055300253530ustar00rootroot00000000000000 Description of sopt_mltb_prox_L1
Home > matlab > prox_operators > sopt_mltb_prox_L1.m

sopt_mltb_prox_L1

PURPOSE ^

sopt_mltb_prox_L1 - Proximal operator with L1 norm

SYNOPSIS ^

function sol = sopt_mltb_prox_L1(x, lambda, param)

DESCRIPTION ^

 sopt_mltb_prox_L1 - Proximal operator with L1 norm

 Compute the L1 proximal operator, i.e. solve

   min_{z} 0.5*||x - z||_2^2 + lambda * ||Psit x||_1 ,

 where x is the input vector and the solution z* is returned as sol.  
 The structure param should contain the following fields:

   - Psit: Sparsifying transform (default = Identity).

   - Psi: Adjoint of Psit (default = Identity).

   - tight: 1 if Psit is a tight frame or 0 otherwise (default = 1).

   - nu: Bound on the norm^2 of the operator Psi, i.e.
       ||Psi x||^2 <= nu * ||x||^2 (default = 1).

   - max_iter: Maximum number of iterations (default = 200).

   - rel_obj: Minimum relative change of the objective value 
       (default = 1e-4).  The algorithm stops if
           | ||x(t)||_1 - ||x(t-1)||_1 | / ||x(t)||_1 < rel_obj,
       where x(t) is the estimate of the solution at iteration t.

   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - weights: Weights for a weighted L1-norm (default = 1).

   - pos: Positivity flag (1 = positive solution,
       0 = general complex case; default = 0).

 References:
 [1] M.J. Fadili and J-L. Starck, "Monotone operator splitting for
 optimization problems in sparse recovery" , IEEE ICIP, Cairo,
 Egypt, 2009.
 [2] Amir Beck and Marc Teboulle, "A Fast Iterative Shrinkage-Thresholding
 Algorithm for Linear Inverse Problems",  SIAM Journal on Imaging Sciences
 2 (2009), no. 1, 183--202.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_prox_L1(x, lambda, param)
0002 % sopt_mltb_prox_L1 - Proximal operator with L1 norm
0003 %
0004 % Compute the L1 proximal operator, i.e. solve
0005 %
0006 %   min_{z} 0.5*||x - z||_2^2 + lambda * ||Psit x||_1 ,
0007 %
0008 % where x is the input vector and the solution z* is returned as sol.
0009 % The structure param should contain the following fields:
0010 %
0011 %   - Psit: Sparsifying transform (default = Identity).
0012 %
0013 %   - Psi: Adjoint of Psit (default = Identity).
0014 %
0015 %   - tight: 1 if Psit is a tight frame or 0 otherwise (default = 1).
0016 %
0017 %   - nu: Bound on the norm^2 of the operator Psi, i.e.
0018 %       ||Psi x||^2 <= nu * ||x||^2 (default = 1).
0019 %
0020 %   - max_iter: Maximum number of iterations (default = 200).
0021 %
0022 %   - rel_obj: Minimum relative change of the objective value
0023 %       (default = 1e-4).  The algorithm stops if
0024 %           | ||x(t)||_1 - ||x(t-1)||_1 | / ||x(t)||_1 < rel_obj,
0025 %       where x(t) is the estimate of the solution at iteration t.
0026 %
0027 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0028 %       2 = print main steps; default = 1).
0029 %
0030 %   - weights: Weights for a weighted L1-norm (default = 1).
0031 %
0032 %   - pos: Positivity flag (1 = positive solution,
0033 %       0 = general complex case; default = 0).
0034 %
0035 % References:
0036 % [1] M.J. Fadili and J-L. Starck, "Monotone operator splitting for
0037 % optimization problems in sparse recovery" , IEEE ICIP, Cairo,
0038 % Egypt, 2009.
0039 % [2] Amir Beck and Marc Teboulle, "A Fast Iterative Shrinkage-Thresholding
0040 % Algorithm for Linear Inverse Problems",  SIAM Journal on Imaging Sciences
0041 % 2 (2009), no. 1, 183--202.
0042 
0043 % Optional input arguments
0044 if ~isfield(param, 'verbose'), param.verbose = 1; end
0045 if ~isfield(param, 'Psit'), param.Psi = @(x) x; param.Psit = @(x) x; end
0046 if ~isfield(param, 'tight'), param.tight = 1; end
0047 if ~isfield(param, 'nu'), param.nu = 1; end
0048 if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end
0049 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0050 if ~isfield(param, 'Psit'), param.Psit = @(x) x; end
0051 if ~isfield(param, 'Psi'), param.Psi = @(x) x; end
0052 if ~isfield(param, 'weights'), param.weights = 1; end
0053 if ~isfield(param, 'pos'), param.pos = 0; end
0054 
0055 % Useful functions
0056 soft = @(z, T) sign(z).*max(abs(z)-T, 0);
0057 
0058 % Projection
0059 if param.tight && ~param.pos % TIGHT FRAME CASE
0060     
0061     temp = param.Psit(x);
0062     sol = x + 1/param.nu * param.Psi(soft(temp, ...
0063         lambda*param.nu*param.weights)-temp);
0064     crit_L1 = 'REL_OBJ'; iter_L1 = 1;
0065     dummy = param.Psit(sol);
0066     norm_l1 = sum(param.weights(:).*abs(dummy(:)));
0067     
0068 else % NON TIGHT FRAME CASE OR CONSTRAINT INVOLVED
0069     
0070     % Initializations
0071     u_l1 = zeros(size(param.Psit(x)));
0072     sol = x - param.Psi(u_l1);
0073     prev_l1 = 0; iter_L1 = 0;
0074     
0075     % Soft-thresholding
0076     % Init
0077     if param.verbose > 1
0078         fprintf('  Proximal l1 operator:\n');
0079     end
0080     while 1
0081         
0082         % L1 norm of the estimate
0083         dummy = param.Psit(sol);
0084         
0085         norm_l1 = .5*norm(x(:) - sol(:), 2)^2 + lambda * ...
0086             sum(param.weights(:).*abs(dummy(:)));
0087         rel_l1 = abs(norm_l1-prev_l1)/norm_l1;
0088         
0089         % Log
0090         if param.verbose>1
0091             fprintf('   Iter %i, ||Psit x||_1 = %e, rel_l1 = %e\n', ...
0092                 iter_L1, norm_l1, rel_l1);
0093         end
0094         
0095         % Stopping criterion
0096         if (rel_l1 < param.rel_obj)
0097             crit_L1 = 'REL_OB'; break;
0098         elseif iter_L1 >= param.max_iter
0099             crit_L1 = 'MAX_IT'; break;
0100         end
0101         
0102         % Soft-thresholding
0103         res = u_l1*param.nu + param.Psit(sol);
0104         dummy = soft(res, lambda*param.nu*param.weights);
0105         if param.pos
0106             dummy = real(dummy); dummy(dummy<0) = 0;
0107         end
0108         u_l1 = 1/param.nu * (res - dummy);
0109         sol = x - param.Psi(u_l1);
0110         
0111         % Update
0112         prev_l1 = norm_l1;
0113         iter_L1 = iter_L1 + 1;
0114         
0115     end
0116 end
0117 
0118 % Log after the projection onto the L2-ball
0119 if param.verbose >= 1
0120     fprintf(['  prox_L1: ||Psi x||_1 = %e,', ...
0121         ' %s, iter = %i\n'], norm_l1, crit_L1, iter_L1);
0122 end
0123 
0124 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/prox_operators/sopt_mltb_prox_TV.html000066400000000000000000000200761277570055300254230ustar00rootroot00000000000000 Description of sopt_mltb_prox_TV
Home > matlab > prox_operators > sopt_mltb_prox_TV.m

sopt_mltb_prox_TV

PURPOSE ^

sopt_mltb_prox_TV - Total variation proximal operator

SYNOPSIS ^

function sol = sopt_mltb_prox_TV(x, lambda, param)

DESCRIPTION ^

 sopt_mltb_prox_TV - Total variation proximal operator

 Compute the TV proximal operator, i.e. solve

   min_{z} ||x - z||_2^2 + lambda * ||z||_{TV}

 where x is the input vector and the solution z* is returned as sol.  
 The structure param should contain the following fields:

   - max_iter: Maximum number of iterations (default = 200).

   - rel_obj: Minimum relative change of the objective value 
       (default = 1e-4).  The algorithm stops if
           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj,
       where x(t) is the estimate of the solution at iteration t.

   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

 Reference:
 [1] A. Beck and  M. Teboulle, "Fast gradient-based algorithms for
 constrained Total Variation Image Denoising and Deblurring Problems", 
 IEEE Transactions on Image Processing, VOL. 18, NO. 11, 2419-2434, 
 November 2009.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_prox_TV(x, lambda, param)
0002 % sopt_mltb_prox_TV - Total variation proximal operator
0003 %
0004 % Compute the TV proximal operator, i.e. solve
0005 %
0006 %   min_{z} ||x - z||_2^2 + lambda * ||z||_{TV}
0007 %
0008 % where x is the input vector and the solution z* is returned as sol.
0009 % The structure param should contain the following fields:
0010 %
0011 %   - max_iter: Maximum number of iterations (default = 200).
0012 %
0013 %   - rel_obj: Minimum relative change of the objective value
0014 %       (default = 1e-4).  The algorithm stops if
0015 %           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj,
0016 %       where x(t) is the estimate of the solution at iteration t.
0017 %
0018 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0019 %       2 = print main steps; default = 1).
0020 %
0021 % Reference:
0022 % [1] A. Beck and  M. Teboulle, "Fast gradient-based algorithms for
0023 % constrained Total Variation Image Denoising and Deblurring Problems",
0024 % IEEE Transactions on Image Processing, VOL. 18, NO. 11, 2419-2434,
0025 % November 2009.
0026 
0027 % Optional input arguments
0028 if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end
0029 if ~isfield(param, 'verbose'), param.verbose = 1; end
0030 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0031 
0032 % Initializations
0033 [r, s] = sopt_mltb_gradient_op(x*0);
0034 pold = r; qold = s;
0035 told = 1; prev_obj = 0;
0036 
0037 % Main iterations
0038 if param.verbose > 1
0039     fprintf('  Proximal TV operator:\n');
0040 end
0041 for iter = 1:param.max_iter
0042     
0043     % Current solution
0044     sol = x - lambda*sopt_mltb_div_op(r, s);
0045     
0046     % Objective function value
0047     obj = .5*norm(x(:)-sol(:), 2)^2 + lambda * sopt_mltb_TV_norm(sol, 0);
0048     rel_obj = abs(obj-prev_obj)/obj;
0049     prev_obj = obj;
0050     
0051     % Stopping criterion
0052     if param.verbose>1
0053         fprintf('   Iter %i, obj = %e, rel_obj = %e\n', ...
0054             iter, obj, rel_obj);
0055     end
0056     if rel_obj < param.rel_obj
0057         crit_TV = 'TOL_EPS'; break;
0058     end
0059     
0060     % Udpate divergence vectors and project
0061     [dx, dy] = sopt_mltb_gradient_op(sol);
0062     r = r - 1/(8*lambda) * dx; s = s - 1/(8*lambda) * dy;
0063     weights = max(1, sqrt(abs(r).^2+abs(s).^2));
0064     p = r./weights; q = s./weights;
0065     
0066     % FISTA update
0067     t = (1+sqrt(4*told^2))/2;
0068     r = p + (told-1)/t * (p - pold); pold = p;
0069     s = q + (told-1)/t * (q - qold); qold = q;
0070     told = t;
0071     
0072 end
0073 
0074 % Log after the minimization
0075 if ~exist('crit_TV', 'var'), crit_TV = 'MAX_IT'; end
0076 if param.verbose >= 1
0077     fprintf(['  Prox_TV: obj = %e, rel_obj = %e,' ...
0078         ' %s, iter = %i\n'], obj, rel_obj, crit_TV, iter);
0079 end
0080 
0081 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/prox_operators/sopt_mltb_prox_TVoA.html000066400000000000000000000306471277570055300257100ustar00rootroot00000000000000 Description of sopt_mltb_prox_TVoA
Home > matlab > prox_operators > sopt_mltb_prox_TVoA.m

sopt_mltb_prox_TVoA

PURPOSE ^

sopt_mltb_prox_TVoA - Agumented total variation proximal operator

SYNOPSIS ^

function sol = sopt_mltb_prox_TVoA(b, lambda, param)

DESCRIPTION ^

 sopt_mltb_prox_TVoA - Agumented total variation proximal operator

 Compute the TV proximal operator when an additional linear operator A is
 incorporated in the TV norm, i.e. solve

   min_{x} ||y - x||_2^2 + lambda * ||A x||_{TV}

 where x is the input vector and the solution z* is returned as sol.  
 The structure param should contain the following fields:

   - max_iter: Maximum number of iterations (default = 200).

   - rel_obj: Minimum relative change of the objective value 
       (default = 1e-4).  The algorithm stops if
           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_1 < rel_obj,
       where x(t) is the estimate of the solution at iteration t.

   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - A: Forward transform (default = Identity).

   - At: Adjoint of At (default = Identity).

   - nu: Bound on the norm^2 of the operator A, i.e.
       ||A x||^2 <= nu * ||x||^2 (default = 1)

 Reference:
 [1] A. Beck and  M. Teboulle, "Fast gradient-based algorithms for
 constrained Total Variation Image Denoising and Deblurring Problems",
 IEEE Transactions on Image Processing, VOL. 18, NO. 11, 2419-2434,
 November 2009.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_prox_TVoA(b, lambda, param)
0002 % sopt_mltb_prox_TVoA - Agumented total variation proximal operator
0003 %
0004 % Compute the TV proximal operator when an additional linear operator A is
0005 % incorporated in the TV norm, i.e. solve
0006 %
0007 %   min_{x} ||y - x||_2^2 + lambda * ||A x||_{TV}
0008 %
0009 % where x is the input vector and the solution z* is returned as sol.
0010 % The structure param should contain the following fields:
0011 %
0012 %   - max_iter: Maximum number of iterations (default = 200).
0013 %
0014 %   - rel_obj: Minimum relative change of the objective value
0015 %       (default = 1e-4).  The algorithm stops if
0016 %           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_1 < rel_obj,
0017 %       where x(t) is the estimate of the solution at iteration t.
0018 %
0019 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0020 %       2 = print main steps; default = 1).
0021 %
0022 %   - A: Forward transform (default = Identity).
0023 %
0024 %   - At: Adjoint of At (default = Identity).
0025 %
0026 %   - nu: Bound on the norm^2 of the operator A, i.e.
0027 %       ||A x||^2 <= nu * ||x||^2 (default = 1)
0028 %
0029 % Reference:
0030 % [1] A. Beck and  M. Teboulle, "Fast gradient-based algorithms for
0031 % constrained Total Variation Image Denoising and Deblurring Problems",
0032 % IEEE Transactions on Image Processing, VOL. 18, NO. 11, 2419-2434,
0033 % November 2009.
0034 
0035 % Optional input arguments
0036 if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end
0037 if ~isfield(param, 'verbose'), param.verbose = 1; end
0038 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0039 if ~isfield(param, 'At'), param.At = @(x) x; end
0040 if ~isfield(param, 'A'), param.A = @(x) x; end
0041 if ~isfield(param, 'nu'), param.nu = 1; end
0042 
0043 % Advanced input arguments (not exposed in documentation)
0044 if ~isfield(param, 'weights_dx'), param.weights_dx = 1; end
0045 if ~isfield(param, 'weights_dy'), param.weights_dy = 1; end
0046 if ~isfield(param, 'zero_weights_flag'), param.zero_weights_flag = 1; end
0047 if ~isfield(param, 'identical_weights_flag')
0048     param.identical_weights_flag = 0; 
0049 end
0050 if ~isfield(param, 'sphere_flag'), param.sphere_flag = 0; end
0051 if ~isfield(param, 'incNP'), param.incNP = 0; end
0052 
0053 % Set grad and div operators to planar or spherical case and also
0054 % include weights or not (depending on parameter flags).
0055 if (param.sphere_flag)
0056    G = @sopt_mltb_gradient_op_sphere;
0057    D = @sopt_mltb_div_op_sphere;
0058 else
0059    G = @sopt_mltb_gradient_op;
0060    D = @sopt_mltb_div_op;
0061 end
0062 
0063 if (~param.identical_weights_flag && param.zero_weights_flag)
0064     grad = @(x) G(x, param.weights_dx, param.weights_dy);
0065     div = @(r, s) D(r, s, param.weights_dx, param.weights_dy);
0066     max_weights = max([abs(param.weights_dx(:)); ...
0067         abs(param.weights_dy(:))])^2;
0068 else
0069     grad = @(x) G(x);
0070     div = @(r, s) D(r, s);
0071 end
0072 
0073 % Initializations
0074 [r, s] = grad(param.A(b*0));
0075 pold = r; qold = s;
0076 told = 1; prev_obj = 0;
0077 
0078 % Main iterations
0079 if param.verbose > 1
0080     fprintf('  Proximal TV operator:\n');
0081 end
0082 for iter = 1:param.max_iter
0083     
0084     % Current solution
0085     sol = b - lambda*param.At(div(r, s));
0086     
0087     % Objective function value
0088     obj = .5*norm(b(:)-sol(:), 2) + lambda * ...
0089         sopt_mltb_TV_norm(param.A(sol), param.weights_dx, param.weights_dy);
0090     rel_obj = abs(obj-prev_obj)/obj;
0091     prev_obj = obj;
0092     
0093     % Stopping criterion
0094     if param.verbose>1
0095         fprintf('   Iter %i, obj = %e, rel_obj = %e\n', ...
0096             iter, obj, rel_obj);
0097     end
0098     if rel_obj < param.rel_obj
0099         crit_TV = 'TOL_EPS'; break;
0100     end
0101     
0102     % Udpate divergence vectors and project
0103     [dx, dy] = grad(param.A(sol));
0104     if (param.identical_weights_flag)
0105         r = r - 1/(8*lambda*param.nu) * dx;
0106         s = s - 1/(8*lambda*param.nu) * dy;
0107         weights = max(param.weights_dx, sqrt(abs(r).^2+abs(s).^2));
0108         p = r./weights.*param.weights_dx; q = s./weights.*param.weights_dx;
0109     else
0110         if (~param.zero_weights_flag)
0111             r = r - 1/(8*lambda*param.nu) * dx;
0112             s = s - 1/(8*lambda*param.nu) * dy;
0113             weights = max(1, sqrt(abs(r./param.weights_dx).^2+...
0114                 abs(s./param.weights_dy).^2));
0115             p = r./weights; q = s./weights;
0116         else
0117             % Weights go into grad and div operators so usual update
0118             r = r - 1/(8*lambda*param.nu*max_weights) * dx;
0119             s = s - 1/(8*lambda*param.nu*max_weights) * dy;
0120             weights = max(1, sqrt(abs(r).^2+abs(s).^2));
0121             p = r./weights; q = s./weights;
0122         end
0123     end
0124     
0125     % FISTA update
0126     t = (1+sqrt(4*told^2))/2;
0127     r = p + (told-1)/t * (p - pold); pold = p;
0128     s = q + (told-1)/t * (q - qold); qold = q;
0129     told = t;
0130     
0131 end
0132 
0133 % Log after the minimization
0134 if ~exist('crit_TV', 'var'), crit_TV = 'MAX_IT'; end
0135 if param.verbose >= 1
0136     fprintf(['  Prox_TV: obj = %e, rel_obj = %e,' ...
0137         ' %s, iter = %i\n'], obj, rel_obj, crit_TV, iter);
0138 end
0139 
0140 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/sopt_mltb_solve_BPDN.html000066400000000000000000000411461277570055300226700ustar00rootroot00000000000000 Description of sopt_mltb_solve_BPDN
Home > matlab > sopt_mltb_solve_BPDN.m

sopt_mltb_solve_BPDN

PURPOSE ^

sopt_mltb_solve_BPDN - Solve BPDN problem

SYNOPSIS ^

function sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param)

DESCRIPTION ^

 sopt_mltb_solve_BPDN - Solve BPDN problem

 Solve the Basis Pursuit Denoising (BPDN) problem

   min ||Psit x||_1   s.t.  ||y-A x||_2 < epsilon

 where y contains the measurements, A is the forward measurement operator 
 and At the associated adjoint operator, Psit is a sparfying transform and
 Psi its adjoint. The structure param should contain the following fields:

   General parameters:
 
   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - max_iter: Maximum number of iterations (default = 200).

   - rel_obj: Minimum relative change of the objective value 
       (default = 1e-4).  The algorithm stops if
           | ||x(t)||_1 - ||x(t-1)||_1 | / ||x(t)||_1 < rel_obj,
       where x(t) is the estimate of the solution at iteration t.

   - gamma: Convergence speed (weighting of L1 norm when solving for
       L1 proximal operator) (default = 1e-1).

   - initsol: Initial solution for a warmstart.
 
   Projection onto the L2-ball:

   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
 
   - nu_B2: Bound on the norm of the operator A, i.e.
       ||A x||^2 <= nu * ||x||^2 (default = 1).

   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
       stops if
         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
       (default = 1e-3).

   - max_iter_B2: Maximum number of iterations for the projection onto the
       L2 ball (default = 200).

   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
       default = 0).

   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
       default = 0).
 
   Proximal L1 operator:

   - rel_obj_L1: Used as stopping criterion for the proximal L1
       operator. Minimum relative change of the objective value between
       two successive estimates.

   - max_iter_L1: Used as stopping criterion for the proximal L1
       operator. Maximun number of iterations.
 
   - param.nu_L1: Bound on the norm^2 of the operator Psi, i.e.
       ||Psi x||^2 <= nu * ||x||^2 (default = 1).
 
   - param.tight_L1: 1 if Psit is a tight frame, 0 otherwise 
       (default = 1).
 
   - param.weights: Weights for a weighted L1-norm defined
       by sum_i{weights_i.*abs(x_i)} (default = 1). 

   - pos_l1: Positivity flag (1 to impose positivity, 0 otherwise;
       default = 0).

 References:
 [1] P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting 
 Approach to Nonsmooth Convex Variational Signal Recovery", IEEE Journal
 of Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param)
0002 % sopt_mltb_solve_BPDN - Solve BPDN problem
0003 %
0004 % Solve the Basis Pursuit Denoising (BPDN) problem
0005 %
0006 %   min ||Psit x||_1   s.t.  ||y-A x||_2 < epsilon
0007 %
0008 % where y contains the measurements, A is the forward measurement operator
0009 % and At the associated adjoint operator, Psit is a sparfying transform and
0010 % Psi its adjoint. The structure param should contain the following fields:
0011 %
0012 %   General parameters:
0013 %
0014 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0015 %       2 = print main steps; default = 1).
0016 %
0017 %   - max_iter: Maximum number of iterations (default = 200).
0018 %
0019 %   - rel_obj: Minimum relative change of the objective value
0020 %       (default = 1e-4).  The algorithm stops if
0021 %           | ||x(t)||_1 - ||x(t-1)||_1 | / ||x(t)||_1 < rel_obj,
0022 %       where x(t) is the estimate of the solution at iteration t.
0023 %
0024 %   - gamma: Convergence speed (weighting of L1 norm when solving for
0025 %       L1 proximal operator) (default = 1e-1).
0026 %
0027 %   - initsol: Initial solution for a warmstart.
0028 %
0029 %   Projection onto the L2-ball:
0030 %
0031 %   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
0032 %
0033 %   - nu_B2: Bound on the norm of the operator A, i.e.
0034 %       ||A x||^2 <= nu * ||x||^2 (default = 1).
0035 %
0036 %   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
0037 %       stops if
0038 %         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
0039 %       (default = 1e-3).
0040 %
0041 %   - max_iter_B2: Maximum number of iterations for the projection onto the
0042 %       L2 ball (default = 200).
0043 %
0044 %   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
0045 %       default = 0).
0046 %
0047 %   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
0048 %       default = 0).
0049 %
0050 %   Proximal L1 operator:
0051 %
0052 %   - rel_obj_L1: Used as stopping criterion for the proximal L1
0053 %       operator. Minimum relative change of the objective value between
0054 %       two successive estimates.
0055 %
0056 %   - max_iter_L1: Used as stopping criterion for the proximal L1
0057 %       operator. Maximun number of iterations.
0058 %
0059 %   - param.nu_L1: Bound on the norm^2 of the operator Psi, i.e.
0060 %       ||Psi x||^2 <= nu * ||x||^2 (default = 1).
0061 %
0062 %   - param.tight_L1: 1 if Psit is a tight frame, 0 otherwise
0063 %       (default = 1).
0064 %
0065 %   - param.weights: Weights for a weighted L1-norm defined
0066 %       by sum_i{weights_i.*abs(x_i)} (default = 1).
0067 %
0068 %   - pos_l1: Positivity flag (1 to impose positivity, 0 otherwise;
0069 %       default = 0).
0070 %
0071 % References:
0072 % [1] P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting
0073 % Approach to Nonsmooth Convex Variational Signal Recovery", IEEE Journal
0074 % of Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.
0075 
0076 % Optional input arguments
0077 if ~isfield(param, 'verbose'), param.verbose = 1; end
0078 if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end
0079 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0080 if ~isfield(param, 'gamma'), param.gamma = 1e-2; end
0081 if ~isfield(param, 'pos_l1'), param.pos_l1 = 0; end
0082 
0083 % Input arguments for projection onto the L2 ball
0084 param_B2.A = A; param_B2.At = At;
0085 param_B2.y = y; param_B2.epsilon = epsilon;
0086 param_B2.verbose = param.verbose;
0087 if isfield(param, 'nu_B2'), param_B2.nu = param.nu_B2; end
0088 if isfield(param, 'tol_B2'), param_B2.tol = param.tol_B2; end
0089 if isfield(param, 'tight_B2'), param_B2.tight = param.tight_B2; end
0090 if isfield(param, 'max_iter_B2')
0091     param_B2.max_iter = param.max_iter_B2;
0092 end
0093 if isfield(param,'pos_B2'), param_B2.pos=param.pos_B2; end
0094 if isfield(param,'real_B2'), param_B2.real=param.real_B2; end
0095 
0096 % Input arguments for prox L1
0097 param_L1.Psi = Psi; param_L1.Psit = Psit; param_L1.pos = param.pos_l1;
0098 param_L1.verbose = param.verbose; 
0099 %param_L1.verbose = 2;
0100 param_L1.rel_obj = param.rel_obj;
0101 if isfield(param, 'nu_L1')
0102     param_L1.nu = param.nu_L1;
0103 end
0104 if isfield(param, 'tight_L1')
0105     param_L1.tight = param.tight_L1;
0106 end
0107 if isfield(param, 'max_iter_L1')
0108     param_L1.max_iter = param.max_iter_L1;
0109 end
0110 if isfield(param, 'rel_obj_L1')
0111     param_L1.rel_obj = param.rel_obj_L1;
0112 end
0113 if isfield(param, 'weights')
0114     param_L1.weights = param.weights;
0115 else
0116     param_L1.weights = 1;
0117 end
0118 
0119 % Initialization
0120 if isfield(param,'initsol')
0121     xhat = param.initsol;
0122 else
0123     xhat = At(y); 
0124 end
0125 
0126 iter = 1; prev_norm = 0;
0127 
0128 % Main loop
0129 while 1
0130     
0131     if param.verbose >= 1
0132         fprintf('Iteration %i:\n', iter);
0133     end
0134     
0135     % Projection onto the L2-ball
0136     [sol, param_B2.u] = sopt_mltb_fast_proj_B2(xhat, param_B2);
0137     
0138     % Global stopping criterion
0139     dummy = Psit(sol);
0140     curr_norm = sum(param_L1.weights(:).*abs(dummy(:)));    
0141     rel_norm = abs(curr_norm - prev_norm)/curr_norm;
0142     if param.verbose >= 1
0143         fprintf('  ||x||_1 = %e, rel_norm = %e\n', ...
0144             curr_norm, rel_norm);
0145     end
0146     if (rel_norm < param.rel_obj)
0147         crit_BPDN = 'REL_NORM';
0148         break;
0149     elseif iter >= param.max_iter
0150         crit_BPDN = 'MAX_IT';
0151         break;
0152     end
0153     
0154     % Proximal L1 operator
0155     xhat = 2*sol - xhat;
0156     temp = sopt_mltb_prox_L1(xhat, param.gamma, param_L1);
0157     xhat = temp + sol - xhat;
0158     
0159     % Update variables
0160     iter = iter + 1;
0161     prev_norm = curr_norm;
0162     
0163 end
0164 
0165 % Log
0166 if param.verbose >= 1
0167   
0168     % L1 norm
0169     fprintf('\n Solution found:\n');
0170     fprintf(' Final L1 norm: %e\n', curr_norm);
0171     
0172     % Residual
0173     dummy = A(sol); res = norm(y(:)-dummy(:), 2);
0174     fprintf(' epsilon = %e, ||y-Ax||_2=%e\n', epsilon, res);
0175     
0176     % Stopping criterion
0177     fprintf(' %i iterations\n', iter);
0178     fprintf(' Stopping criterion: %s \n\n', crit_BPDN);
0179     
0180 end
0181 
0182 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/sopt_mltb_solve_L2DN.html000066400000000000000000000314111277570055300226360ustar00rootroot00000000000000 Description of sopt_mltb_solve_L2DN
Home > matlab > sopt_mltb_solve_L2DN.m

sopt_mltb_solve_L2DN

PURPOSE ^

sopt_mltb_solve_L2DN - Solve L2DN problem.

SYNOPSIS ^

function sol = sopt_mltb_solve_L2DN(y, epsilon, A, At, param)

DESCRIPTION ^

 sopt_mltb_solve_L2DN - Solve L2DN problem.

 Solve the L2 denoising problem

   min ||x||_2   s.t.  ||y-A x||_2 < epsilon

 where y contains the measurements, A is the forward measurement operator 
 and At the associated adjoint operator. The structure param should 
 contain the following fields:

   General parameters:
 
   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - max_iter: Maximum number of iterations (default = 200).

   - rel_obj: Minimum relative change of the objective value 
       (default = 1e-4).  The algorithm stops if
           | ||x(t)||_2 - ||x(t-1)||_2 | / ||x(t)||_2 < rel_obj,
       where x(t) is the estimate of the solution at iteration t.

   - gamma: Convergence speed (weighting of L1 norm when solving for
       L1 proximal operator) (default = 1e-1).

   - param.weights: weightsfor a weighted L2-norm defined
       by norm(weights_i.*x_i,2) (default = 1).

   - initsol: Initial solution for a warmstart.

   Projection onto the L2-ball:

   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
 
   - nu_B2: Bound on the norm of the operator A, i.e.
       ||A x||^2 <= nu * ||x||^2 (default = 1).

   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
       stops if
         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
       (default = 1e-3).

   - max_iter_B2: Maximum number of iterations for the projection onto the
       L2 ball (default = 200).

   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
       default = 0).

   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
       default = 0).

 References:
 [1] P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting 
 Approach to Nonsmooth Convex Variational Signal Recovery", IEEE Journal
 of Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_solve_L2DN(y, epsilon, A, At, param)
0002 % sopt_mltb_solve_L2DN - Solve L2DN problem.
0003 %
0004 % Solve the L2 denoising problem
0005 %
0006 %   min ||x||_2   s.t.  ||y-A x||_2 < epsilon
0007 %
0008 % where y contains the measurements, A is the forward measurement operator
0009 % and At the associated adjoint operator. The structure param should
0010 % contain the following fields:
0011 %
0012 %   General parameters:
0013 %
0014 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0015 %       2 = print main steps; default = 1).
0016 %
0017 %   - max_iter: Maximum number of iterations (default = 200).
0018 %
0019 %   - rel_obj: Minimum relative change of the objective value
0020 %       (default = 1e-4).  The algorithm stops if
0021 %           | ||x(t)||_2 - ||x(t-1)||_2 | / ||x(t)||_2 < rel_obj,
0022 %       where x(t) is the estimate of the solution at iteration t.
0023 %
0024 %   - gamma: Convergence speed (weighting of L1 norm when solving for
0025 %       L1 proximal operator) (default = 1e-1).
0026 %
0027 %   - param.weights: weightsfor a weighted L2-norm defined
0028 %       by norm(weights_i.*x_i,2) (default = 1).
0029 %
0030 %   - initsol: Initial solution for a warmstart.
0031 %
0032 %   Projection onto the L2-ball:
0033 %
0034 %   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
0035 %
0036 %   - nu_B2: Bound on the norm of the operator A, i.e.
0037 %       ||A x||^2 <= nu * ||x||^2 (default = 1).
0038 %
0039 %   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
0040 %       stops if
0041 %         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
0042 %       (default = 1e-3).
0043 %
0044 %   - max_iter_B2: Maximum number of iterations for the projection onto the
0045 %       L2 ball (default = 200).
0046 %
0047 %   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
0048 %       default = 0).
0049 %
0050 %   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
0051 %       default = 0).
0052 %
0053 % References:
0054 % [1] P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting
0055 % Approach to Nonsmooth Convex Variational Signal Recovery", IEEE Journal
0056 % of Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.
0057 
0058 % Optional input arguments
0059 if ~isfield(param, 'verbose'), param.verbose = 1; end
0060 if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end
0061 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0062 if ~isfield(param, 'gamma'), param.gamma = 1e-2; end
0063 if ~isfield(param, 'weights'), param.weights = 1; end
0064 
0065 % Input arguments for projection onto the L2 ball
0066 param_B2.A = A; param_B2.At = At;
0067 param_B2.y = y; param_B2.epsilon = epsilon;
0068 param_B2.verbose = param.verbose;
0069 if isfield(param, 'nu_B2'), param_B2.nu = param.nu_B2; end
0070 if isfield(param, 'tol_B2'), param_B2.tol = param.tol_B2; end
0071 if isfield(param, 'tight_B2'), param_B2.tight = param.tight_B2; end
0072 if isfield(param, 'max_iter_B2')
0073     param_B2.max_iter = param.max_iter_B2;
0074 end
0075 if isfield(param,'pos_B2'), param_B2.pos=param.pos_B2; end
0076 if isfield(param,'real_B2'), param_B2.real=param.real_B2; end
0077 
0078 % Initialization
0079 if isfield(param,'initsol')
0080     xhat = param.initsol;
0081 else
0082     xhat = At(y); 
0083 end
0084 
0085 iter = 1; prev_norm = 0;
0086 
0087 % Main loop
0088 while 1
0089     
0090     %
0091     if param.verbose >= 1
0092         fprintf('Iteration %i:\n', iter);
0093     end
0094     
0095     % Projection onto the L2-ball
0096     [sol, param_B2.u] = sopt_mltb_fast_proj_B2(xhat, param_B2);
0097     
0098     % Global stopping criterion
0099     dummy = sol;
0100     curr_norm = norm(param.weights(:).*dummy(:));    
0101     rel_norm = abs(curr_norm - prev_norm)/curr_norm;
0102     if param.verbose >= 1
0103         fprintf('  ||x||_1 = %e, rel_norm = %e\n', ...
0104             curr_norm, rel_norm);
0105     end
0106     if (rel_norm < param.rel_obj)
0107         crit_BPDN = 'REL_NORM';
0108         break;
0109     elseif iter >= param.max_iter
0110         crit_BPDN = 'MAX_IT';
0111         break;
0112     end
0113     
0114     % Proximal L2 operator
0115     xhat = 2*sol - xhat;   
0116     temp = xhat ./ (1 + 2*param.weights);    
0117     xhat = temp + sol - xhat;
0118     
0119     % Update variables
0120     iter = iter + 1;
0121     prev_norm = curr_norm;
0122     
0123 end
0124 
0125 % Log
0126 if param.verbose >= 1
0127   
0128     % L1 norm
0129     fprintf('\n Solution found:\n');
0130     fprintf(' Final L1 norm: %e\n', curr_norm);
0131     
0132     % Residual
0133     dummy = A(sol); res = norm(y(:)-dummy(:), 2);
0134     fprintf(' epsilon = %e, ||y-Ax||_2=%e\n', epsilon, res);
0135     
0136     % Stopping criterion
0137     fprintf(' %i iterations\n', iter);
0138     fprintf(' Stopping criterion: %s \n\n', crit_BPDN);
0139     
0140 end
0141 
0142 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/sopt_mltb_solve_TVDN.html000066400000000000000000000330451277570055300227170ustar00rootroot00000000000000 Description of sopt_mltb_solve_TVDN
Home > matlab > sopt_mltb_solve_TVDN.m

sopt_mltb_solve_TVDN

PURPOSE ^

sopt_mltb_solve_TVDN - Solve TVDN problem

SYNOPSIS ^

function sol = sopt_mltb_solve_TVDN(y, epsilon, A, At, param)

DESCRIPTION ^

 sopt_mltb_solve_TVDN - Solve TVDN problem

 Solve the total variation denoising (TVDN) problem

   min ||x||_TV   s.t.  ||y - A x||_2 < epsilon

 where y contains the measurements, A is the forward measurement operator 
 and At the associated adjoint operator. The structure param should
 contain the following fields:

   General parameters:
 
   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - max_iter: Maximum number of iterations (default = 200).

   - rel_obj: Minimum relative change of the objective value 
       (default = 1e-4).  The algorithm stops if
           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj,
       where x(t) is the estimate of the solution at iteration t.

   - gamma: Convergence speed (weighting of TV norm when solving for
       TV proximal operator) (default = 1e-1).

   - initsol: Initial solution for a warmstart.
 
   Projection onto the L2-ball:

   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
 
   - nu_B2: Bound on the norm of the operator A, i.e.
       ||A x||^2 <= nu * ||x||^2 (default = 1).

   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
       stops if
         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
       (default = 1e-3).

   - max_iter_B2: Maximum number of iterations for the projection onto the
       L2 ball (default = 200).

   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
       default = 0).

   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
       default = 0).
 
   Proximal L1 operator:

   - max_iter_TV: Used as stopping criterion for the proximal TV
       operator. Maximun number of iterations (default = 200).

 References:
 P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting Approach
 to Nonsmooth Convex Variational Signal Recovery", IEEE Journal of
 Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_solve_TVDN(y, epsilon, A, At, param)
0002 % sopt_mltb_solve_TVDN - Solve TVDN problem
0003 %
0004 % Solve the total variation denoising (TVDN) problem
0005 %
0006 %   min ||x||_TV   s.t.  ||y - A x||_2 < epsilon
0007 %
0008 % where y contains the measurements, A is the forward measurement operator
0009 % and At the associated adjoint operator. The structure param should
0010 % contain the following fields:
0011 %
0012 %   General parameters:
0013 %
0014 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0015 %       2 = print main steps; default = 1).
0016 %
0017 %   - max_iter: Maximum number of iterations (default = 200).
0018 %
0019 %   - rel_obj: Minimum relative change of the objective value
0020 %       (default = 1e-4).  The algorithm stops if
0021 %           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj,
0022 %       where x(t) is the estimate of the solution at iteration t.
0023 %
0024 %   - gamma: Convergence speed (weighting of TV norm when solving for
0025 %       TV proximal operator) (default = 1e-1).
0026 %
0027 %   - initsol: Initial solution for a warmstart.
0028 %
0029 %   Projection onto the L2-ball:
0030 %
0031 %   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
0032 %
0033 %   - nu_B2: Bound on the norm of the operator A, i.e.
0034 %       ||A x||^2 <= nu * ||x||^2 (default = 1).
0035 %
0036 %   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
0037 %       stops if
0038 %         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
0039 %       (default = 1e-3).
0040 %
0041 %   - max_iter_B2: Maximum number of iterations for the projection onto the
0042 %       L2 ball (default = 200).
0043 %
0044 %   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
0045 %       default = 0).
0046 %
0047 %   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
0048 %       default = 0).
0049 %
0050 %   Proximal L1 operator:
0051 %
0052 %   - max_iter_TV: Used as stopping criterion for the proximal TV
0053 %       operator. Maximun number of iterations (default = 200).
0054 %
0055 % References:
0056 % P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting Approach
0057 % to Nonsmooth Convex Variational Signal Recovery", IEEE Journal of
0058 % Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.
0059 
0060 % Optional input arguments
0061 if ~isfield(param, 'verbose'), param.verbose = 1; end
0062 if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end
0063 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0064 if ~isfield(param, 'gamma'), param.gamma = 1e-2; end
0065 
0066 % Input arguments for projection onto the L2 ball
0067 param_B2.A = A; param_B2.At = At;
0068 param_B2.y = y; param_B2.epsilon = epsilon;
0069 param_B2.verbose = param.verbose;
0070 if isfield(param, 'nu_B2'), param_B2.nu = param.nu_B2; end
0071 if isfield(param, 'tol_B2'), param_B2.tol = param.tol_B2; end
0072 if isfield(param, 'tight_B2'), param_B2.tight = param.tight_B2; end
0073 if isfield(param, 'max_iter_B2')
0074     param_B2.max_iter = param.max_iter_B2;
0075 end
0076 if isfield(param,'pos_B2'), param_B2.pos=param.pos_B2; end
0077 if isfield(param,'real_B2'), param_B2.real=param.real_B2; end
0078 
0079 % Input arguments for prox TV
0080 param_TV.verbose = param.verbose; param_TV.rel_obj = param.rel_obj;
0081 if isfield(param, 'max_iter_TV')
0082     param_TV.max_iter= param.max_iter_TV;
0083 end
0084 
0085 % Initialization
0086 if isfield(param,'initsol')
0087     xhat = param.initsol;
0088 else
0089     xhat = At(y); 
0090 end 
0091 
0092 iter = 1; prev_norm = 0;
0093 
0094 % Main loop
0095 while 1
0096     
0097     if param.verbose>=1
0098         fprintf('Iteration %i:\n', iter);
0099     end
0100     
0101     % Projection onto the L2-ball
0102     [sol, param_B2.u] = sopt_mltb_fast_proj_B2(xhat, param_B2);
0103     
0104     % Global stopping criterion
0105     curr_norm = sopt_mltb_TV_norm(sol, 0);
0106     rel_norm = abs(curr_norm - prev_norm)/curr_norm;
0107     if param.verbose >= 1
0108         fprintf('  ||x||_TV = %e, rel_norm = %e\n', ...
0109             curr_norm, rel_norm);
0110     end
0111     if (rel_norm < param.rel_obj)
0112         crit_BPDN = 'REL_NORM';
0113         break;
0114     elseif iter >= param.max_iter
0115         crit_BPDN = 'MAX_IT';
0116         break;
0117     end
0118     
0119     % Proximal L1 operator
0120     xhat = 2*sol - xhat;
0121     temp = sopt_mltb_prox_TV(xhat, param.gamma, param_TV);
0122     xhat = temp + sol - xhat;
0123     
0124     % Update variables
0125     iter = iter + 1;
0126     prev_norm = curr_norm;
0127     
0128 end
0129 
0130 % Log
0131 if param.verbose >= 1
0132   
0133     % L1 norm
0134     fprintf('\n Solution found:\n');
0135     fprintf(' Final TV norm: %e\n', curr_norm);
0136     
0137     % Residual
0138     temp = A(sol);
0139     fprintf(' epsilon = %e, ||y-Ax||_2=%e\n', epsilon, ...
0140         norm(y(:)-temp(:)));
0141     
0142     % Stopping criterion
0143     fprintf(' %i iterations\n', iter);
0144     fprintf(' Stopping criterion: %s \n\n', crit_BPDN);
0145     
0146 end
0147 
0148 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/sopt_mltb_solve_TVDNoA.html000066400000000000000000000411271277570055300231770ustar00rootroot00000000000000 Description of sopt_mltb_solve_TVDNoA
Home > matlab > sopt_mltb_solve_TVDNoA.m

sopt_mltb_solve_TVDNoA

PURPOSE ^

sopt_mltb_solve_TVDNoA - Solve augmented TVDN problem

SYNOPSIS ^

function sol = sopt_mltb_solve_TVDNoA(y, epsilon, A, At, S, St, param)

DESCRIPTION ^

 sopt_mltb_solve_TVDNoA - Solve augmented TVDN problem

 Solve the total variation denoising (TVDN) problem when an additional 
 linear operator S is incorporated in the TV norm, i.e. solve

   min ||S x||_TV   s.t.  ||y - A x||_2 < epsilon

 where y contains the measurements, A is the forward measurement operator 
 and At the associated adjoint operator, S is the operator appearing in
 the TV norm and St the associated adjoint operator.  The structure param 
 should contain the following fields:

   General parameters:
 
   - verbose: Verbosity level (0 = no log, 1 = summary at convergence, 
       2 = print main steps; default = 1).

   - max_iter: Maximum number of iterations (default = 200).

   - rel_obj: Minimum relative change of the objective value 
       (default = 1e-4).  The algorithm stops if
           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj,
       where x(t) is the estimate of the solution at iteration t.

   - gamma: Convergence speed (weighting of TV norm when solving for
       TV proximal operator) (default = 1e-1).
 
   - nu_TV: Bound on the norm of the operator S, i.e.
       ||S x||^2 <= nu * ||x||^2 (default = 1).

   - initsol: Initial solution for a warmstart.
 
   Projection onto the L2-ball:

   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
 
   - nu_B2: Bound on the norm of the operator A, i.e.
       ||A x||^2 <= nu * ||x||^2 (default = 1).

   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
       stops if
         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
       (default = 1e-3).

   - max_iter_B2: Maximum number of iterations for the projection onto the
       L2 ball (default = 200).

   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
       default = 0).

   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
       default = 0).
 
   Proximal L1 operator:

   - max_iter_TV: Used as stopping criterion for the proximal TV
       operator. Maximun number of iterations (default = 200).

   - param.weights_dx_TV: Weights for a weighted TV-norm in the x
       direction (default = 1).

   - param.weights_dy_TV: Weights for a weighted TV-norm in the y
       direction (default = 1).

 References:
 P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting Approach 
 to Nonsmooth Convex Variational Signal Recovery", IEEE Journal of 
 Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.

CROSS-REFERENCE INFORMATION ^

This function calls:
This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_solve_TVDNoA(y, epsilon, A, At, S, St, param)
0002 % sopt_mltb_solve_TVDNoA - Solve augmented TVDN problem
0003 %
0004 % Solve the total variation denoising (TVDN) problem when an additional
0005 % linear operator S is incorporated in the TV norm, i.e. solve
0006 %
0007 %   min ||S x||_TV   s.t.  ||y - A x||_2 < epsilon
0008 %
0009 % where y contains the measurements, A is the forward measurement operator
0010 % and At the associated adjoint operator, S is the operator appearing in
0011 % the TV norm and St the associated adjoint operator.  The structure param
0012 % should contain the following fields:
0013 %
0014 %   General parameters:
0015 %
0016 %   - verbose: Verbosity level (0 = no log, 1 = summary at convergence,
0017 %       2 = print main steps; default = 1).
0018 %
0019 %   - max_iter: Maximum number of iterations (default = 200).
0020 %
0021 %   - rel_obj: Minimum relative change of the objective value
0022 %       (default = 1e-4).  The algorithm stops if
0023 %           | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj,
0024 %       where x(t) is the estimate of the solution at iteration t.
0025 %
0026 %   - gamma: Convergence speed (weighting of TV norm when solving for
0027 %       TV proximal operator) (default = 1e-1).
0028 %
0029 %   - nu_TV: Bound on the norm of the operator S, i.e.
0030 %       ||S x||^2 <= nu * ||x||^2 (default = 1).
0031 %
0032 %   - initsol: Initial solution for a warmstart.
0033 %
0034 %   Projection onto the L2-ball:
0035 %
0036 %   - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1).
0037 %
0038 %   - nu_B2: Bound on the norm of the operator A, i.e.
0039 %       ||A x||^2 <= nu * ||x||^2 (default = 1).
0040 %
0041 %   - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms
0042 %       stops if
0043 %         epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol)
0044 %       (default = 1e-3).
0045 %
0046 %   - max_iter_B2: Maximum number of iterations for the projection onto the
0047 %       L2 ball (default = 200).
0048 %
0049 %   - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise;
0050 %       default = 0).
0051 %
0052 %   - real_B2: Reality flag (1 to impose reality, 0 otherwise;
0053 %       default = 0).
0054 %
0055 %   Proximal L1 operator:
0056 %
0057 %   - max_iter_TV: Used as stopping criterion for the proximal TV
0058 %       operator. Maximun number of iterations (default = 200).
0059 %
0060 %   - param.weights_dx_TV: Weights for a weighted TV-norm in the x
0061 %       direction (default = 1).
0062 %
0063 %   - param.weights_dy_TV: Weights for a weighted TV-norm in the y
0064 %       direction (default = 1).
0065 %
0066 % References:
0067 % P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting Approach
0068 % to Nonsmooth Convex Variational Signal Recovery", IEEE Journal of
0069 % Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007.
0070 
0071 % Optional input arguments
0072 if ~isfield(param, 'verbose'), param.verbose = 1; end
0073 if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end
0074 if ~isfield(param, 'max_iter'), param.max_iter = 200; end
0075 if ~isfield(param, 'gamma'), param.gamma = 1e-2; end
0076 if ~isfield(param, 'weights_dx_TV'), param.weights_dx_TV = 1.0; end
0077 if ~isfield(param, 'weights_dy_TV'), param.weights_dy_TV = 1.0; end
0078 if ~isfield(param, 'incNP'), param.incNP = false; end
0079 if ~isfield(param, 'sphere_flag'), param.sphere_flag = false; end
0080 
0081 % Input arguments for projection onto the L2 ball
0082 param_B2.A = A; param_B2.At = At;
0083 param_B2.y = y; param_B2.epsilon = epsilon;
0084 param_B2.verbose = param.verbose;
0085 if isfield(param, 'nu_B2'), param_B2.nu = param.nu_B2; end
0086 if isfield(param, 'tol_B2'), param_B2.tol = param.tol_B2; end
0087 if isfield(param, 'tight_B2'), param_B2.tight = param.tight_B2; end
0088 if isfield(param, 'max_iter_B2')
0089     param_B2.max_iter = param.max_iter_B2;
0090 end
0091 if isfield(param,'pos_B2'), param_B2.pos=param.pos_B2; end
0092 if isfield(param,'real_B2'), param_B2.real=param.real_B2; end
0093 
0094 % Input arguments for prox TVoA
0095 param_TV.A = S; param_TV.At = St;
0096 param_TV.verbose = param.verbose; 
0097 param_TV.rel_obj = param.rel_obj;
0098 param_TV.weights_dx = param.weights_dx_TV;
0099 param_TV.weights_dy = param.weights_dy_TV;
0100 if isfield(param, 'nu_TV')
0101    param_TV.nu = param.nu_TV; 
0102 end
0103 if isfield(param, 'max_iter_TV')
0104     param_TV.max_iter= param.max_iter_TV;
0105 end
0106 if isfield(param, 'zero_weights_flag_TV')
0107    param_TV.zero_weights_flag = param.zero_weights_flag_TV; 
0108 end
0109 if isfield(param, 'identical_weights_flag_TV'), 
0110    param_TV.identical_weights_flag = param.identical_weights_flag_TV;
0111 end
0112 if isfield(param, 'sphere_flag'), 
0113    param_TV.sphere_flag = param.sphere_flag;
0114    param_TV.incNP = param.incNP;
0115 end
0116 
0117 
0118 % Initialization
0119 if isfield(param,'initsol')
0120     xhat = param.initsol;
0121 else
0122     xhat = At(y); 
0123 end
0124 
0125 iter = 1; prev_norm = 0;
0126 
0127 % Main loop
0128 while 1
0129     
0130     %
0131     if param.verbose>=1
0132         fprintf('Iteration %i:\n', iter);
0133     end
0134     
0135     % Projection onto the L2-ball
0136     [sol, param_B2.u] = sopt_mltb_fast_proj_B2(xhat, param_B2);    
0137     
0138     % Global stopping criterion
0139     curr_norm = sopt_mltb_TV_norm(S(sol), param_TV.sphere_flag, param_TV.incNP, param_TV.weights_dx, param_TV.weights_dy);
0140     rel_norm = abs(curr_norm - prev_norm)/curr_norm;
0141     if param.verbose >= 1
0142         fprintf('  ||x||_TV = %e, rel_norm = %e\n', ...
0143             curr_norm, rel_norm);
0144     end
0145     if (rel_norm < param.rel_obj)
0146         crit_BPDN = 'REL_NORM';
0147         break;
0148     elseif iter >= param.max_iter
0149         crit_BPDN = 'MAX_IT';
0150         break;
0151     end
0152     
0153     % Proximal L1 operator
0154     xhat = 2*sol - xhat;
0155     temp = sopt_mltb_prox_TVoA(xhat, param.gamma, param_TV);
0156     xhat = temp + sol - xhat;
0157     
0158     % Update variables
0159     iter = iter + 1;
0160     prev_norm = curr_norm;
0161     
0162 end
0163 
0164 % Log
0165 if param.verbose>=1
0166     % L1 norm
0167     fprintf('\n Solution found:\n');
0168     fprintf(' Final TV norm: %e\n', curr_norm);
0169     
0170     % Residual
0171     temp = A(sol);
0172     fprintf(' epsilon = %e, ||y-Ax||_2=%e\n', epsilon, ...
0173         norm(y(:)-temp(:)));
0174     
0175     % Stopping criterion
0176     fprintf(' %i iterations\n', iter);
0177     fprintf(' Stopping criterion: %s \n\n', crit_BPDN);
0178     
0179 end
0180 
0181 end
Generated on Fri 22-Feb-2013 15:54:47 by m2html © 2005
sopt-2.0.0/matlab/doc/matlab/sopt_mltb_solve_rwBPDN.html000066400000000000000000000224301277570055300232340ustar00rootroot00000000000000 Description of sopt_mltb_solve_rwBPDN
Home > matlab > sopt_mltb_solve_rwBPDN.m

sopt_mltb_solve_rwBPDN

PURPOSE ^

sopt_mltb_solve_rwBPDN - Solve reweighted BPDN problem

SYNOPSIS ^

function sol = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi, Psit,paramT, sigma, tol, maxiter, initsol)

DESCRIPTION ^

 sopt_mltb_solve_rwBPDN - Solve reweighted BPDN problem

 Solve the reweighted L1 minimization function using an homotopy
 continuation method to approximate the L0 norm.  At each iteration the 
 following problem is solved:

   min_x ||W Psit x||_1   s.t.  ||y-A x||_2 < epsilon

 where W is a diagonal matrix with diagonal elements given by

   [W]_ii = delta(t)/(delta(t)+|[Psit x(t)]_i|).

 The input parameters are defined as follows.

   - y: Input data (measurements).

   - epsilon: Noise bound.

   - A: Forward measurement operator.

   - At: Adjoint measurement operator.

   - Psi: Synthesis sparsity transform.

   - Psit: Analysis sparsity transform.

   - paramT: Structure containing parameters for the L1 solver (see 
       documentation for sopt_mltb_solve_BPDN).  

   - sigma: Noise standard deviation in the analysis domain.

   - tol: Minimum relative change in the solution.
       The algorithm stops if 
           ||x(t)-x(t-1)||_2/||x(t-1)||_2 < tol.
       where x(t) is the estimate of the solution at iteration t.

   - maxiter: Maximum number of iterations of the reweighted algorithm.

   - initsol: Initial solution for a warmstart.

 References:
 [1] R. E. Carrillo, J. D. McEwen, D. Van De Ville, J.-Ph. Thiran, and 
 Y. Wiaux. Sparsity averaging for compressive imaging. IEEE Sig. Proc. 
 Let., in press, 2013.
 [2] R. E. Carrillo, J. D. McEwen, and Y. Wiaux. Sparsity Averaging 
 Reweighted Analysis (SARA): a novel algorithm for radio-interferometric 
 imaging. Mon. Not. Roy. Astron. Soc., 426(2):1223-1234, 2012.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi, Psit, ...
0002   paramT, sigma, tol, maxiter, initsol)
0003 % sopt_mltb_solve_rwBPDN - Solve reweighted BPDN problem
0004 %
0005 % Solve the reweighted L1 minimization function using an homotopy
0006 % continuation method to approximate the L0 norm.  At each iteration the
0007 % following problem is solved:
0008 %
0009 %   min_x ||W Psit x||_1   s.t.  ||y-A x||_2 < epsilon
0010 %
0011 % where W is a diagonal matrix with diagonal elements given by
0012 %
0013 %   [W]_ii = delta(t)/(delta(t)+|[Psit x(t)]_i|).
0014 %
0015 % The input parameters are defined as follows.
0016 %
0017 %   - y: Input data (measurements).
0018 %
0019 %   - epsilon: Noise bound.
0020 %
0021 %   - A: Forward measurement operator.
0022 %
0023 %   - At: Adjoint measurement operator.
0024 %
0025 %   - Psi: Synthesis sparsity transform.
0026 %
0027 %   - Psit: Analysis sparsity transform.
0028 %
0029 %   - paramT: Structure containing parameters for the L1 solver (see
0030 %       documentation for sopt_mltb_solve_BPDN).
0031 %
0032 %   - sigma: Noise standard deviation in the analysis domain.
0033 %
0034 %   - tol: Minimum relative change in the solution.
0035 %       The algorithm stops if
0036 %           ||x(t)-x(t-1)||_2/||x(t-1)||_2 < tol.
0037 %       where x(t) is the estimate of the solution at iteration t.
0038 %
0039 %   - maxiter: Maximum number of iterations of the reweighted algorithm.
0040 %
0041 %   - initsol: Initial solution for a warmstart.
0042 %
0043 % References:
0044 % [1] R. E. Carrillo, J. D. McEwen, D. Van De Ville, J.-Ph. Thiran, and
0045 % Y. Wiaux. Sparsity averaging for compressive imaging. IEEE Sig. Proc.
0046 % Let., in press, 2013.
0047 % [2] R. E. Carrillo, J. D. McEwen, and Y. Wiaux. Sparsity Averaging
0048 % Reweighted Analysis (SARA): a novel algorithm for radio-interferometric
0049 % imaging. Mon. Not. Roy. Astron. Soc., 426(2):1223-1234, 2012.
0050 
0051 param=paramT;
0052 iter=0;
0053 rel_dist=1;
0054 
0055 if nargin<11
0056     fprintf('RW iteration: %i\n', iter);
0057     sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0058 else
0059     sol = initsol;
0060 end
0061 
0062 temp=Psit(sol);
0063 delta=std(temp(:));
0064 
0065 while (rel_dist>tol && iter<maxiter)
0066   
0067     iter=iter+1;
0068     delta=max(sigma/10,delta);
0069     fprintf('RW iteration: %i\n', iter);
0070     fprintf('delta = %e\n', delta);
0071     
0072     % Weights
0073     weights=abs(Psit(sol));
0074     param.weights=delta./(delta+weights);
0075     
0076     % Warm start
0077     param.initsol=sol;
0078     param.gamma=1e-1*max(weights(:));
0079     sol1=sol;
0080     
0081     % Weighted L1 problem
0082     sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param);
0083     
0084     % Relative distance
0085     rel_dist=norm(sol(:)-sol1(:))/norm(sol1(:));
0086     fprintf('Relative distance = %e\n\n', rel_dist);
0087     delta = delta/10;
0088     
0089 end
0090
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sopt-2.0.0/matlab/doc/matlab/sopt_mltb_solve_rwTVDN.html000066400000000000000000000217451277570055300232740ustar00rootroot00000000000000 Description of sopt_mltb_solve_rwTVDN
Home > matlab > sopt_mltb_solve_rwTVDN.m

sopt_mltb_solve_rwTVDN

PURPOSE ^

sopt_mltb_solve_rwTVDN - Solve reweighted TVDN problem

SYNOPSIS ^

function sol = sopt_mltb_solve_rwTVDN(y, epsilon, A, At, paramT,sigma, tol, maxiter, initsol)

DESCRIPTION ^

 sopt_mltb_solve_rwTVDN - Solve reweighted TVDN problem

 Solve the reweighted TV minimization function using an homotopy
 continuation method to approximate the L0 norm of the magnitude of the 
 gradient.  At each iteration the following problem is solved:

   min_x ||W x||_TV   s.t.  ||y-A x||_2 < epsilon

 where W is a diagonal matrix with diagonal elements given by esentially
 the inverse of the graident.
   
 The input parameters are defined as follows.

   - y: Input data (measurements).

   - epsilon: Noise bound.

   - A: Forward measurement operator.

   - At: Adjoint measurement operator.

   - paramT: Structure containing parameters for the TV solver (see 
       documentation for sopt_mltb_solve_TVDN).  

   - sigma: Noise standard deviation in the analysis domain.

   - tol: Minimum relative change in the solution.
       The algorithm stops if 
           ||x(t)-x(t-1)||_2/||x(t-1)||_2 < tol.
       where x(t) is the estimate of the solution at iteration t.

   - maxiter: Maximum number of iterations of the reweighted algorithm.

   - initsol: Initial solution for a warmstart.

 References:
 [1] R. E. Carrillo, J. D. McEwen, D. Van De Ville, J.-Ph. Thiran, and 
 Y. Wiaux. Sparsity averaging for compressive imaging. IEEE Sig. Proc. 
 Let., in press, 2013.
 [2] R. E. Carrillo, J. D. McEwen, and Y. Wiaux. Sparsity Averaging 
 Reweighted Analysis (SARA): a novel algorithm for radio-interferometric 
 imaging. Mon. Not. Roy. Astron. Soc., 426(2):1223-1234, 2012.

CROSS-REFERENCE INFORMATION ^

This function calls: This function is called by:

SOURCE CODE ^

0001 function sol = sopt_mltb_solve_rwTVDN(y, epsilon, A, At, paramT, ...
0002   sigma, tol, maxiter, initsol)
0003 % sopt_mltb_solve_rwTVDN - Solve reweighted TVDN problem
0004 %
0005 % Solve the reweighted TV minimization function using an homotopy
0006 % continuation method to approximate the L0 norm of the magnitude of the
0007 % gradient.  At each iteration the following problem is solved:
0008 %
0009 %   min_x ||W x||_TV   s.t.  ||y-A x||_2 < epsilon
0010 %
0011 % where W is a diagonal matrix with diagonal elements given by esentially
0012 % the inverse of the graident.
0013 %
0014 % The input parameters are defined as follows.
0015 %
0016 %   - y: Input data (measurements).
0017 %
0018 %   - epsilon: Noise bound.
0019 %
0020 %   - A: Forward measurement operator.
0021 %
0022 %   - At: Adjoint measurement operator.
0023 %
0024 %   - paramT: Structure containing parameters for the TV solver (see
0025 %       documentation for sopt_mltb_solve_TVDN).
0026 %
0027 %   - sigma: Noise standard deviation in the analysis domain.
0028 %
0029 %   - tol: Minimum relative change in the solution.
0030 %       The algorithm stops if
0031 %           ||x(t)-x(t-1)||_2/||x(t-1)||_2 < tol.
0032 %       where x(t) is the estimate of the solution at iteration t.
0033 %
0034 %   - maxiter: Maximum number of iterations of the reweighted algorithm.
0035 %
0036 %   - initsol: Initial solution for a warmstart.
0037 %
0038 % References:
0039 % [1] R. E. Carrillo, J. D. McEwen, D. Van De Ville, J.-Ph. Thiran, and
0040 % Y. Wiaux. Sparsity averaging for compressive imaging. IEEE Sig. Proc.
0041 % Let., in press, 2013.
0042 % [2] R. E. Carrillo, J. D. McEwen, and Y. Wiaux. Sparsity Averaging
0043 % Reweighted Analysis (SARA): a novel algorithm for radio-interferometric
0044 % imaging. Mon. Not. Roy. Astron. Soc., 426(2):1223-1234, 2012.
0045 
0046 Psit = @(x) x; Psi=@(x) x;
0047 
0048 param=paramT;
0049 iter=0;
0050 rel_dist=1;
0051 
0052 if nargin<9
0053     fprintf('RW iteration: %i\n', iter);
0054     sol = sopt_mltb_solve_TVoA_B2(y, epsilon, A, At, Psi, Psit, param);
0055 else
0056     sol = initsol;
0057 end
0058 
0059 
0060 delta=std(sol(:));
0061 
0062 while (rel_dist>tol && iter<maxiter)
0063   
0064     iter=iter+1;
0065     delta=max(sigma/10,delta);
0066     fprintf('RW iteration: %i\n', iter);
0067     fprintf('delta = %e\n', delta);
0068     
0069     %Warm start
0070     param.initsol=sol;
0071     sol1=sol;
0072     
0073     % Weights
0074     [param.weights_dx_TV param.weights_dy_TV] = sopt_mltb_gradient_op(real(sol));
0075     param.weights_dx_TV = delta./(abs(param.weights_dx_TV)+delta);
0076     param.weights_dy_TV = delta./(abs(param.weights_dy_TV)+delta);
0077     param.identical_weights_flag_TV = 0;
0078     param.gamma=1e-1*max(abs(sol1(:)));
0079     
0080     %Weighted TV problem
0081     sol = sopt_mltb_solve_TVDNoA(y, epsilon, A, At, Psi, Psit, param);
0082 
0083     %Relative distance
0084     rel_dist=norm(sol(:)-sol1(:))/norm(sol1(:)); 
0085     fprintf('relative distance = %e\n\n', rel_dist);
0086     delta = delta/10;
0087     
0088 end
0089
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All rights reserved.!,r$p~d 2dE2j…2$ KR!?0cSL4~o?3a͠-jtМK5TҥQz5kƈX;sopt-2.0.0/matlab/doc/solaris.png000066400000000000000000000004361277570055300167030ustar00rootroot00000000000000PNG  IHDRPIDATxWm - GgWѻuڱ2^B_޳|h@D j'2㺮yiv1̨ۙQ.gFzԏf Pf 47NE.n򦔖e1#GO J݊Py/2kn{kX; 4AshMEV'$| +\jԹ'IO$QO醿ys?~/#tr^J,IENDB`sopt-2.0.0/matlab/doc/up.png000066400000000000000000000002421277570055300156460ustar00rootroot00000000000000PNG  IHDR PLTEfstRNS0JbKGD- pHYsHHFk>'IDATxc```P†@!!EF@5  QIENDB`sopt-2.0.0/matlab/doc/windows.png000066400000000000000000000004361277570055300167210ustar00rootroot00000000000000PNG  IHDRPIDATxWQ0Gӫ*x7F `ӥ[cOV "HE IBk?θm[?χ03ܾ^QJò,юs*O87ERaZ SoJ鐈8O:D"zcb^BIȖpTr0 0) fprintf('Iter = %i, norm = %e \n',k,val); end if (rel_var < tol) break; end init_val=val; x=x/val; end end sopt-2.0.0/matlab/misc/sopt_mltb_SNR.m000066400000000000000000000010751277570055300176120ustar00rootroot00000000000000function snr = sopt_mltb_SNR(map_init, map_recon) % SNR - Compute the SNR between two images % % C omputes the SNR between the maps map_init and map_recon. The SNR is % computed by % 10 * log10( var(MAP_INIT) / var(MAP_INIT-MAP_NOISY) ) % where var stands for the matlab built-in function that computes the % variance. % % Inputs: % % - map_init: Initial image. % % - map_recon: Reconstructed image. % % Outputs: % % - snr: SNR. noise = map_init(:)-map_recon(:); var_init = var(map_init(:)); var_den = var(noise(:)); snr = 10 * log10(var_init/var_den); end sopt-2.0.0/matlab/misc/sopt_mltb_TV_norm.m000066400000000000000000000020651277570055300205340ustar00rootroot00000000000000function y = sopt_mltb_TV_norm(u, sphere_flag, ... incNP, weights_dx, weights_dy) % sopt_mltb_TV_norm - Compute TV norm % % Compute the TV of an image on the plane or sphere. % % Inputs: % % - u: Image to compute TV norm of. % % - sphere_flag: Flag indicating whether to compute the TV norm on the % sphere (1 = on sphere, 2 = on plane). % % - includeNP: Flag indicating whether the North pole is included % in the sampling grid (1 = North pole included, 0 = North pole not % included). % % - weights_dx: Weights in the x (phi) direction. % % - weights_dy: Weights in the y (theta) direction. % % Outputs: % % - y: TV norm of image. if sphere_flag if nargin>3 [dx, dy] = sopt_mltb_gradient_op_sphere(u, incNP, weights_dx, weights_dy); else [dx, dy] = sopt_mltb_gradient_op_sphere(u, incNP); end else if nargin>3 [dx, dy] = sopt_mltb_gradient_op(u, weights_dx, weights_dy); else [dx, dy] = sopt_mltb_gradient_op(u); end end temp = sqrt(abs(dx).^2 + abs(dy).^2); y = sum(temp(:)); end sopt-2.0.0/matlab/misc/sopt_mltb_adjcurvelet.m000066400000000000000000000015341277570055300214600ustar00rootroot00000000000000function restim = sopt_mltb_adjcurvelet(coef, Mod, real) % sopt_mltb_adjcurvelet - Adjoint curvelet transform % % Compute the adjoint curvelet transform from the curvelet % coefficient vector. % % Inputs: % % - coef: Input curvelet coefficient vector. % % - Mod: Data structure that stores sizes of the curvelet transform % generated from the CurveLab toolbox. % % - real: Flag indicating if the transform is real or complex (1 = real, % 0 = complex). % % Outputs: % % - restim: Estimated image from the Curvelet coefficients. CR=cell(size(Mod)); marker=0; for s=1:length(Mod) CR{s} = cell(size(Mod{s})); for w=1:length(Mod{s}) m=Mod{s}{w}(1); n=Mod{s}{w}(2); CR{s}{w}=reshape(coef(marker+1:marker+(m*n)),m,n); marker=marker+m*n; end end % Apply adjoint curvelet transform restim = ifdct_usfft(CR,real); sopt-2.0.0/matlab/misc/sopt_mltb_div_op.m000066400000000000000000000012121277570055300204210ustar00rootroot00000000000000function I = sopt_mltb_div_op(dx, dy, weights_dx, weights_dy) % sopt_mltb_div_op - Compute divergence % % Compute the divergence (adjoint of the gradient) of a two dimensional % signal from the horizontal and vertical gradients. % % Inputs: % % - dx: Gradient in x. % % - dy: Gradient in y. % % - weights_dx: Weights in the x direction. % % - weights_dy: Weights in the y direction. % % Outputs: % % - I: Divergence. if nargin > 2 dx = dx .* conj(weights_dx); dy = dy .* conj(weights_dy); end I = [dx(1, :) ; dx(2:end-1, :)-dx(1:end-2, :) ; -dx(end-1, :)]; I = I + [dy(:, 1) , dy(:, 2:end-1)-dy(:, 1:end-2) , -dy(:, end-1)]; end sopt-2.0.0/matlab/misc/sopt_mltb_div_op_sphere.m000066400000000000000000000021321277570055300217710ustar00rootroot00000000000000function I = sopt_mltb_div_op_sphere(dx, dy, includeNorthpole, ... weights_dx, weights_dy) % sopt_mltb_div_op_sphere - Compute divergence on sphere % % Compute the divergence (adjoint of the gradient) of a two dimensional % signal on the sphere. The phi direction (x) is periodic, while the theta % direction (y) is not. % % Inputs: % % - dx: Gradient in x. % % - dy: Gradient in y. % % - includeNorthPole: Flag indicating whether the North pole is included % in the sampling grid (1 = North pole included, 0 = North pole not % included). % % - weights_dx: Weights in the x (phi) direction. % % - weights_dy: Weights in the y (theta) direction. % % Outputs: % % - I: Divergence. if nargin > 3 dx = dx .* conj(weights_dx); dy = dy .* conj(weights_dy); end if(includeNorthpole) I = [zeros(1, size(dx,2)) ; dx(2, :); dx(3:end-1, :)-dx(2:end-2, :); -dx(end-1, :)]; I = I + [dy(:, 1) - dy(:,end) , dy(:, 2:end)-dy(:, 1:end-1)]; else I = [dx(1, :) ; dx(2:end-1, :)-dx(1:end-2, :) ; -dx(end-1, :)]; I = I + [dy(:, 1) - dy(:,end) , dy(:, 2:end)-dy(:, 1:end-1)]; end end sopt-2.0.0/matlab/misc/sopt_mltb_fwdcurvelet.m000066400000000000000000000010551277570055300215000ustar00rootroot00000000000000function coef = sopt_mltb_fwdcurvelet(im,real) % sopt_mltb_fwdcurvelet - Forward curvelet transform % % Compute the forward curvelet transform of an image and stores it in % vector coef. % % Inputs: % % - im: Input image. % % - real: Flag indicating if the transform is real or complex (1 = real, % 0 = complex). % % Outputs: % % - coef: Curvelet coefficients. C = fdct_usfft(im,real); % Compute vector version of curvelets coeffcients coef = []; for s=1:length(C) for w=1:length(C{s}) A = C{s}{w}; coef= [coef; A(:)]; end end sopt-2.0.0/matlab/misc/sopt_mltb_genmask.m000066400000000000000000000007421277570055300205750ustar00rootroot00000000000000function mask = sopt_mltb_genmask(pdf, seed) % sopt_mltb_genmask - Generate mask % % Generate a binary mask with variable density sampling. It is used % in sopt_mltb_vdsmask in the generation of variable density sampling % profiles. % % Inputs: % % - pdf: Sampling profile function. % % - seed: Seed for the random number generator. Optional. % % Outputs: % % - mask: Binary mask. if nargin==2 rand('seed', seed); end mask = rand(size(pdf))1 dx = dx .* weights_dx; dy = dy .* weights_dy; end end sopt-2.0.0/matlab/misc/sopt_mltb_gradient_op_sphere.m000066400000000000000000000023701277570055300230100ustar00rootroot00000000000000function [dx, dy] = sopt_mltb_gradient_op_sphere(I, includeNorthpole, ... weights_dx, weights_dy) % sopt_mltb_gradient_op_sphere - Compute gradient on sphere % % Compute the gradientof a signal on the sphere. The phi direction (x) is % periodic, while the theta direction (y) is not. % % Inputs: % % - I: Input two dimesional signal on the sphere. % % - includeNorthPole: Flag indicating whether the North pole is included % in the sampling grid (1 = North pole included, 0 = North pole not % included). % % - weights_dx: Weights in the x (phi) direction. % % - weights_dy: Weights in the y (theta) direction. % % Outputs: % % - dx: Gradient in x (phi) direction. % % - dy: Gradient in y (theta) direction. dx = zeros(size(I, 1),size(I, 2)); I_big = zeros(size(I, 1)+2,size(I, 2)); I_big(2:end-1, 1:end) = I; % Theta direction if(includeNorthpole) dx = [I(2:end, :)-I(1:end-1, :) ; zeros(1, size(I, 2))]; dx(1,:) = zeros(); else dx = [I(2:end, :)-I(1:end-1, :) ; zeros(1, size(I, 2))]; end % Phi direction if(includeNorthpole) dy = [I(:, 2:end)-I(:, 1:end-1) , I(:, 1)-I(:, end)]; else dy = [I(:, 2:end)-I(:, 1:end-1) , I(:, 1)-I(:, end)]; end if nargin>2 dx = dx .* weights_dx; dy = dy .* weights_dy; end end sopt-2.0.0/matlab/misc/sopt_mltb_modifypdf.m000066400000000000000000000016651277570055300211360ustar00rootroot00000000000000function [new_pdf, alpha] = sopt_mltb_modifypdf(pdf, nb_meas) % sopt_mltb_modifypdf - Modify PDF of sampling profile % % Checks PDF of the sampling profile and normalizes it. It is used % in sopt_mltb_vdsmask in the generation of variable density sampling % profiles. % % Inputs: % % - pdf: Sampling profile function. % % - nb_meas: Number of measurements. % % Outputs: % % - new_pdf: New sampling profile function. % % - alpha: DC level for the required number of samples. if sum(pdf(:)<0)>0 error('PDF contains negative values'); end pdf = pdf/max(pdf(:)); % Find alpha alpha_min = -1; alpha_max = 1; while 1 alpha = (alpha_min + alpha_max)/2; new_pdf = pdf + alpha; new_pdf(new_pdf>1) = 1; new_pdf(new_pdf<0) = 0; M = round(sum(new_pdf(:))); if M > nb_meas alpha_max = alpha; elseif M < nb_meas alpha_min = alpha; else break; end end sopt-2.0.0/matlab/misc/sopt_mltb_struct2size.m000066400000000000000000000007751277570055300214570ustar00rootroot00000000000000function Mod = sopt_mltb_struct2size(C) %sopt_mltb_struct2size - Compute curvelet data structure % % Compute data structure to store the sizes of the curvelet transform % generated from the CurveLab toolbox. % % Inputs: % % - C: Curvelet coefficients arranged in the original format. % % Outputs: % % - Mod: Data structure with the appropiate sizes. Mod=cell(size(C)); for s=1:length(C) Mod{s} = cell(size(C{s})); for w=1:length(C{s}) [m,n]=size(C{s}{w}); Mod{s}{w}=[m,n]; end end sopt-2.0.0/matlab/misc/sopt_mltb_vdsmask.m000066400000000000000000000026621277570055300206230ustar00rootroot00000000000000function mask = sopt_mltb_vdsmask(N,M,p) % sopt_mltb_vdsmask - Create variable density sampling profile % % Creates a binary mask generated from a variable density sampling % profile for two dimensional images in the frequency domain. The mask % contains on average p*N*M ones's. % % Inputs: % % - N and M: Size of mask. % % - p: Coverage percentage. % % Note: The undersampling ratio is passed as 2*p and then the function % only takes samples in half plane to account for signal reality. The range % of p is 0 < p <= 0.5. % % Outputs: % % - mask: Binary mask with the sampling profile in the frequency domain. p = 2*p; if p==1 mask=ones(N,M); else nb_meas=round(p*N*M); tol=ceil(p*N*M)-floor(p*N*M); d=1; [x,y] = meshgrid(linspace(-1, 1, M), linspace(-1, 1, N)); % Cartesian grid r = sqrt(x.^2+y.^2); r = r/max(r(:)); % Polar grid alpha=-1; it=0; maxit=20; while (alpha<-0.01 || alpha>0.01) && itnb_meas+tol || sum(mask(:)) 1 fprintf(' Proj. B2:\n'); end while true % Residual res = param.A(sol) - param.y; norm_res = norm(res(:), 2); % Scaling for the projection res = u*param.nu + res; norm_proj = norm(res(:), 2); % Log if param.verbose>1 fprintf(' Iter %i, epsilon = %e, ||y - Ax||_2 = %e\n', ... iter, param.epsilon, norm_res); end % Stopping criterion if (norm_res>=epsilon_low && norm_res<=epsilon_up) crit_B2 = 'TOL_EPS'; break; elseif iter >= param.max_iter crit_B2 = 'MAX_IT'; break; end % Projection onto the L2 ball t = (1+sqrt(1+4*told^2))/2; ratio = min(1, param.epsilon/norm_proj); u = v; v = 1/param.nu * (res - res*ratio); u = v + (told-1)/t * (v - u); % Current estimate sol = x - param.At(u); % Projection onto the non-negative orthant (positivity constraint) if (param.pos) sol = real(sol); sol(sol<0) = 0; end % Projection onto the real orthant (reality constraint) if (param.real) sol = real(sol); end % Update number of iteration told = t; % Update number of iterations iter = iter + 1; end end % Log after the projection onto the L2-ball if param.verbose >= 1 temp = param.A(sol); fprintf([' Proj. B2: epsilon = %e, ||y-Ax||_2 = %e,', ... ' %s, iter = %i\n'], param.epsilon, norm(param.y(:)-temp(:)), ... crit_B2, iter); end end sopt-2.0.0/matlab/prox_operators/sopt_mltb_prox_L1.m000066400000000000000000000103141277570055300226230ustar00rootroot00000000000000function [sol, norm_l1] = sopt_mltb_prox_L1(x, lambda, param) % sopt_mltb_prox_L1 - Proximal operator with L1 norm % % Compute the L1 proximal operator, i.e. solve % % min_{z} 0.5*||x - z||_2^2 + lambda * ||Psit x||_1 , % % where x is the input vector and the solution z* is returned as sol. % The structure param should contain the following fields: % % - Psit: Sparsifying transform (default = Identity). % % - Psi: Adjoint of Psit (default = Identity). % % - tight: 1 if Psit is a tight frame or 0 otherwise (default = 1). % % - nu: Bound on the norm^2 of the operator Psi, i.e. % ||Psi x||^2 <= nu * ||x||^2 (default = 1). % % - max_iter: Maximum number of iterations (default = 200). % % - rel_obj: Minimum relative change of the objective value % (default = 1e-4). The algorithm stops if % | ||x(t)||_1 - ||x(t-1)||_1 | / ||x(t)||_1 < rel_obj, % where x(t) is the estimate of the solution at iteration t. % % - verbose: Verbosity level (0 = no log, 1 = summary at convergence, % 2 = print main steps; default = 1). % % - weights: Weights for a weighted L1-norm (default = 1). % % - pos: Positivity flag (1 = positive solution, % 0 = general complex case; default = 0). % % References: % [1] M.J. Fadili and J-L. Starck, "Monotone operator splitting for % optimization problems in sparse recovery" , IEEE ICIP, Cairo, % Egypt, 2009. % [2] Amir Beck and Marc Teboulle, "A Fast Iterative Shrinkage-Thresholding % Algorithm for Linear Inverse Problems", SIAM Journal on Imaging Sciences % 2 (2009), no. 1, 183--202. % Optional input arguments if ~isfield(param, 'verbose'), param.verbose = 1; end if ~isfield(param, 'Psit'), param.Psi = @(x) x; param.Psit = @(x) x; end if ~isfield(param, 'tight'), param.tight = 1; end if ~isfield(param, 'nu'), param.nu = 1; end if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end if ~isfield(param, 'max_iter'), param.max_iter = 200; end if ~isfield(param, 'Psit'), param.Psit = @(x) x; end if ~isfield(param, 'Psi'), param.Psi = @(x) x; end if ~isfield(param, 'weights'), param.weights = 1; end if ~isfield(param, 'pos'), param.pos = 0; end if ~isfield(param, 'real'), param.real = 0; end % Useful functions soft = @(z, T) sign(z).*max(abs(z)-T, 0); % Projection if param.tight && ~param.pos && ~param.real % TIGHT FRAME CASE temp = param.Psit(x); sol = x + 1/param.nu * param.Psi(soft(temp, ... lambda*param.nu*param.weights)-temp); crit_L1 = 'REL_OBJ'; iter_L1 = 1; dummy = param.Psit(sol); %coef=soft(temp,lambda*param.nu*param.weights); norm_l1 = sum(param.weights(:).*abs(dummy(:))); else % NON TIGHT FRAME CASE OR CONSTRAINT INVOLVED % Initializations sol = x; if param.pos || param.real sol = real(sol); end dummy = param.Psit(sol); u_l1 = zeros(size(dummy)); prev_obj = 0; iter_L1 = 0; % Soft-thresholding % Init if param.verbose > 1 fprintf(' Proximal L1 operator:\n'); end while 1 % L1 norm of the estimate norm_l1 = sum(param.weights(:).*abs(dummy(:))); obj = .5*norm(x(:) - sol(:), 2)^2 + lambda * norm_l1; rel_obj = abs(obj-prev_obj)/obj; % Log if param.verbose>1 fprintf(' Iter %i, prox_fval = %e, rel_fval = %e\n', ... iter_L1, obj, rel_obj); end % Stopping criterion if (rel_obj < param.rel_obj) crit_L1 = 'REL_OB'; break; elseif iter_L1 >= param.max_iter crit_L1 = 'MAX_IT'; break; end % Soft-thresholding res = u_l1*param.nu + dummy; dummy = soft(res, lambda*param.nu*param.weights); u_l1 = 1/param.nu * (res - dummy); sol = x - param.Psi(u_l1); if param.pos sol = real(sol); sol(sol<0) = 0; end if param.real sol = real(sol); end % Update prev_obj = obj; iter_L1 = iter_L1 + 1; dummy = param.Psit(sol); end end % Log after the projection onto the L2-ball if param.verbose >= 1 fprintf([' prox_L1: prox_fval = %e,', ... ' %s, iter = %i\n'], norm_l1, crit_L1, iter_L1); end endsopt-2.0.0/matlab/prox_operators/sopt_mltb_prox_L1v2.m000066400000000000000000000102471277570055300231000ustar00rootroot00000000000000function [sol, norm_l1] = prox_L1(x, lambda, param) % PROJ_L1 - Proximal operator with L1 norm % % sol = prox_L1(x, lambda, param) solves: % % min_{z} 0.5*||x - z||_2^2 + lambda * ||Psit x||_1 % % The input argument param contains the following fields: % % - Psit: Sparsifying transform (default: Id). % % - Psi: Adjoint of Psit (default: Id). % % - tight: 1 if Psit is a tight frame or 0 if not (default = 1) % % - nu: bound on the norm of the operator A, i.e. % ||Psi x||^2 <= nu * ||x||^2 (default: 1) % % - max_iter: max. nb. of iterations (default: 200). % % - rel_obj: minimum relative change of the objective value (default: % 1e-4) % The algorithm stops if % | ||x(t)||_1 - ||x(t-1)||_1 | / ||x(t)||_1 < rel_obj, % where x(t) is the estimate of the solution at iteration t. % % - verbose: 0 no log, 1 a summary at convergence, 2 print main % steps (default: 1) % % - weights: weights for a weighted L1-norm (default = 1) % % % % References: % [1] M.J. Fadili and J-L. Starck, "Monotone operator splitting for % optimization problems in sparse recovery" , IEEE ICIP, Cairo, % Egypt, 2009. % [2] Amir Beck and Marc Teboulle, "A Fast Iterative Shrinkage-Thresholding % Algorithm for Linear Inverse Problems", SIAM Journal on Imaging Sciences % 2 (2009), no. 1, 183--202. % % % % Author: Gilles Puy % E-mail: gilles.puy@epfl.ch % Date: Dec. 1, 2010 % % Optional input arguments if ~isfield(param, 'verbose'), param.verbose = 1; end if ~isfield(param, 'Psit'), param.Psi = @(x) x; param.Psit = @(x) x; end if ~isfield(param, 'tight'), param.tight = 1; end if ~isfield(param, 'nu'), param.nu = 1; end if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end if ~isfield(param, 'max_iter'), param.max_iter = 200; end if ~isfield(param, 'Psit'), param.Psit = @(x) x; end if ~isfield(param, 'Psi'), param.Psi = @(x) x; end if ~isfield(param, 'weights'), param.weights = 1; end if ~isfield(param, 'pos'), param.pos = 0; end if ~isfield(param, 'real'), param.real = 0; end % Useful functions soft = @(z, T) sign(z).*max(abs(z)-T, 0); % Projection if param.tight && ~param.pos && ~param.real % TIGHT FRAME CASE temp = param.Psit(x); sol = x + 1/param.nu * param.Psi(soft(temp, ... lambda*param.nu*param.weights)-temp); crit_L1 = 'REL_OBJ'; iter_L1 = 1; dummy = param.Psit(sol); %coef=soft(temp,lambda*param.nu*param.weights); norm_l1 = sum(param.weights(:).*abs(dummy(:))); else % NON TIGHT FRAME CASE OR CONSTRAINT INVOLVED % Initializations sol = x; if param.pos || param.real sol = real(sol); end dummy = param.Psit(sol); u_l1 = zeros(size(dummy)); prev_obj = 0; iter_L1 = 0; % Soft-thresholding % Init if param.verbose > 1 fprintf(' Proximal L1 operator:\n'); end while 1 % L1 norm of the estimate norm_l1 = sum(param.weights(:).*abs(dummy(:))); obj = .5*norm(x(:) - sol(:), 2)^2 + lambda * norm_l1; rel_obj = abs(obj-prev_obj)/obj; % Log if param.verbose>1 fprintf(' Iter %i, prox_fval = %e, rel_fval = %e\n', ... iter_L1, obj, rel_obj); end % Stopping criterion if (rel_obj < param.rel_obj) crit_L1 = 'REL_OB'; break; elseif iter_L1 >= param.max_iter crit_L1 = 'MAX_IT'; break; end % Soft-thresholding res = u_l1*param.nu + dummy; dummy = soft(res, lambda*param.nu*param.weights); u_l1 = 1/param.nu * (res - dummy); sol = x - param.Psi(u_l1); if param.pos sol = real(sol); sol(sol<0) = 0; end if param.real sol = real(sol); end % Update prev_obj = obj; iter_L1 = iter_L1 + 1; dummy = param.Psit(sol); end end % Log after the projection onto the L2-ball if param.verbose >= 1 fprintf([' prox_L1: prox_fval = %e,', ... ' %s, iter = %i\n'], norm_l1, crit_L1, iter_L1); end endsopt-2.0.0/matlab/prox_operators/sopt_mltb_prox_TV.m000066400000000000000000000047411277570055300227070ustar00rootroot00000000000000function sol = sopt_mltb_prox_TV(x, lambda, param) % sopt_mltb_prox_TV - Total variation proximal operator % % Compute the TV proximal operator, i.e. solve % % min_{z} ||x - z||_2^2 + lambda * ||z||_{TV} % % where x is the input vector and the solution z* is returned as sol. % The structure param should contain the following fields: % % - max_iter: Maximum number of iterations (default = 200). % % - rel_obj: Minimum relative change of the objective value % (default = 1e-4). The algorithm stops if % | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj, % where x(t) is the estimate of the solution at iteration t. % % - verbose: Verbosity level (0 = no log, 1 = summary at convergence, % 2 = print main steps; default = 1). % % Reference: % [1] A. Beck and M. Teboulle, "Fast gradient-based algorithms for % constrained Total Variation Image Denoising and Deblurring Problems", % IEEE Transactions on Image Processing, VOL. 18, NO. 11, 2419-2434, % November 2009. % Optional input arguments if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end if ~isfield(param, 'verbose'), param.verbose = 1; end if ~isfield(param, 'max_iter'), param.max_iter = 200; end % Initializations [r, s] = sopt_mltb_gradient_op(x*0); pold = r; qold = s; told = 1; prev_obj = 0; % Main iterations if param.verbose > 1 fprintf(' Proximal TV operator:\n'); end for iter = 1:param.max_iter % Current solution sol = x - lambda*sopt_mltb_div_op(r, s); % Objective function value obj = .5*norm(x(:)-sol(:), 2)^2 + lambda * sopt_mltb_TV_norm(sol, 0); rel_obj = abs(obj-prev_obj)/obj; prev_obj = obj; % Stopping criterion if param.verbose>1 fprintf(' Iter %i, obj = %e, rel_obj = %e\n', ... iter, obj, rel_obj); end if rel_obj < param.rel_obj crit_TV = 'TOL_EPS'; break; end % Udpate divergence vectors and project [dx, dy] = sopt_mltb_gradient_op(sol); r = r - 1/(8*lambda) * dx; s = s - 1/(8*lambda) * dy; weights = max(1, sqrt(abs(r).^2+abs(s).^2)); p = r./weights; q = s./weights; % FISTA update t = (1+sqrt(4*told^2))/2; r = p + (told-1)/t * (p - pold); pold = p; s = q + (told-1)/t * (q - qold); qold = q; told = t; end % Log after the minimization if ~exist('crit_TV', 'var'), crit_TV = 'MAX_IT'; end if param.verbose >= 1 fprintf([' Prox_TV: obj = %e, rel_obj = %e,' ... ' %s, iter = %i\n'], obj, rel_obj, crit_TV, iter); end end sopt-2.0.0/matlab/prox_operators/sopt_mltb_prox_TVoA.m000066400000000000000000000113111277570055300231560ustar00rootroot00000000000000function sol = sopt_mltb_prox_TVoA(b, lambda, param) % sopt_mltb_prox_TVoA - Agumented total variation proximal operator % % Compute the TV proximal operator when an additional linear operator A is % incorporated in the TV norm, i.e. solve % % min_{x} ||y - x||_2^2 + lambda * ||A x||_{TV} % % where x is the input vector and the solution z* is returned as sol. % The structure param should contain the following fields: % % - max_iter: Maximum number of iterations (default = 200). % % - rel_obj: Minimum relative change of the objective value % (default = 1e-4). The algorithm stops if % | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_1 < rel_obj, % where x(t) is the estimate of the solution at iteration t. % % - verbose: Verbosity level (0 = no log, 1 = summary at convergence, % 2 = print main steps; default = 1). % % - A: Forward transform (default = Identity). % % - At: Adjoint of At (default = Identity). % % - nu: Bound on the norm^2 of the operator A, i.e. % ||A x||^2 <= nu * ||x||^2 (default = 1) % % Reference: % [1] A. Beck and M. Teboulle, "Fast gradient-based algorithms for % constrained Total Variation Image Denoising and Deblurring Problems", % IEEE Transactions on Image Processing, VOL. 18, NO. 11, 2419-2434, % November 2009. % Optional input arguments if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end if ~isfield(param, 'verbose'), param.verbose = 1; end if ~isfield(param, 'max_iter'), param.max_iter = 200; end if ~isfield(param, 'At'), param.At = @(x) x; end if ~isfield(param, 'A'), param.A = @(x) x; end if ~isfield(param, 'nu'), param.nu = 1; end % Advanced input arguments (not exposed in documentation) if ~isfield(param, 'weights_dx'), param.weights_dx = 1; end if ~isfield(param, 'weights_dy'), param.weights_dy = 1; end if ~isfield(param, 'zero_weights_flag'), param.zero_weights_flag = 1; end if ~isfield(param, 'identical_weights_flag') param.identical_weights_flag = 0; end if ~isfield(param, 'sphere_flag'), param.sphere_flag = 0; end if ~isfield(param, 'incNP'), param.incNP = 0; end % Set grad and div operators to planar or spherical case and also % include weights or not (depending on parameter flags). if (param.sphere_flag) G = @sopt_mltb_gradient_op_sphere; D = @sopt_mltb_div_op_sphere; else G = @sopt_mltb_gradient_op; D = @sopt_mltb_div_op; end if (~param.identical_weights_flag && param.zero_weights_flag) grad = @(x) G(x, param.weights_dx, param.weights_dy); div = @(r, s) D(r, s, param.weights_dx, param.weights_dy); max_weights = max([abs(param.weights_dx(:)); ... abs(param.weights_dy(:))])^2; else grad = @(x) G(x); div = @(r, s) D(r, s); end % Initializations [r, s] = grad(param.A(b*0)); pold = r; qold = s; told = 1; prev_obj = 0; % Main iterations if param.verbose > 1 fprintf(' Proximal TV operator:\n'); end for iter = 1:param.max_iter % Current solution sol = b - lambda*param.At(div(r, s)); % Objective function value obj = .5*norm(b(:)-sol(:), 2) + lambda * ... sopt_mltb_TV_norm(param.A(sol), param.weights_dx, param.weights_dy); rel_obj = abs(obj-prev_obj)/obj; prev_obj = obj; % Stopping criterion if param.verbose>1 fprintf(' Iter %i, obj = %e, rel_obj = %e\n', ... iter, obj, rel_obj); end if rel_obj < param.rel_obj crit_TV = 'TOL_EPS'; break; end % Udpate divergence vectors and project [dx, dy] = grad(param.A(sol)); if (param.identical_weights_flag) r = r - 1/(8*lambda*param.nu) * dx; s = s - 1/(8*lambda*param.nu) * dy; weights = max(param.weights_dx, sqrt(abs(r).^2+abs(s).^2)); p = r./weights.*param.weights_dx; q = s./weights.*param.weights_dx; else if (~param.zero_weights_flag) r = r - 1/(8*lambda*param.nu) * dx; s = s - 1/(8*lambda*param.nu) * dy; weights = max(1, sqrt(abs(r./param.weights_dx).^2+... abs(s./param.weights_dy).^2)); p = r./weights; q = s./weights; else % Weights go into grad and div operators so usual update r = r - 1/(8*lambda*param.nu*max_weights) * dx; s = s - 1/(8*lambda*param.nu*max_weights) * dy; weights = max(1, sqrt(abs(r).^2+abs(s).^2)); p = r./weights; q = s./weights; end end % FISTA update t = (1+sqrt(4*told^2))/2; r = p + (told-1)/t * (p - pold); pold = p; s = q + (told-1)/t * (q - qold); qold = q; told = t; end % Log after the minimization if ~exist('crit_TV', 'var'), crit_TV = 'MAX_IT'; end if param.verbose >= 1 fprintf([' Prox_TV: obj = %e, rel_obj = %e,' ... ' %s, iter = %i\n'], obj, rel_obj, crit_TV, iter); end end sopt-2.0.0/matlab/sopt_mltb_admm_bpcon.m000066400000000000000000000113651277570055300203170ustar00rootroot00000000000000function [xsol, z] = sopt_mltb_admm_bpcon(y, epsilon, A, At, Psi, Psit, param) % % sol = admm_bpcon(y, epsilon, A, At, Psi, Psit, param) solves: % % min ||Psit x||_1 s.t. ||y-A x||_2 <= epsilon % % % y contains the measurements. A is the forward measurement operator and % At the associated adjoint operator. Psit is a sparfying transform and Psi % its adjoint. PARAM a Matlab structure containing the following fields: % % General parameters: % % - verbose: 0 no log, 1 print main steps, 2 print all steps. % % - max_iter: max. nb. of iterations (default: 200). % % - rel_obj: minimum relative change of the objective value (default: % 1e-4) % The algorithm stops if % | ||x(t)||_1 - ||x(t-1)||_1 | / ||x(t)||_1 < rel_obj, % where x(t) is the estimate of the solution at iteration t. % % - gamma: control the converge speed (default: 1e-2). % % % Proximal L1 operator: % % - rel_obj_L1: Used as stopping criterion for the proximal L1 % operator. Min. relative change of the objective value between two % successive estimates. % % - max_iter_L1: Used as stopping criterion for the proximal L1 % operator. Maximun number of iterations. % % - param.nu_L1: bound on the norm^2 of the operator Psi, i.e. % ||Psi x||^2 <= nu * ||x||^2 (default: 1) % % - param.tight_L1: 1 if Psit is a tight frame or 0 if not (default = 1) % % - param.weights: weights (default = 1) for a weighted L1-norm defined % as sum_i{weights_i.*abs(x_i)} % % % Author: Rafael Carrillo % E-mail: rafael.carrillo@epfl.ch % Date: Nov. 22, 2014 % % Optional input arguments if ~isfield(param, 'verbose'), param.verbose = 1; end if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end if ~isfield(param, 'max_iter'), param.max_iter = 200; end if ~isfield(param, 'gamma'), param.gamma = 1e-2; end if ~isfield(param, 'nu'), param.nu = 1; end % Input arguments for prox L1 function param_L1.Psi = Psi; param_L1.Psit = Psit; if isfield(param, 'nu_L1') param_L1.nu = param.nu_L1; end if isfield(param, 'tight_L1') param_L1.tight = param.tight_L1; end if isfield(param, 'max_iter_L1') param_L1.max_iter = param.max_iter_L1; end if isfield(param, 'rel_obj_L1') param_L1.rel_obj = param.rel_obj_L1; else param_L1.rel_obj = param.rel_obj; end if isfield(param, 'weights') param_L1.weights = param.weights; else param_L1.weights = 1; end if isfield(param, 'pos_L1') param_L1.pos = param.pos_L1; else param_L1.pos = 0; end if isfield(param, 'verbose_L1') param_L1.verbose = param.verbose_L1; else param_L1.verbose = param.verbose; end % Useful functions for the projection sc = @(z) z*min(epsilon/norm(z(:)), 1); % scaling %Initializations. %Initial solution if isfield(param,'initsol') xsol = param.initsol; else xsol = 1/param.nu*At(y); end %Initial dual variables if isfield(param, 'initz') z = param.initz; else z = zeros(size(y)); end %Initial residual res = A(xsol) - y; %Flags initialization dummy = Psit(xsol); fval = sum(param_L1.weights(:).*abs(dummy(:))); flag = 0; %Step sizes computation %Step size primal mu = 1.0/param.nu; %Step size for the dual variables beta = 0.9; %Main loop. Sequential. for t = 1:param.max_iter %Slack variable update s = sc(- z - res); %Gradient formation r = At(z + res + s); %Gradient decend r = xsol - mu*r; %Prox L1 norm (global solution) prev_fval = fval; [xsol, fval] = sopt_mltb_prox_L1(r, param.gamma*mu, param_L1); %Residual res = A(xsol) - y; %Lagrange multipliers update z = z + beta*(res + s); %Check feasibility res1 = norm(res(:)); %Check relative change of objective function rel_fval = abs(fval - prev_fval)/fval; %Log if (param.verbose >= 1) fprintf('Iter %i\n',t); fprintf(' L1 norm = %e, rel_fval = %e\n', ... fval, rel_fval); fprintf(' epsilon = %e, residual = %e\n\n', epsilon, res1); end %Global stopping criteria if (rel_fval < param.rel_obj && res1 <= epsilon*1.001) flag = 1; break; end end %Final log if (param.verbose > 0) if (flag == 1) fprintf('Solution found\n'); fprintf(' Objective function = %e\n', fval); fprintf(' Final residual = %e\n', res1); else fprintf('Maximum number of iterations reached\n'); fprintf(' Objective function = %e\n', fval); fprintf(' Relative variation = %e\n', rel_fval); fprintf(' Final residual = %e\n', res1); fprintf(' epsilon = %e\n', epsilon); end end end sopt-2.0.0/matlab/sopt_mltb_admm_bpconw.m000066400000000000000000000115521277570055300205040ustar00rootroot00000000000000function [xsol, z] = sopt_mltb_admm_bpconw(y, epsilon, A, At, Psi, Psit, w, param) % % sol = admm_bpconw(y, epsilon, A, At, Psi, Psit, w, param) solves: % % min ||Psit x||_1 s.t. ||W*(y-A x)||_2 <= epsilon % % % y contains the measurements. A is the forward measurement operator and % At the associated adjoint operator. Psit is a sparfying transform and Psi % its adjoint. w is a vector containing the weights for the L2 norm. % PARAM a Matlab structure containing the following fields: % % General parameters: % % - verbose: 0 no log, 1 print main steps, 2 print all steps. % % - max_iter: max. nb. of iterations (default: 200). % % - rel_obj: minimum relative change of the objective value (default: % 1e-4) % The algorithm stops if % | ||x(t)||_1 - ||x(t-1)||_1 | / ||x(t)||_1 < rel_obj, % where x(t) is the estimate of the solution at iteration t. % % - gamma: control the converge speed (default: 1e-1). % % % Proximal L1 operator: % % - rel_obj_L1: Used as stopping criterion for the proximal L1 % operator. Min. relative change of the objective value between two % successive estimates. % % - max_iter_L1: Used as stopping criterion for the proximal L1 % operator. Maximun number of iterations. % % - param.nu_L1: bound on the norm^2 of the operator Psi, i.e. % ||Psi x||^2 <= nu * ||x||^2 (default: 1) % % - param.tight_L1: 1 if Psit is a tight frame or 0 if not (default = 1) % % - param.weights: weights (default = 1) for a weighted L1-norm defined % as sum_i{weights_i.*abs(x_i)} % % % Author: Rafael Carrillo % E-mail: rafael.carrillo@epfl.ch % Date: Nov. 22, 2014 % % Optional input arguments if ~isfield(param, 'verbose'), param.verbose = 1; end if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end if ~isfield(param, 'max_iter'), param.max_iter = 200; end if ~isfield(param, 'gamma'), param.gamma = 1e-2; end if ~isfield(param, 'nu'), param.nu = 1; end % Input arguments for prox L1 param_L1.Psi = Psi; param_L1.Psit = Psit; if isfield(param, 'nu_L1') param_L1.nu = param.nu_L1; end if isfield(param, 'tight_L1') param_L1.tight = param.tight_L1; end if isfield(param, 'max_iter_L1') param_L1.max_iter = param.max_iter_L1; end if isfield(param, 'rel_obj_L1') param_L1.rel_obj = param.rel_obj_L1; else param_L1.rel_obj = param.rel_obj; end if isfield(param, 'weights') param_L1.weights = param.weights; else param_L1.weights = 1; end if isfield(param, 'pos_L1') param_L1.pos = param.pos_L1; else param_L1.pos = 0; end if isfield(param, 'verbose_L1') param_L1.verbose = param.verbose_L1; else param_L1.verbose = param.verbose; end % Useful functions for the projection sc = @(z) z*min(epsilon/norm(z(:)), 1); % scaling %Initializations. %Initial solution if isfield(param,'initsol') xsol = param.initsol; else xsol = 1/param.nu*At(y); end %Initial dual variables if isfield(param, 'initz') z = param.initz; else z = zeros(size(y)); end %Initial residual res = A(xsol) - y; %Flags initialization dummy = Psit(xsol); fval = sum(param_L1.weights(:).*abs(dummy(:))); flag = 0; %Step sizes computation %Step size primal mu = 1.0/param.nu; %Step size for the dual variables beta = 0.9; %Main loop. Sequential. for t = 1:param.max_iter if (param.verbose >= 1) fprintf('Iter %i\n',t); end %Slack variable update s = sc(-w.*(z + res))./w; %Gradient formation r = At(z + res + s); %Gradient decend r = xsol - mu*r; %Prox L1 norm (global solution) prev_fval = fval; [xsol, fval] = sopt_mltb_prox_L1(r, param.gamma*mu, param_L1); %Residual res = A(xsol) - y; %Lagrange multipliers update z = z + beta*(res + s); %Check feasibility res1 = norm(res(:).*w); %Check relative change of objective function rel_fval = abs(fval - prev_fval)/fval; %Log if (param.verbose >= 1) fprintf(' L1 norm = %e, rel_fval = %e\n', ... fval, rel_fval); fprintf(' epsilon = %e, residual = %e\n\n', epsilon, res1); end %Global stopping criteria if (rel_fval < param.rel_obj && res1 <= epsilon*1.001) flag = 1; break; end end %Final log if (param.verbose > 0) if (flag == 1) fprintf('Solution found\n'); fprintf(' Objective function = %e\n', fval); fprintf(' Final residual = %e\n', res1); else fprintf('Maximum number of iterations reached\n'); fprintf(' Objective function = %e\n', fval); fprintf(' Relative variation = %e\n', rel_fval); fprintf(' Final residual = %e\n', res1); fprintf(' epsilon = %e\n', epsilon); end end end sopt-2.0.0/matlab/sopt_mltb_solve_BPDN.m000066400000000000000000000127701277570055300201540ustar00rootroot00000000000000function sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param) % sopt_mltb_solve_BPDN - Solve BPDN problem % % Solve the Basis Pursuit Denoising (BPDN) problem % % min ||Psit x||_1 s.t. ||y-A x||_2 < epsilon % % where y contains the measurements, A is the forward measurement operator % and At the associated adjoint operator, Psit is a sparfying transform and % Psi its adjoint. The structure param should contain the following fields: % % General parameters: % % - verbose: Verbosity level (0 = no log, 1 = summary at convergence, % 2 = print main steps; default = 1). % % - max_iter: Maximum number of iterations (default = 200). % % - rel_obj: Minimum relative change of the objective value % (default = 1e-4). The algorithm stops if % | ||x(t)||_1 - ||x(t-1)||_1 | / ||x(t)||_1 < rel_obj, % where x(t) is the estimate of the solution at iteration t. % % - gamma: Convergence speed (weighting of L1 norm when solving for % L1 proximal operator) (default = 1e-1). % % - initsol: Initial solution for a warmstart. % % Projection onto the L2-ball: % % - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1). % % - nu_B2: Bound on the norm of the operator A, i.e. % ||A x||^2 <= nu * ||x||^2 (default = 1). % % - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms % stops if % epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol) % (default = 1e-3). % % - max_iter_B2: Maximum number of iterations for the projection onto the % L2 ball (default = 200). % % - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise; % default = 0). % % - real_B2: Reality flag (1 to impose reality, 0 otherwise; % default = 0). % % Proximal L1 operator: % % - rel_obj_L1: Used as stopping criterion for the proximal L1 % operator. Minimum relative change of the objective value between % two successive estimates. % % - max_iter_L1: Used as stopping criterion for the proximal L1 % operator. Maximun number of iterations. % % - param.nu_L1: Bound on the norm^2 of the operator Psi, i.e. % ||Psi x||^2 <= nu * ||x||^2 (default = 1). % % - param.tight_L1: 1 if Psit is a tight frame, 0 otherwise % (default = 1). % % - param.weights: Weights for a weighted L1-norm defined % by sum_i{weights_i.*abs(x_i)} (default = 1). % % - pos_l1: Positivity flag (1 to impose positivity, 0 otherwise; % default = 0). % % References: % [1] P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting % Approach to Nonsmooth Convex Variational Signal Recovery", IEEE Journal % of Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007. % Optional input arguments if ~isfield(param, 'verbose'), param.verbose = 1; end if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end if ~isfield(param, 'max_iter'), param.max_iter = 200; end if ~isfield(param, 'gamma'), param.gamma = 1e-2; end if ~isfield(param, 'pos_l1'), param.pos_l1 = 0; end % Input arguments for projection onto the L2 ball param_B2.A = A; param_B2.At = At; param_B2.y = y; param_B2.epsilon = epsilon; param_B2.verbose = param.verbose; if isfield(param, 'nu_B2'), param_B2.nu = param.nu_B2; end if isfield(param, 'tol_B2'), param_B2.tol = param.tol_B2; end if isfield(param, 'tight_B2'), param_B2.tight = param.tight_B2; end if isfield(param, 'max_iter_B2') param_B2.max_iter = param.max_iter_B2; end if isfield(param,'pos_B2'), param_B2.pos=param.pos_B2; end if isfield(param,'real_B2'), param_B2.real=param.real_B2; end % Input arguments for prox L1 param_L1.Psi = Psi; param_L1.Psit = Psit; param_L1.pos = param.pos_l1; param_L1.verbose = param.verbose; %param_L1.verbose = 2; param_L1.rel_obj = param.rel_obj; if isfield(param, 'nu_L1') param_L1.nu = param.nu_L1; end if isfield(param, 'tight_L1') param_L1.tight = param.tight_L1; end if isfield(param, 'max_iter_L1') param_L1.max_iter = param.max_iter_L1; end if isfield(param, 'rel_obj_L1') param_L1.rel_obj = param.rel_obj_L1; end if isfield(param, 'weights') param_L1.weights = param.weights; else param_L1.weights = 1; end % Initialization if isfield(param,'initsol') xhat = param.initsol; else xhat = At(y); end iter = 1; prev_norm = 0; % Main loop while 1 if param.verbose >= 1 fprintf('Iteration %i:\n', iter); end % Projection onto the L2-ball [sol, param_B2.u] = sopt_mltb_fast_proj_B2(xhat, param_B2); % Global stopping criterion dummy = Psit(sol); curr_norm = sum(param_L1.weights(:).*abs(dummy(:))); rel_norm = abs(curr_norm - prev_norm)/curr_norm; if param.verbose >= 1 fprintf(' ||x||_1 = %e, rel_norm = %e\n', ... curr_norm, rel_norm); end if (rel_norm < param.rel_obj) crit_BPDN = 'REL_NORM'; break; elseif iter >= param.max_iter crit_BPDN = 'MAX_IT'; break; end % Proximal L1 operator xhat = 2*sol - xhat; [temp,~] = sopt_mltb_prox_L1(xhat, param.gamma, param_L1); xhat = temp + sol - xhat; % Update variables iter = iter + 1; prev_norm = curr_norm; end % Log if param.verbose >= 1 % L1 norm fprintf('\n Solution found:\n'); fprintf(' Final L1 norm: %e\n', curr_norm); % Residual dummy = A(sol); res = norm(y(:)-dummy(:), 2); fprintf(' epsilon = %e, ||y-Ax||_2=%e\n', epsilon, res); % Stopping criterion fprintf(' %i iterations\n', iter); fprintf(' Stopping criterion: %s \n\n', crit_BPDN); end end sopt-2.0.0/matlab/sopt_mltb_solve_L2DN.m000066400000000000000000000103241277570055300201210ustar00rootroot00000000000000function sol = sopt_mltb_solve_L2DN(y, epsilon, A, At, param) % sopt_mltb_solve_L2DN - Solve L2DN problem. % % Solve the L2 denoising problem % % min ||x||_2 s.t. ||y-A x||_2 < epsilon % % where y contains the measurements, A is the forward measurement operator % and At the associated adjoint operator. The structure param should % contain the following fields: % % General parameters: % % - verbose: Verbosity level (0 = no log, 1 = summary at convergence, % 2 = print main steps; default = 1). % % - max_iter: Maximum number of iterations (default = 200). % % - rel_obj: Minimum relative change of the objective value % (default = 1e-4). The algorithm stops if % | ||x(t)||_2 - ||x(t-1)||_2 | / ||x(t)||_2 < rel_obj, % where x(t) is the estimate of the solution at iteration t. % % - gamma: Convergence speed (weighting of L1 norm when solving for % L1 proximal operator) (default = 1e-1). % % - param.weights: weightsfor a weighted L2-norm defined % by norm(weights_i.*x_i,2) (default = 1). % % - initsol: Initial solution for a warmstart. % % Projection onto the L2-ball: % % - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1). % % - nu_B2: Bound on the norm of the operator A, i.e. % ||A x||^2 <= nu * ||x||^2 (default = 1). % % - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms % stops if % epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol) % (default = 1e-3). % % - max_iter_B2: Maximum number of iterations for the projection onto the % L2 ball (default = 200). % % - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise; % default = 0). % % - real_B2: Reality flag (1 to impose reality, 0 otherwise; % default = 0). % % References: % [1] P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting % Approach to Nonsmooth Convex Variational Signal Recovery", IEEE Journal % of Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007. % Optional input arguments if ~isfield(param, 'verbose'), param.verbose = 1; end if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end if ~isfield(param, 'max_iter'), param.max_iter = 200; end if ~isfield(param, 'gamma'), param.gamma = 1e-2; end if ~isfield(param, 'weights'), param.weights = 1; end % Input arguments for projection onto the L2 ball param_B2.A = A; param_B2.At = At; param_B2.y = y; param_B2.epsilon = epsilon; param_B2.verbose = param.verbose; if isfield(param, 'nu_B2'), param_B2.nu = param.nu_B2; end if isfield(param, 'tol_B2'), param_B2.tol = param.tol_B2; end if isfield(param, 'tight_B2'), param_B2.tight = param.tight_B2; end if isfield(param, 'max_iter_B2') param_B2.max_iter = param.max_iter_B2; end if isfield(param,'pos_B2'), param_B2.pos=param.pos_B2; end if isfield(param,'real_B2'), param_B2.real=param.real_B2; end % Initialization if isfield(param,'initsol') xhat = param.initsol; else xhat = At(y); end iter = 1; prev_norm = 0; % Main loop while 1 % if param.verbose >= 1 fprintf('Iteration %i:\n', iter); end % Projection onto the L2-ball [sol, param_B2.u] = sopt_mltb_fast_proj_B2(xhat, param_B2); % Global stopping criterion dummy = sol; curr_norm = norm(param.weights(:).*dummy(:)); rel_norm = abs(curr_norm - prev_norm)/curr_norm; if param.verbose >= 1 fprintf(' ||x||_1 = %e, rel_norm = %e\n', ... curr_norm, rel_norm); end if (rel_norm < param.rel_obj) crit_BPDN = 'REL_NORM'; break; elseif iter >= param.max_iter crit_BPDN = 'MAX_IT'; break; end % Proximal L2 operator xhat = 2*sol - xhat; temp = xhat ./ (1 + 2*param.weights); xhat = temp + sol - xhat; % Update variables iter = iter + 1; prev_norm = curr_norm; end % Log if param.verbose >= 1 % L1 norm fprintf('\n Solution found:\n'); fprintf(' Final L1 norm: %e\n', curr_norm); % Residual dummy = A(sol); res = norm(y(:)-dummy(:), 2); fprintf(' epsilon = %e, ||y-Ax||_2=%e\n', epsilon, res); % Stopping criterion fprintf(' %i iterations\n', iter); fprintf(' Stopping criterion: %s \n\n', crit_BPDN); end end sopt-2.0.0/matlab/sopt_mltb_solve_TVDN.m000066400000000000000000000105731277570055300202030ustar00rootroot00000000000000function sol = sopt_mltb_solve_TVDN(y, epsilon, A, At, param) % sopt_mltb_solve_TVDN - Solve TVDN problem % % Solve the total variation denoising (TVDN) problem % % min ||x||_TV s.t. ||y - A x||_2 < epsilon % % where y contains the measurements, A is the forward measurement operator % and At the associated adjoint operator. The structure param should % contain the following fields: % % General parameters: % % - verbose: Verbosity level (0 = no log, 1 = summary at convergence, % 2 = print main steps; default = 1). % % - max_iter: Maximum number of iterations (default = 200). % % - rel_obj: Minimum relative change of the objective value % (default = 1e-4). The algorithm stops if % | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj, % where x(t) is the estimate of the solution at iteration t. % % - gamma: Convergence speed (weighting of TV norm when solving for % TV proximal operator) (default = 1e-1). % % - initsol: Initial solution for a warmstart. % % Projection onto the L2-ball: % % - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1). % % - nu_B2: Bound on the norm of the operator A, i.e. % ||A x||^2 <= nu * ||x||^2 (default = 1). % % - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms % stops if % epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol) % (default = 1e-3). % % - max_iter_B2: Maximum number of iterations for the projection onto the % L2 ball (default = 200). % % - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise; % default = 0). % % - real_B2: Reality flag (1 to impose reality, 0 otherwise; % default = 0). % % Proximal L1 operator: % % - max_iter_TV: Used as stopping criterion for the proximal TV % operator. Maximun number of iterations (default = 200). % % References: % P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting Approach % to Nonsmooth Convex Variational Signal Recovery", IEEE Journal of % Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007. % Optional input arguments if ~isfield(param, 'verbose'), param.verbose = 1; end if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end if ~isfield(param, 'max_iter'), param.max_iter = 200; end if ~isfield(param, 'gamma'), param.gamma = 1e-2; end % Input arguments for projection onto the L2 ball param_B2.A = A; param_B2.At = At; param_B2.y = y; param_B2.epsilon = epsilon; param_B2.verbose = param.verbose; if isfield(param, 'nu_B2'), param_B2.nu = param.nu_B2; end if isfield(param, 'tol_B2'), param_B2.tol = param.tol_B2; end if isfield(param, 'tight_B2'), param_B2.tight = param.tight_B2; end if isfield(param, 'max_iter_B2') param_B2.max_iter = param.max_iter_B2; end if isfield(param,'pos_B2'), param_B2.pos=param.pos_B2; end if isfield(param,'real_B2'), param_B2.real=param.real_B2; end % Input arguments for prox TV param_TV.verbose = param.verbose; param_TV.rel_obj = param.rel_obj; if isfield(param, 'max_iter_TV') param_TV.max_iter= param.max_iter_TV; end % Initialization if isfield(param,'initsol') xhat = param.initsol; else xhat = At(y); end iter = 1; prev_norm = 0; % Main loop while 1 if param.verbose>=1 fprintf('Iteration %i:\n', iter); end % Projection onto the L2-ball [sol, param_B2.u] = sopt_mltb_fast_proj_B2(xhat, param_B2); % Global stopping criterion curr_norm = sopt_mltb_TV_norm(sol, 0); rel_norm = abs(curr_norm - prev_norm)/curr_norm; if param.verbose >= 1 fprintf(' ||x||_TV = %e, rel_norm = %e\n', ... curr_norm, rel_norm); end if (rel_norm < param.rel_obj) crit_BPDN = 'REL_NORM'; break; elseif iter >= param.max_iter crit_BPDN = 'MAX_IT'; break; end % Proximal L1 operator xhat = 2*sol - xhat; temp = sopt_mltb_prox_TV(xhat, param.gamma, param_TV); xhat = temp + sol - xhat; % Update variables iter = iter + 1; prev_norm = curr_norm; end % Log if param.verbose >= 1 % L1 norm fprintf('\n Solution found:\n'); fprintf(' Final TV norm: %e\n', curr_norm); % Residual temp = A(sol); fprintf(' epsilon = %e, ||y-Ax||_2=%e\n', epsilon, ... norm(y(:)-temp(:))); % Stopping criterion fprintf(' %i iterations\n', iter); fprintf(' Stopping criterion: %s \n\n', crit_BPDN); end end sopt-2.0.0/matlab/sopt_mltb_solve_TVDNoA.m000066400000000000000000000133401277570055300204560ustar00rootroot00000000000000function sol = sopt_mltb_solve_TVDNoA(y, epsilon, A, At, S, St, param) % sopt_mltb_solve_TVDNoA - Solve augmented TVDN problem % % Solve the total variation denoising (TVDN) problem when an additional % linear operator S is incorporated in the TV norm, i.e. solve % % min ||S x||_TV s.t. ||y - A x||_2 < epsilon % % where y contains the measurements, A is the forward measurement operator % and At the associated adjoint operator, S is the operator appearing in % the TV norm and St the associated adjoint operator. The structure param % should contain the following fields: % % General parameters: % % - verbose: Verbosity level (0 = no log, 1 = summary at convergence, % 2 = print main steps; default = 1). % % - max_iter: Maximum number of iterations (default = 200). % % - rel_obj: Minimum relative change of the objective value % (default = 1e-4). The algorithm stops if % | ||x(t)||_TV - ||x(t-1)||_TV | / ||x(t)||_TV < rel_obj, % where x(t) is the estimate of the solution at iteration t. % % - gamma: Convergence speed (weighting of TV norm when solving for % TV proximal operator) (default = 1e-1). % % - nu_TV: Bound on the norm of the operator S, i.e. % ||S x||^2 <= nu * ||x||^2 (default = 1). % % - initsol: Initial solution for a warmstart. % % Projection onto the L2-ball: % % - param.tight_B2: 1 if A is a tight frame or 0 otherwise (default = 1). % % - nu_B2: Bound on the norm of the operator A, i.e. % ||A x||^2 <= nu * ||x||^2 (default = 1). % % - tol_B2: Tolerance for the projection onto the L2 ball. The algorithms % stops if % epsilon/(1-tol) <= ||y - A z||_2 <= epsilon/(1+tol) % (default = 1e-3). % % - max_iter_B2: Maximum number of iterations for the projection onto the % L2 ball (default = 200). % % - pos_B2: Positivity flag (1 to impose positivity, 0 otherwise; % default = 0). % % - real_B2: Reality flag (1 to impose reality, 0 otherwise; % default = 0). % % Proximal L1 operator: % % - max_iter_TV: Used as stopping criterion for the proximal TV % operator. Maximun number of iterations (default = 200). % % - param.weights_dx_TV: Weights for a weighted TV-norm in the x % direction (default = 1). % % - param.weights_dy_TV: Weights for a weighted TV-norm in the y % direction (default = 1). % % References: % P. L. Combettes and J-C. Pesquet, "A Douglas-Rachford Splitting Approach % to Nonsmooth Convex Variational Signal Recovery", IEEE Journal of % Selected Topics in Signal Processing, vol. 1, no. 4, pp. 564-574, 2007. % Optional input arguments if ~isfield(param, 'verbose'), param.verbose = 1; end if ~isfield(param, 'rel_obj'), param.rel_obj = 1e-4; end if ~isfield(param, 'max_iter'), param.max_iter = 200; end if ~isfield(param, 'gamma'), param.gamma = 1e-2; end if ~isfield(param, 'weights_dx_TV'), param.weights_dx_TV = 1.0; end if ~isfield(param, 'weights_dy_TV'), param.weights_dy_TV = 1.0; end if ~isfield(param, 'incNP'), param.incNP = false; end if ~isfield(param, 'sphere_flag'), param.sphere_flag = false; end % Input arguments for projection onto the L2 ball param_B2.A = A; param_B2.At = At; param_B2.y = y; param_B2.epsilon = epsilon; param_B2.verbose = param.verbose; if isfield(param, 'nu_B2'), param_B2.nu = param.nu_B2; end if isfield(param, 'tol_B2'), param_B2.tol = param.tol_B2; end if isfield(param, 'tight_B2'), param_B2.tight = param.tight_B2; end if isfield(param, 'max_iter_B2') param_B2.max_iter = param.max_iter_B2; end if isfield(param,'pos_B2'), param_B2.pos=param.pos_B2; end if isfield(param,'real_B2'), param_B2.real=param.real_B2; end % Input arguments for prox TVoA param_TV.A = S; param_TV.At = St; param_TV.verbose = param.verbose; param_TV.rel_obj = param.rel_obj; param_TV.weights_dx = param.weights_dx_TV; param_TV.weights_dy = param.weights_dy_TV; if isfield(param, 'nu_TV') param_TV.nu = param.nu_TV; end if isfield(param, 'max_iter_TV') param_TV.max_iter= param.max_iter_TV; end if isfield(param, 'zero_weights_flag_TV') param_TV.zero_weights_flag = param.zero_weights_flag_TV; end if isfield(param, 'identical_weights_flag_TV'), param_TV.identical_weights_flag = param.identical_weights_flag_TV; end if isfield(param, 'sphere_flag'), param_TV.sphere_flag = param.sphere_flag; param_TV.incNP = param.incNP; end % Initialization if isfield(param,'initsol') xhat = param.initsol; else xhat = At(y); end iter = 1; prev_norm = 0; % Main loop while 1 % if param.verbose>=1 fprintf('Iteration %i:\n', iter); end % Projection onto the L2-ball [sol, param_B2.u] = sopt_mltb_fast_proj_B2(xhat, param_B2); % Global stopping criterion curr_norm = sopt_mltb_TV_norm(S(sol), param_TV.sphere_flag, param_TV.incNP, param_TV.weights_dx, param_TV.weights_dy); rel_norm = abs(curr_norm - prev_norm)/curr_norm; if param.verbose >= 1 fprintf(' ||x||_TV = %e, rel_norm = %e\n', ... curr_norm, rel_norm); end if (rel_norm < param.rel_obj) crit_BPDN = 'REL_NORM'; break; elseif iter >= param.max_iter crit_BPDN = 'MAX_IT'; break; end % Proximal L1 operator xhat = 2*sol - xhat; temp = sopt_mltb_prox_TVoA(xhat, param.gamma, param_TV); xhat = temp + sol - xhat; % Update variables iter = iter + 1; prev_norm = curr_norm; end % Log if param.verbose>=1 % L1 norm fprintf('\n Solution found:\n'); fprintf(' Final TV norm: %e\n', curr_norm); % Residual temp = A(sol); fprintf(' epsilon = %e, ||y-Ax||_2=%e\n', epsilon, ... norm(y(:)-temp(:))); % Stopping criterion fprintf(' %i iterations\n', iter); fprintf(' Stopping criterion: %s \n\n', crit_BPDN); end end sopt-2.0.0/matlab/sopt_mltb_solve_rwBPDN.m000066400000000000000000000047701277570055300205260ustar00rootroot00000000000000function sol = sopt_mltb_solve_rwBPDN(y, epsilon, A, At, Psi, Psit, ... paramT, sigma, tol, maxiter, initsol) % sopt_mltb_solve_rwBPDN - Solve reweighted BPDN problem % % Solve the reweighted L1 minimization function using an homotopy % continuation method to approximate the L0 norm. At each iteration the % following problem is solved: % % min_x ||W Psit x||_1 s.t. ||y-A x||_2 < epsilon % % where W is a diagonal matrix with diagonal elements given by % % [W]_ii = delta(t)/(delta(t)+|[Psit x(t)]_i|). % % The input parameters are defined as follows. % % - y: Input data (measurements). % % - epsilon: Noise bound. % % - A: Forward measurement operator. % % - At: Adjoint measurement operator. % % - Psi: Synthesis sparsity transform. % % - Psit: Analysis sparsity transform. % % - paramT: Structure containing parameters for the L1 solver (see % documentation for sopt_mltb_solve_BPDN). % % - sigma: Noise standard deviation in the analysis domain. % % - tol: Minimum relative change in the solution. % The algorithm stops if % ||x(t)-x(t-1)||_2/||x(t-1)||_2 < tol. % where x(t) is the estimate of the solution at iteration t. % % - maxiter: Maximum number of iterations of the reweighted algorithm. % % - initsol: Initial solution for a warmstart. % % References: % [1] R. E. Carrillo, J. D. McEwen, D. Van De Ville, J.-Ph. Thiran, and % Y. Wiaux. Sparsity averaging for compressive imaging. IEEE Sig. Proc. % Let., in press, 2013. % [2] R. E. Carrillo, J. D. McEwen, and Y. Wiaux. Sparsity Averaging % Reweighted Analysis (SARA): a novel algorithm for radio-interferometric % imaging. Mon. Not. Roy. Astron. Soc., 426(2):1223-1234, 2012. param=paramT; iter=0; rel_dist=1; if nargin<11 fprintf('RW iteration: %i\n', iter); sol = sopt_mltb_solve_BPDN(y, epsilon, A, At, Psi, Psit, param); else sol = initsol; end temp=Psit(sol); delta=std(temp(:)); while (rel_dist>tol && itertol && iter-pi)); sf2=find((v1-pi)); sf=intersect(sf1,sf2); v=v1(sf); u=u1(sf); M = length(u); clear v1 u1 %% Measurement operator initialization %Oversampling factors for nufft ox = 2; oy = 2; %Number of neighbours for nufft Kx = 8; Ky = 8; %Initialize nufft parameters fprintf('Initializing the NUFFT operator\n\n'); tstart1=tic; st = nufft_init([v u],[Ny Nx],[Ky Kx],[oy*Ny ox*Nx], [Ny/2 Nx/2]); tend1=toc(tstart1); fprintf('Time for the initialization: %e\n\n', tend1); %Operator functions A = @(x) nufft(x, st); At = @(x) nufft_adj(x, st); %Maximum eigenvalue of operato A^TA eval = pow_method(A, At, [Ny,Nx], 1e-4, 100, 1); y0 = A(im); % Add Gaussian i.i.d. noise %sigma_noise = 10^(-input_snr/20)*norm(im(:))/sqrt(N); sigma_noise = 10^(-input_snr/20)*norm(y0)/sqrt(M); noise = (randn(size(y0)) + 1i*randn(size(y0)))*sigma_noise/sqrt(2); y = y0 + noise; epsilon = sqrt(M + 2*sqrt(M))*sigma_noise; %Dirty image dirty = At(y); dirty1 = 2*real(dirty)/eval; %dirty1(dirty1<0) = 0; %% Sparsity operator definition %Wavelets parameters nlevel=4; dwtmode('per'); % Sparsity operator for SARA [C1,S1]=wavedec2(dirty1,nlevel,'db1'); ncoef1=length(C1); [C2,S2]=wavedec2(dirty1,nlevel,'db2'); ncoef2=length(C2); [C3,S3]=wavedec2(dirty1,nlevel,'db3'); ncoef3=length(C3); [C4,S4]=wavedec2(dirty1,nlevel,'db4'); ncoef4=length(C4); [C5,S5]=wavedec2(dirty1,nlevel,'db5'); ncoef5=length(C5); [C6,S6]=wavedec2(dirty1,nlevel,'db6'); ncoef6=length(C6); [C7,S7]=wavedec2(dirty1,nlevel,'db7'); ncoef7=length(C7); [C8,S8]=wavedec2(dirty1,nlevel,'db8'); ncoef8=length(C8); clear C1 C2 C3 C4 C5 C6 C7 C8 Psit = @(x) [wavedec2(x,nlevel,'db1')'; wavedec2(x,nlevel,'db2')';... wavedec2(x,nlevel,'db3')';wavedec2(x,nlevel,'db4')';... wavedec2(x,nlevel,'db5')'; wavedec2(x,nlevel,'db6')';... wavedec2(x,nlevel,'db7')'; wavedec2(x,nlevel,'db8')'; x(:)]/sqrt(9); Psi = @(x) (waverec2(x(1:ncoef1),S1,'db1')+... waverec2(x(ncoef1+1:ncoef1+ncoef2),S2,'db2')+... waverec2(x(2*ncoef1+1:2*ncoef1+ncoef2),S3,'db3')+... waverec2(x(3*ncoef1+1:3*ncoef1+ncoef2),S4,'db4')+... waverec2(x(4*ncoef1+1:4*ncoef1+ncoef2),S5,'db5')+... waverec2(x(5*ncoef1+1:5*ncoef1+ncoef2),S6,'db6')+... waverec2(x(6*ncoef1+1:6*ncoef1+ncoef2),S7,'db7')+... waverec2(x(7*ncoef1+1:7*ncoef1+ncoef2),S8,'db8')+... reshape(x(8*ncoef1+1:8*ncoef1+ncoef2), [Ny Nx]))/sqrt(9); % Parameters for BPDN param1.verbose = 1; % Print log or not param1.gamma = 1e-2; % Converge parameter param1.rel_obj = 1e-4; % Stopping criterion for the L1 problem param1.max_iter = 500; % Max. number of iterations for the L1 problem param1.nu = eval; % Bound on the norm of the operator A param1.tight_L1 = 0; % Indicate if Psit is a tight frame (1) or not (0) param1.max_iter_L1 = 100; param1.rel_obj_L1 = 1e-2; param1.pos_L1 = 1; param1.nu_L1 = 1; param1.verbose_L1 = 0; param1.initsol = dirty1; %param1.initz = z; %Solve BPDN tstart = tic; [sol, z] = sopt_mltb_admm_bpcon(y, epsilon, A, At, Psi, Psit, param1); tend = toc(tstart) error = im - sol; SNR = 20*log10(norm(im(:))/norm(error(:))) residual = At(y - A(sol)); DR = eval*max(sol(:))/(norm(residual(:))/sqrt(N)) % save('results/par_results_1.mat','sol','residual','tend','DR','param1',... % 'nlevel','SNR','z'); soln = sol/max(sol(:)); dirtyn = dirty1 - min(dirty1(:)); dirtyn = dirtyn/max(dirtyn(:)); figure, imagesc(log10(soln + 1e-4)), colorbar, axis image figure, imagesc(log10(dirtyn + 1e-4)), colorbar, axis image figure, imagesc(real(residual)/eval), colorbar, axis image sopt-2.0.0/matlab/test_images/000077500000000000000000000000001277570055300162555ustar00rootroot00000000000000sopt-2.0.0/matlab/test_images/Brain_low_res.mat000066400000000000000000021514501277570055300215550ustar00rootroot00000000000000MATLAB 5.0 MAT-file, Platform: MACI, Created on: Thu Mar 3 11:18:53 2011 IMx,w4!daWxY {+(ZVTdE^TTF $#|s{׽ܳc]yWmvoM6;&JCivE|̏)yA]xn u+n00_vT;q܅T}u tKk.ut&R 9)5|ɴd#Gq!q.oF[C ?$լ]SҜtEHe p:=0,H.-9. ܍t~Էv_}:DE#QnC$|Sam檭z~P 0WʹKyJaau=SqxG W~;u2r 1R bN^5Rjz)+>x#15fj;./t%hy`&Uϝc܂!)r)wXZGɪdp9 {7E˞e`Į8&5MJx GUU{ʼ1t&m')qukVxYqR qR3tKj">e(S7ٹz xSxoDi/`ekx祶J džbZZEAltuJ`*ڡP8Q>K>ɺ[ׂ($OcJOeC'W xmKha0 \&&8v[LQis?CϨ_60& ڂXSA3ml˓ U´1Wyt1)84"ސCg]~W˺\"]UܳN;#|Ep8^9ces%X#JquHRJnjە}Ҋ`ڮ7T^n 0_G?V{6-xO}f'bbP\=F'N=wV ~ȱOޙO _Y dZKzTm(),]6c@ #=_ð$(C\A6a_B` |;{|:n^䘖Us ܡG~ƀ;j|#a#`Y&_[( \V-.|GyhQV[#M"R8uvȧ$ 7f:J>[g"=o( YsgPHxHA2/Ul9Px~:>"CYwJS!Lp%:Ϝ9 wc|RmS$r{/(gö!kY} /FЪg3]7l^҉ۢް봪I,;ќ 31E `csE0~G..=n# yc׬+`24hϭр?LwyiW(5?/Iu7;w鑜 e'츹<Z:)c۝y*i:Uy8"i- !t_v^TԩLĢ!oGQ~ohv#4lGb^7OESmR!UkBw$)h4 s}Bgg/J!4ϡ}dQe1]`{G'7-[(Q|DQf3rm2 (ڙa 5sŧ1QLY4~EGCm=3`溇Mo@|O|26,"?ɍ8&tiB "=p.I1CWvX~IQ"Sf <֘bNf\β|^6OTբ?4 rlS g׏(i7 4Km.X>-FRR3:' *Bģ'Fo3 ~.!vF2kB.?^%L&+gE4~ն,-QJT8Cg6nN1܏Y @Hl KKS2%X08p342J-_OozFf*ǜâZeL,\0IN/2& ~f)KB |BDfs}t<mEt>{}J:8inx?F"vxbg3X(^dc8ty7˞מA~B=[G]`~Y|prZH,7n/V[$(F7u/1ɒmY] ^X1>R@]js̓ef2{ܻR~0 Þ}$x]|4 Չ~zAeY3gŠ:fppde(L GkJ,ޠٰ*R,3< &u hEf^=7s#"M{ѶN52-n"I0"Z%~Cfw4OS ̭eԤ@5F)4Ӗ((5E3:wkq *X &Y=y8Ց8]Gg8 8of_a\П{4M=O7y2l\eYw RVƵi;"0lc:5$wGwUǐ`T3H33ӈIW\6dSi/xj}()\n&K^zVܼ2׵pw%4 WVUl.{7v%'Y" 90f?`n,׭ TF_a=dHZwY]9vp)Q\mthre1=3gܱNKj  C)z} |8nh A4 !fk)8ӊ෾:yGG_ɡ" y~5%ŶKhF[=Iz}Լ8,5%E*׭-UpS, gU響tP!kv,uQyM|hbq]9;.k@03V6IխTnt=wOwjsV##}-ԓn^?QnVE40iKƑ qKX$NփϺ ΂0j[Rcǘ`T]+|zsV< 1Nn81 0 E׹MSHoL?c\YT7 kmYq;^*PSAAʲU Ĵ|wu~qثJ^ :.DDmyã ( PeGL@>}pҒ`b5Q= YD-tdO<'6F^31- ҾJc}NUr Ta?EZ?[kcBH*SE!R$\z_Ws!́ yC6T55ѝb||$|8 clAPvA#f8H_\K^dc }4 N {mJF(d[HiY z5"y؈a50)d[{5t46Z:S ~*KCT-ީU M>d O);j)-ݥ9љgybAxt|u]Cݖ̝rf1*ᑹ[l)Z80 r?Nu=F)s&!ls l nNYЊ@~ܼt5 YZh 9#}+Nu@ wqGpb1LK:w%bJ< '-eCjU 4r~}ΰ?[ser: MFG,{]!O`Pf@H<)8 )2kbpqg):^dOFF8Mq>L.9>29s&O;v#TŵxsI7 HQ XMUB4=ڜefYZm iVdfcR?&0{Kƪr%笰Y+R٢YWol|~}?iU 7P][{o e MRxvN Mյf^ #~e(Q u=TQ7[v|k 8zYAxƾ (ǐE+OU5AorHs/o3d>G- WN#GK<&gc$%@e)/1RMi6{F a\Dy-c0jwGNj9F>:wRN7W;oo| / Z\++"4i:Ev80eJ˓V\- ?7i^pR".=go ; 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INSTRUME OBSERVER= 'BRAUN' DATE-OBS= ' 26/09/8' DATE-MAP= '04/05/91' BSCALE = 1.00000000000E+00 / BZERO = 0.00000000000E+00 / BUNIT = 'JY/PIXEL' EPOCH = 1.987699951E+03 / DATAMAX = 1.006458163E+00 / DATAMIN = -1.003974790E-12 / CTYPE1 = 'RA---SIN' CRVAL1 = 0.00000000000E+00 / CDELT1 = -2.777777845E-04 / CRPIX1 = 1.280000000E+02 / CROTA1 = 0.000000000E+00 / CTYPE2 = 'DEC--SIN' CRVAL2 = 3.50000000000E+01 / CDELT2 = 2.777777845E-04 / CRPIX2 = 1.280000000E+02 / CROTA2 = 0.000000000E+00 / CTYPE3 = 'FREQ' CRVAL3 = 2.30000000000E+11 / CDELT3 = 2.300000051E+10 / CRPIX3 = 1.000000000E+00 / CROTA3 = 0.000000000E+00 / ORIGIN = 'NRAO sparc1 SDE' DATE = '04/05/91' HISTORY ---- /Begin "HISTORY" information found in fits tape he HISTORY BLOCKED = T /TAPE MAY BE BLOCKE HISTORY /------------------------------------------------- HISTORY ----- /BEGIN "HISTORY" INFORMATION FOUND IN FITS TAPE HE HISTORY BLOCKED = T /TAPE MAY BE BLOCKE HISTORY / HISTORY OF INPUT HISTORY / HISTORY OF INPUT HISTORY /------------------------------------------------- HISTORY ----- /BEGIN "HISTORY" INFORMATION FOUND IN FITS TAPE HE HISTORY OBSERVAT= 'KPNO' / origin of data HISTORY CCDPICNO= 218 / original ccd pict HISTORY EXPTIME = 1200 / actual integratio HISTORY s) DARKTIME= 1200 / total elapsed tim HISTORY IMAGETYP= 'OBJECT ' / object,dark,bias, HISTORY RA = ' 00:43:35' / right ascension ( HISTORY DEC = ' 41:15:06' / declination (tele HISTORY ZD = ' 11:52:00' / zenith distance HISTORY UT = ' 07:14:19' / universal time HISTORY ST = ' 00:06:13' / sidereal time HISTORY DETECTOR= 'TEK1 ' / detector (ccd typ HISTORY ter, etcCONTINUED:) HISTORY CAMTEMP = -107.88 / camera temperatur HISTORY DEWTEMP = -180.46 / dewar temperature HISTORY FILTERS = ' 2 0 ' / filter bolt posit HISTORY OVERSCAN= 520 / overscan subtract HISTORY ZEROCOR = 100 / zero level subtra HISTORY CCDSUM = ' 1 1 ' / on chip summation HISTORY TAPENUM = 2 HISTORY LICK = 'FITS2 HISTORY CONTINUED: ' HISTORY /END FITS TAPE HEADER "HISTORY" INFORMATION HISTORY /------------------------------------------------- HISTORY ----- IMLOD OUTNAME =' ' OUTCLASS =' HISTORY IMLOD OUTSEQ = -1 INTAPE = 2 OUTDISK= 1 HISTORY IMLOD RELEASE = '15JUL88' HISTORY RENAM INNAME='x13y11.5 ' INCLASS='IMAP ' HISTORY RENAM INSEQ= 2 INDISK= 1 HISTORY RENAM OUTNAME='x13y11.5 ' OUTCLASS='HA ' HISTORY RENAM OUTSEQ= 1 OUTDISK= 1 HISTORY 11 TRANS INNAME='x13y11.5 ' INCLASS='HA ' HISTORY TRANS INSEQ= 1 INDISK= 1 HISTORY TRANS OUTNAME='X13Y11 ' OUTCLASS='HA ' HISTORY TRANS OUTSEQ= 1 OUTDISK= 2 HISTORY TRANS BLC= 1., 1., 1., 1., 1., 1., HISTORY GE TRANS TRC= 508., 508., 1., 1., 1., 1., HISTORY GE TRANS TRANCOD='21 ' / OUTPUT AXIS ORDER HISTORY PUTVALUE PIXXY= 478, 29, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= -1.000000E+00 HISTORY PUTVALUE / NEW VALUE= 4.785877E+01 HISTORY PUTVALUE PIXXY= 361, 185, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 3.358684E+00 HISTORY PUTVALUE / NEW VALUE= 5.015400E+01 HISTORY PUTVALUE / OLD VALUE= 2.030912E+01 HISTORY PUTVALUE / NEW VALUE= 4.857830E+01 HISTORY PUTVALUE PIXXY= 245, 183, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 3.532237E+01 HISTORY PUTVALUE / NEW VALUE= 6.312109E+01 HISTORY PUTVALUE PIXXY= 270, 450, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 6.264473E+00 HISTORY PUTVALUE / NEW VALUE= 5.098941E+01 HISTORY PUTVALUE PIXXY= 100, 274, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 2.418351E+01 HISTORY PUTVALUE / NEW VALUE= 4.643355E+01 HISTORY PUTVALUE PIXXY= 238, 267, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 3.047939E+01 HISTORY NINER RELEASE ='15JUL88 ' /********* START 01-JUL HISTORY 55 NINER INNAME='X13Y11 ' INCLASS='HA ' HISTORY NINER INSEQ= 1 INDISK= 2 HISTORY NINER OUTNAME='X13Y11 ' OUTCLASS='HA ' HISTORY NINER OUTSEQ= 3 OUTDISK= 3 HISTORY NINER BLC = 1., 1., 1., 1., 1., HISTORY NINER TRC = 508., 508., 1., 1., 1., HISTORY NINER OPCODE = 'NINE' HISTORY HGEOM RELEASE ='15JUL88 ' /********* START 07-JUL HISTORY 29 HGEOM INNAME='X13Y11 ' INCLASS='HA ' HISTORY HGEOM INSEQ= 3 INDISK= 3 HISTORY HGEOM IN2NAME='X13Y11 ' IN2CLASS='RC ' HISTORY HGEOM IN2SEQ= 3 IN2DISK= 3 HISTORY HGEOM OUTSEQ= 1 OUTDISK= 3 HISTORY HGEOM BLC = 1, 1, 1, 1, 1, 1, 1 HISTORY HGEOM TRC = 508, 508, 1, 1, 1, 1, 1 HISTORY HGEOM IMSIZE = 508, 508 / OUTPUT IMAGE HISTORY HGEOM / INTERPOLATION ORDER USED WAS BIQUINTIC HISTORY HGEOM / INDETERMINATE PIXELS FILLED WITH ZEROS HISTORY HGEOM / 2028 PIXELS BLANKED DUE TO MEMORY L HISTORY TRY HGEOM / 0 PIXELS BLANKED DUE TO INPUT BL HISTORY ADDBEAM BMAJ= 9.30556E-04 BMIN= 7.16667E-04 BPA= HISTORY PUTHEAD NITER = 0 / OLD HISTORY PUTHEAD NITER = 1 / NEW HISTORY ADDBEAM BMAJ= 9.30556E-04 BMIN= 7.16667E-04 BPA= HISTORY ADDBEAM BMAJ= 7.22111E-04 BMIN= 5.56133E-04 BPA= HISTORY 49 CONVL INNAME='X13Y11 ' INCLASS='HH ' HISTORY CONVL INSEQ= 1 INDISK= 3 HISTORY CONVL OUTNAME='X13Y11 ' OUTCLASS='HHC ' HISTORY CONVL OUTSEQ= 1 OUTDISK= 2 HISTORY CONVL BLC= 1. 1. 1. 1. 1. 1. HISTORY CONVL TRC= 508. 508. 1. 1. 1. 1. HISTORY CONVL BMAJ= 2.8100 BMIN= 2.8100 BPA= 0.0/OUTP HISTORY CONVL CVBMAJ= 1.9717 CVBMIN= 1.0669 CVBPA= 148. HISTORY CONVL FACTOR= 1.00000E+00 /UNITS SCALING FACTOR HISTORY / HISTORY OF INPUT HISTORY /------------------------------------------------- HISTORY ----- /BEGIN "HISTORY" INFORMATION FOUND IN FITS TAPE HE HISTORY OBSERVAT= 'KPNO' / origin of data HISTORY EXPTIME = 1200 / actual integratio HISTORY s) DARKTIME= 1200 / total elapsed tim HISTORY OTIME = 1200 / shutter open time HISTORY IMAGETYP= 'OBJECT ' / object,dark,bias, HISTORY RA = ' 00:43:35' / right ascension ( HISTORY DEC = ' 41:15:10' / declination (tele HISTORY ZD = ' 09:51:00' / zenith distance HISTORY UT = ' 07:35:14' / universal time HISTORY ST = ' 00:27:12' / sidereal time HISTORY DETECTOR= 'TEK1 ' / detector (ccd typ HISTORY ter, etcCONTINUED:) HISTORY CAMTEMP = -107.88 / camera temperatur HISTORY DEWTEMP = -180.36 / dewar temperature HISTORY OVERSCAN= 521 / overscan subtract HISTORY ZEROCOR = 100 / zero level subtra HISTORY ,bias) FLATCOR = 103 / flat field correc HISTORY CCDSUM = ' 1 1 ' / on chip summation HISTORY TAPENUM = 5 HISTORY LICK = 'FITS2 HISTORY CONTINUED: ' HISTORY /END FITS TAPE HEADER "HISTORY" INFORMATION HISTORY /------------------------------------------------- HISTORY ----- IMLOD OUTNAME =' ' OUTCLASS =' HISTORY IMLOD OUTSEQ = -1 INTAPE = 2 OUTDISK= 1 HISTORY IMLOD RELEASE = '15JUL88' HISTORY RENAM INNAME='x13y11.5 ' INCLASS='IMAP ' HISTORY RENAM OUTNAME='x13y11.5 ' OUTCLASS='HA ' HISTORY RENAM OUTSEQ= 2 OUTDISK= 1 HISTORY TRANS RELEASE ='15JUL88 ' /********* START 01-JUL HISTORY 48 TRANS INNAME='x13y11.5 ' INCLASS='HA ' HISTORY TRANS INSEQ= 2 INDISK= 1 HISTORY TRANS OUTNAME='X13Y11 ' OUTCLASS='HA ' HISTORY TRANS OUTSEQ= 2 OUTDISK= 2 HISTORY TRANS BLC= 1., 1., 1., 1., 1., 1., HISTORY GE TRANS TRC= 508., 508., 1., 1., 1., 1., HISTORY GE TRANS TRANCOD='21 ' / OUTPUT AXIS ORDER HISTORY PUTVALUE PIXXY= 245, 183, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 2.708539E+01 HISTORY PUTVALUE / NEW VALUE= 6.688857E+01 HISTORY PUTVALUE / OLD VALUE= 5.076327E+00 HISTORY PUTVALUE / NEW VALUE= 4.791787E+01 HISTORY PUTVALUE PIXXY= 362, 185, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 1.919535E+01 HISTORY PUTVALUE / NEW VALUE= 4.768552E+01 HISTORY PUTVALUE PIXXY= 478, 29, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 3.415265E+00 HISTORY PUTVALUE / NEW VALUE= 4.977964E+01 HISTORY PUTVALUE PIXXY= 270, 450, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 4.245796E+00 HISTORY PUTVALUE / NEW VALUE= 5.090547E+01 HISTORY PUTVALUE PIXXY= 100, 274, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 2.127168E+01 HISTORY NINER RELEASE ='15JUL88 ' /********* START 01-JUL HISTORY 01 NINER INNAME='X13Y11 ' INCLASS='HA ' HISTORY NINER INSEQ= 2 INDISK= 2 HISTORY NINER OUTNAME='X13Y11 ' OUTCLASS='HA ' HISTORY NINER OUTSEQ= 4 OUTDISK= 3 HISTORY NINER BLC = 1., 1., 1., 1., 1., HISTORY NINER TRC = 508., 508., 1., 1., 1., HISTORY NINER OPCODE = 'NINE' HISTORY HGEOM RELEASE ='15JUL88 ' /********* START 07-JUL HISTORY 18 HGEOM INNAME='X13Y11 ' INCLASS='HA ' HISTORY HGEOM INSEQ= 4 INDISK= 3 HISTORY HGEOM IN2NAME='X13Y11 ' IN2CLASS='RC ' HISTORY HGEOM IN2SEQ= 3 IN2DISK= 3 HISTORY HGEOM OUTSEQ= 2 OUTDISK= 3 HISTORY HGEOM BLC = 1, 1, 1, 1, 1, 1, 1 HISTORY HGEOM TRC = 508, 508, 1, 1, 1, 1, 1 HISTORY HGEOM IMSIZE = 508, 508 / OUTPUT IMAGE HISTORY HGEOM / INTERPOLATION ORDER USED WAS BIQUINTIC HISTORY HGEOM / INDETERMINATE PIXELS FILLED WITH ZEROS HISTORY HGEOM / 1522 PIXELS BLANKED DUE TO MEMORY L HISTORY TRY HGEOM / 0 PIXELS BLANKED DUE TO INPUT BL HISTORY ADDBEAM BMAJ= 1.00278E-03 BMIN= 7.77778E-04 BPA= HISTORY PUTHEAD NITER = 0 / OLD HISTORY PUTHEAD NITER = 1 / NEW HISTORY ADDBEAM BMAJ= 1.00278E-03 BMIN= 7.77778E-04 BPA= HISTORY ADDBEAM BMAJ= 7.77778E-04 BMIN= 6.02778E-04 BPA= HISTORY 51 CONVL INNAME='X13Y11 ' INCLASS='HH ' HISTORY CONVL INSEQ= 2 INDISK= 3 HISTORY CONVL OUTNAME='X13Y11 ' OUTCLASS='HHC ' HISTORY CONVL OUTSEQ= 2 OUTDISK= 2 HISTORY CONVL BLC= 1. 1. 1. 1. 1. 1. HISTORY CONVL TRC= 508. 508. 1. 1. 1. 1. HISTORY CONVL BMAJ= 2.8100 BMIN= 2.8100 BPA= 0.0/OUTP HISTORY CONVL CVBMAJ= 1.7853 CVBMIN= 0.2369 CVBPA= 147. HISTORY CONVL FACTOR= 1.00000E+00 /UNITS SCALING FACTOR HISTORY / END OF OLD HISTOR HISTORY 15L8 HISTORY COMB INNAME='X13Y11 ' INCLASS='HHC ' HISTORY COMB INSEQ= 1 INDISK= 2 HISTORY COMB IN2SEQ= 2 IN2DISK= 2 HISTORY COMB OUTNAME='X13Y11 ' OUTCLASS='H ' HISTORY COMB OUTSEQ= 1 OUTDISK= 3 HISTORY COMB USERID= 1341 HISTORY COMB CTYPE= 1 /LIN.COMB HISTORY COMB BLC= 1 1 1 1 1 1 1 / BO HISTORY ER COMB TRC= 508 508 1 1 1 1 1 / TO HISTORY COMB A( 1)= 5.0000E-01 A( 2)= 5.0000E-01 A( 3 HISTORY COMB / UNDEFINED PIXELS MAGIC-VALUE BLANKED HISTORY / HISTORY OF INPUT HISTORY / HISTORY OF INPUT HISTORY /------------------------------------------------- HISTORY ----- /BEGIN "HISTORY" INFORMATION FOUND IN FITS TAPE HE HISTORY CCDPICNO= 214 / original ccd pict HISTORY EXPTIME = 600 / actual integratio HISTORY s) DARKTIME= 600 / total elapsed tim HISTORY OTIME = 600 / shutter open time HISTORY IMAGETYP= 'OBJECT ' / object,dark,bias, HISTORY RA = ' 00:43:34' / right ascension ( HISTORY DEC = ' 41:15:13' / declination (tele HISTORY ZD = ' 23:21:00' / zenith distance HISTORY UT = ' 06:04:29' / universal time HISTORY ST = ' 22:56:11' / sidereal time HISTORY DETECTOR= 'TEK1 ' / detector (ccd typ HISTORY ter, etcCONTINUED:) HISTORY CAMTEMP = -107.88 / camera temperatur HISTORY FILTERS = ' 4 0 ' / filter bolt posit HISTORY OVERSCAN= 518 / overscan subtract HISTORY ZEROCOR = 100 / zero level subtra HISTORY ,bias) FLATCOR = 102 / flat field correc HISTORY CCDSUM = ' 1 1 ' / on chip summation HISTORY TAPENUM = 3 HISTORY CLIPIX = 2 HISTORY LICK = 'FITS2 HISTORY CONTINUED: ' HISTORY /END FITS TAPE HEADER "HISTORY" INFORMATION HISTORY /------------------------------------------------- HISTORY ----- IMLOD OUTNAME =' ' OUTCLASS =' HISTORY IMLOD OUTSEQ = -1 INTAPE = 2 OUTDISK= 1 HISTORY RENAM INNAME='x13y11.5 ' INCLASS='IMAP ' HISTORY RENAM INSEQ= 3 INDISK= 1 HISTORY RENAM OUTNAME='x13y11.5 ' OUTCLASS='RC ' HISTORY RENAM OUTSEQ= 1 OUTDISK= 1 HISTORY TRANS RELEASE ='15JUL88 ' /********* START 01-JUL HISTORY 45 TRANS INNAME='x13y11.5 ' INCLASS='RC ' HISTORY TRANS INSEQ= 1 INDISK= 1 HISTORY TRANS OUTNAME='X13Y11 ' OUTCLASS='RC ' HISTORY TRANS OUTSEQ= 1 OUTDISK= 2 HISTORY TRANS BLC= 1., 1., 1., 1., 1., 1., HISTORY GE TRANS TRC= 508., 508., 1., 1., 1., 1., HISTORY GE TRANS TRANCOD='21 ' / OUTPUT AXIS ORDER HISTORY PUTVALUE PIXXY= 477, 29, 1, 1, 1 HISTORY PUTVALUE / NEW VALUE= 6.212081E+01 HISTORY PUTVALUE PIXXY= 245, 183, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 2.227195E+01 HISTORY PUTVALUE / NEW VALUE= 5.640461E+01 HISTORY PUTVALUE PIXXY= 100, 274, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 2.128498E+01 HISTORY PUTVALUE / NEW VALUE= 5.084704E+01 HISTORY PUTVALUE PIXXY= 270, 450, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 7.467422E+00 HISTORY PUTVALUE / NEW VALUE= 6.360126E+01 HISTORY PUTVALUE PIXXY= 342, 330, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 4.506516E+00 HISTORY PUTVALUE / NEW VALUE= 3.210023E+04 HISTORY 16 NINER INNAME='X13Y11 ' INCLASS='RC ' HISTORY NINER INSEQ= 1 INDISK= 2 HISTORY NINER OUTNAME='X13Y11 ' OUTCLASS='RC ' HISTORY NINER OUTSEQ= 3 OUTDISK= 3 HISTORY NINER BLC = 1., 1., 1., 1., 1., HISTORY NINER TRC = 508., 508., 1., 1., 1., HISTORY NINER OPCODE = 'NINE' HISTORY ADDBEAM BMAJ= 7.02778E-04 BMIN= 6.19444E-04 BPA= HISTORY RENAM INNAME='X13Y11 ' INCLASS='RC ' HISTORY RENAM INSEQ= 3 INDISK= 3 HISTORY RENAM OUTNAME='X13Y11 ' OUTCLASS='RH ' HISTORY RENAM OUTSEQ= 1 OUTDISK= 3 HISTORY PUTHEAD NITER = 0 / OLD HISTORY ADDBEAM BMAJ= 7.02778E-04 BMIN= 6.19444E-04 BPA= HISTORY ADDBEAM BMAJ= 5.45356E-04 BMIN= 4.80689E-04 BPA= HISTORY CONVL RELEASE ='15JUL88 ' /********* START 11-JUL HISTORY 16 CONVL INNAME='X13Y11 ' INCLASS='RH ' HISTORY CONVL INSEQ= 1 INDISK= 3 HISTORY CONVL OUTNAME='X13Y11 ' OUTCLASS='RHC ' HISTORY CONVL OUTSEQ= 1 OUTDISK= 2 HISTORY CONVL BLC= 1. 1. 1. 1. 1. 1. HISTORY CONVL TRC= 508. 508. 1. 1. 1. 1. HISTORY CONVL BMAJ= 2.8100 BMIN= 2.8100 BPA= 0.0/OUTP HISTORY CONVL CVBMAJ= 2.2139 CVBMIN= 2.0104 CVBPA= 157. HISTORY CONVL FACTOR= 1.00000E+00 /UNITS SCALING FACTOR HISTORY / HISTORY OF INPUT HISTORY ----- /BEGIN "HISTORY" INFORMATION FOUND IN FITS TAPE HE HISTORY OBSERVAT= 'KPNO' / origin of data HISTORY CCDPICNO= 215 / original ccd pict HISTORY EXPTIME = 600 / actual integratio HISTORY s) DARKTIME= 600 / total elapsed tim HISTORY OTIME = 600 / shutter open time HISTORY IMAGETYP= 'OBJECT ' / object,dark,bias, HISTORY RA = ' 00:43:34' / right ascension ( HISTORY DEC = ' 41:15:11' / declination (tele HISTORY ZD = ' 21:16:00' / zenith distance HISTORY UT = ' 06:15:57' / universal time HISTORY ST = ' 23:07:41' / sidereal time HISTORY DETECTOR= 'TEK1 ' / detector (ccd typ HISTORY CAMTEMP = -107.88 / camera temperatur HISTORY DEWTEMP = -180.36 / dewar temperature HISTORY FILTERS = ' 4 0 ' / filter bolt posit HISTORY OVERSCAN= 519 / overscan subtract HISTORY ZEROCOR = 100 / zero level subtra HISTORY ,bias) FLATCOR = 102 / flat field correc HISTORY CCDSUM = ' 1 1 ' / on chip summation HISTORY TAPENUM = 6 HISTORY LICK = 'FITS2 HISTORY CONTINUED: ' HISTORY /END FITS TAPE HEADER "HISTORY" INFORMATION HISTORY /------------------------------------------------- HISTORY ----- IMLOD OUTNAME =' ' OUTCLASS =' HISTORY IMLOD RELEASE = '15JUL88' HISTORY RENAM INNAME='x13y11.5 ' INCLASS='IMAP ' HISTORY RENAM INSEQ= 6 INDISK= 1 HISTORY RENAM OUTNAME='x13y11.5 ' OUTCLASS='RC ' HISTORY RENAM OUTSEQ= 2 OUTDISK= 1 HISTORY TRANS RELEASE ='15JUL88 ' /********* START 01-JUL HISTORY 18 TRANS INNAME='x13y11.5 ' INCLASS='RC ' HISTORY TRANS INSEQ= 2 INDISK= 1 HISTORY TRANS OUTNAME='X13Y11 ' OUTCLASS='RC ' HISTORY TRANS OUTSEQ= 2 OUTDISK= 2 HISTORY TRANS BLC= 1., 1., 1., 1., 1., 1., HISTORY GE TRANS TRC= 508., 508., 1., 1., 1., 1., HISTORY GE TRANS TRANCOD='21 ' / OUTPUT AXIS ORDER HISTORY PUTVALUE / OLD VALUE= -1.000000E+00 HISTORY PUTVALUE / NEW VALUE= 6.017683E+01 HISTORY PUTVALUE PIXXY= 361, 185, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 2.454924E+00 HISTORY PUTVALUE / NEW VALUE= 6.025087E+01 HISTORY PUTVALUE PIXXY= 270, 450, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 6.403409E+00 HISTORY PUTVALUE / NEW VALUE= 6.173703E+01 HISTORY PUTVALUE PIXXY= 362, 185, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 2.812008E+01 HISTORY PUTVALUE / NEW VALUE= 5.791139E+01 HISTORY PUTVALUE PIXXY= 245, 183, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 2.812008E+01 HISTORY PUTVALUE PIXXY= 238, 267, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 2.910720E+01 HISTORY PUTVALUE / NEW VALUE= 5.522478E+01 HISTORY PUTVALUE PIXXY= 100, 274, 1, 1, 1 HISTORY PUTVALUE / OLD VALUE= 2.318447E+01 HISTORY PUTVALUE / NEW VALUE= 5.040434E+01 HISTORY NINER RELEASE ='15JUL88 ' /********* START 01-JUL HISTORY 51 NINER INNAME='X13Y11 ' INCLASS='RC ' HISTORY NINER INSEQ= 2 INDISK= 2 HISTORY NINER OUTNAME='X13Y11 ' OUTCLASS='RC ' HISTORY NINER OUTSEQ= 4 OUTDISK= 3 HISTORY NINER BLC = 1., 1., 1., 1., 1., HISTORY NINER TRC = 508., 508., 1., 1., 1., HISTORY HGEOM RELEASE ='15JUL88 ' /********* START 07-JUL HISTORY 21 HGEOM INNAME='X13Y11 ' INCLASS='RC ' HISTORY HGEOM INSEQ= 4 INDISK= 3 HISTORY HGEOM IN2NAME='X13Y11 ' IN2CLASS='RC ' HISTORY HGEOM IN2SEQ= 3 IN2DISK= 3 HISTORY HGEOM OUTNAME='X13Y11 ' OUTCLASS='RH ' HISTORY HGEOM OUTSEQ= 1 OUTDISK= 3 HISTORY HGEOM BLC = 1, 1, 1, 1, 1, 1, 1 HISTORY HGEOM TRC = 508, 508, 1, 1, 1, 1, 1 HISTORY HGEOM IMSIZE = 508, 508 / OUTPUT IMAGE HISTORY HGEOM / INTERPOLATION ORDER USED WAS BIQUINTIC HISTORY HGEOM / INDETERMINATE PIXELS FILLED WITH ZEROS HISTORY HGEOM / 1015 PIXELS BLANKED DUE TO MEMORY L HISTORY ADDBEAM BMAJ= 7.47222E-04 BMIN= 6.52778E-04 BPA= HISTORY RENAM INNAME='X13Y11 ' INCLASS='RH ' HISTORY RENAM INSEQ= 1 INDISK= 3 HISTORY RENAM OUTNAME='X13Y11 ' OUTCLASS='RH ' HISTORY RENAM OUTSEQ= 2 OUTDISK= 3 HISTORY PUTHEAD NITER = 0 / OLD HISTORY PUTHEAD NITER = 1 / NEW HISTORY ADDBEAM BMAJ= 7.47222E-04 BMIN= 6.52778E-04 BPA= HISTORY ADDBEAM BMAJ= 5.79844E-04 BMIN= 5.06556E-04 BPA= HISTORY CONVL RELEASE ='15JUL88 ' /********* START 11-JUL HISTORY 39 CONVL INNAME='X13Y11 ' INCLASS='RH ' HISTORY CONVL INSEQ= 2 INDISK= 3 HISTORY CONVL OUTNAME='X13Y11 ' OUTCLASS='RHC ' HISTORY CONVL BLC= 1. 1. 1. 1. 1. 1. HISTORY CONVL TRC= 508. 508. 1. 1. 1. 1. HISTORY CONVL BMAJ= 2.8100 BMIN= 2.8100 BPA= 0.0/OUTP HISTORY CONVL CVBMAJ= 2.1379 CVBMIN= 1.8811 CVBPA= 155. HISTORY CONVL FACTOR= 1.00000E+00 /UNITS SCALING FACTOR HISTORY / END OF OLD HISTOR HISTORY 15L8 HISTORY COMB INNAME='X13Y11 ' INCLASS='RHC ' HISTORY COMB INSEQ= 1 INDISK= 2 HISTORY COMB IN2NAME='X13Y11 ' IN2CLASS='RHC ' HISTORY COMB IN2SEQ= 2 IN2DISK= 2 HISTORY COMB OUTNAME='X13Y11 ' OUTCLASS='R ' HISTORY COMB OUTSEQ= 1 OUTDISK= 3 HISTORY COMB CTYPE= 1 /LIN.COMB HISTORY COMB BLC= 1 1 1 1 1 1 1 / BO HISTORY ER COMB TRC= 508 508 1 1 1 1 1 / TO HISTORY COMB A( 1)= 5.0000E-01 A( 2)= 5.0000E-01 A( 3 HISTORY COMB / UNDEFINED PIXELS MAGIC-VALUE BLANKED HISTORY / END OF OLD HISTOR HISTORY 15L8 HISTORY COMB INNAME='X13Y11 ' INCLASS='H ' HISTORY COMB INSEQ= 1 INDISK= 3 HISTORY COMB IN2NAME='X13Y11 ' IN2CLASS='R ' HISTORY COMB IN2SEQ= 1 IN2DISK= 3 HISTORY COMB OUTNAME='X13Y11 ' OUTCLASS='H-R ' HISTORY COMB OUTSEQ= 1 OUTDISK= 3 HISTORY COMB CTYPE= 1 /LIN.COMB HISTORY COMB BLC= 1 1 1 1 1 1 1 / BO HISTORY ER COMB TRC= 508 508 1 1 1 1 1 / TO HISTORY COMB A( 1)= 1.0000E+00 A( 2)= -6.1400E-01 A( 3 HISTORY COMB / UNDEFINED PIXELS MAGIC-VALUE BLANKED HISTORY AIPS IMNAME='X13Y11 ' IMCLASS='H-R ' IMSE HISTORY AIPS USERNO= 1341 / HISTORY AIPS CLEAN BMAJ= 7.8056E-04 BMIN= 7.8056E-04 B HISTORY AIPS CLEAN NITER= 1 PRODUCT=1 / NORMAL HISTORY /END FITS TAPE HEADER "HISTORY" INFORMATION HISTORY /------------------------------------------------- HISTORY ----- IMLOD OUTNAME =' ' OUTCLASS =' HISTORY IMLOD OUTSEQ = 0 INTAPE = 2 OUTDISK= 1 HISTORY SUBIM RELEASE ='15OCT88 ' /********* START 06-OCT HISTORY 51 SUBIM INNAME='X13Y11 ' INCLASS='H-R ' HISTORY SUBIM INSEQ= 1 INDISK= 1 HISTORY SUBIM INTYPE ='MA' USERID= 1341 HISTORY SUBIM OUTNAME='X13Y11 ' OUTCLASS='SUBIM ' HISTORY SUBIM OUTSEQ= 1 OUTDISK= 1 HISTORY SUBIM BLC = 13, 242, 1, 1, 1, 1, 1 HISTORY SUBIM TRC = 214, 449, 1, 1, 1, 1, 1 HISTORY SUBIM XINC = 1 YINC = 1 HISTORY BLANK RELEASE ='15OCT88 ' /********* START 06-OCT HISTORY 38 BLANK INNAME='X13Y11 ' INCLASS='SUBIM ' HISTORY BLANK INSEQ= 1 INDISK= 1 HISTORY BLANK OUTNAME='X13Y11 ' OUTCLASS='BLANK ' HISTORY BLANK BLC = 1., 1., 1., 1., 1., HISTORY BLANK TRC = 202., 208., 1., 1., 1., HISTORY BLANK OPCODE = 'SELC' / OP = SELF HISTORY BLANK DPARM(3) = 3.00000E+02 / KEEP F <= HISTORY BLANK DPARM(4) = 2.50000E+01 / KEEP F >= HISTORY BLANK / WHERE F IS FROM THE INPUT IMAGE ITSELF HISTORY BLANK / BLANKED PIXELS SET TO MAGIC VALUE BLANKIN HISTORY BLANK RELEASE ='15OCT88 ' /********* START 06-OCT HISTORY 41 BLANK INNAME='X13Y11 ' INCLASS='BLANK ' HISTORY BLANK INSEQ= 1 INDISK= 1 HISTORY BLANK OUTNAME='X13Y11 ' OUTCLASS='BLANK ' HISTORY BLANK OUTSEQ= 2 OUTDISK= 1 HISTORY BLANK BLC = 1., 1., 1., 1., 1., HISTORY BLANK OPCODE = 'TVCU' / OP = TV C HISTORY BLANK / BLANK PIXELS OUTSIDE BLOTCH REGIONS HISTORY BLANK / BLANKED PIXELS SET TO MAGIC VALUE BLANKIN HISTORY RESCA INNAME='X13Y11 ' INCLASS='BLANK ' HISTORY RESCA INSEQ= 2 INDISK= 1 HISTORY RESCA FACTOR= 0.10000000E+01 OFFSET= -0.250000 HISTORY REMAG RELEASE ='15OCT88 ' /********* START 06-OCT HISTORY 18 REMAG INNAME='X13Y11 ' INCLASS='BLANK ' HISTORY REMAG INSEQ= 2 INDISK= 1 HISTORY REMAG OUTNAME='X13Y11 ' OUTCLASS='REMAG ' HISTORY REMAG OUTSEQ= 1 OUTDISK= 1 HISTORY REMAG PIXVAL = 0.00000E+00 BLC = ( 1, 1) TRC HISTORY REMAG NUMBER OF REPLACED MAGIC BLANKS = 2.7321000 HISTORY 39 PADIM INNAME='X13Y11 ' INCLASS='REMAG ' HISTORY PADIM INSEQ= 1 INDISK= 1 HISTORY PADIM OUTNAME='X13Y11 ' OUTCLASS='PADIM ' HISTORY PADIM OUTSEQ= 1 OUTDISK= 1 HISTORY PADIM IMSIZE = 512, 512 HISTORY PADIM IMAGE PADDED WITH 0.0000E+00's HISTORY PUTHEAD BUNIT =' ' / OLD HISTORY PUTHEAD BUNIT ='JY/PIXEL' / NEW HISTORY PUTHEAD CROTA2 = 1.12898E+00 / OLD HISTORY PUTHEAD CROTA2 = 0.00000E+00 / NEW HISTORY PUTHEAD OBJECT ='x13y11.5' / OLD HISTORY PUTHEAD OBJECT ='MMA-TEST' / NEW HISTORY PUTHEAD CRVAL1 = 1.038014025E+01 / OLD HISTORY PUTHEAD CRVAL2 = 4.104682085E+01 / OLD HISTORY PUTHEAD CRVAL2 = 0.000000000E+00 / NEW HISTORY PUTHEAD CRPIX1 = 3.97000E+02 / OLD HISTORY PUTHEAD CRPIX1 = 2.56000E+02 / NEW HISTORY PUTHEAD CRPIX2 = 1.65000E+02 / OLD HISTORY PUTHEAD CRPIX2 = 2.56000E+02 / NEW HISTORY PUTHEAD CDELT1 = -2.15482E-04 / OLD HISTORY PUTHEAD CDELT1 = -2.77778E-04 / NEW HISTORY PUTHEAD CDELT2 = 2.15646E-04 / OLD HISTORY PUTHEAD CDELT2 = 2.77778E-04 / NEW HISTORY ADDBEAM BMAJ= 7.80560E-04 BMIN= 7.80560E-04 BPA= HISTORY ADDBEAM BMAJ= 1.00000E-03 BMIN= 1.00000E-03 BPA= HISTORY PUTHEAD CDELT2 = 2.77778E-04 / OLD HISTORY PUTHEAD CDELT1 = -2.77778E-04 / OLD HISTORY PUTHEAD CDELT1 = -2.77778E-05 / NEW HISTORY PUTHEAD CDELT1 = -2.77778E-05 / OLD HISTORY PUTHEAD CDELT1 = -6.94444E-05 / NEW HISTORY PUTHEAD CDELT2 = 2.77778E-05 / OLD HISTORY PUTHEAD CDELT2 = 6.94444E-05 / NEW HISTORY PUTHEAD TELESCOP =' ' / OLD HISTORY PUTHEAD TELESCOP ='MMA ' / NEW HISTORY RESCA INNAME='X13Y11 ' INCLASS='PADIM ' HISTORY RESCA INSEQ= 1 INDISK= 1 HISTORY RESCA FACTOR= 0.39999998E-02 OFFSET= 0.000000 HISTORY SUBIM RELEASE ='15OCT88 ' /********* START 10-OCT HISTORY 01 SUBIM INNAME='X13Y11 ' INCLASS='PADIM ' HISTORY SUBIM INTYPE ='MA' USERID= 1341 HISTORY SUBIM OUTNAME='M31 ' OUTCLASS='MOD ' HISTORY SUBIM OUTSEQ= 1 OUTDISK= 3 HISTORY SUBIM BLC = 1, 1, 1, 1, 1, 1, 1 HISTORY SUBIM TRC = 512, 512, 1, 1, 1, 1, 1 HISTORY SUBIM XINC = 1 YINC = 1 HISTORY SUBIM RELEASE ='15OCT88 ' /********* START 10-OCT HISTORY 47 SUBIM INNAME='M31 ' INCLASS='MOD ' HISTORY SUBIM INSEQ= 1 INDISK= 3 HISTORY SUBIM INTYPE ='MA' USERID= 1341 HISTORY SUBIM OUTNAME='M31 ' OUTCLASS='SUBIM ' HISTORY SUBIM OUTSEQ= 1 OUTDISK= 1 HISTORY SUBIM BLC = 1, 1, 1, 1, 1, 1, 1 HISTORY SUBIM XINC = 1 YINC = 1 HISTORY RENAM INNAME='M31 ' INCLASS='SUBIM ' HISTORY RENAM INSEQ= 1 INDISK= 1 HISTORY RENAM OUTNAME='M31 ' OUTCLASS='TEST ' HISTORY RENAM OUTSEQ= 1 OUTDISK= 1 HISTORY PUTHEAD OBSERVER =' ' / OLD HISTORY PUTHEAD OBSERVER ='BRAUN ' / NEW HISTORY PUTHEAD CDELT1 = -6.94444E-05 / OLD HISTORY PUTHEAD CDELT1 = -2.77778E-04 / NEW HISTORY PUTHEAD CDELT2 = 6.94444E-05 / OLD HISTORY PUTHEAD CDELT2 = 2.77778E-04 / NEW HISTORY SUBIM RELEASE ='15JAN89 ' /********* START 28-OCT HISTORY 06 SUBIM INNAME='M31 ' INCLASS='TEST ' HISTORY SUBIM INTYPE ='MA' USERID= 1341 HISTORY SUBIM OUTNAME='M31 ' OUTCLASS='SUB ' HISTORY SUBIM OUTSEQ= 1 OUTDISK= 1 HISTORY SUBIM BLC = 129, 129, 1, 1, 1, 1, 1 HISTORY SUBIM TRC = 384, 384, 1, 1, 1, 1, 1 HISTORY SUBIM XINC = 1 YINC = 1 HISTORY PUTHEAD CRVAL2 = 0.000000000E+00 / OLD HISTORY PUTHEAD CRVAL2 = 3.500000000E+01 / NEW HISTORY AIPS IMNAME='M31 ' IMCLASS='SUB ' IMSE HISTORY AIPS USERNO= 1341 / HISTORY AIPS CLEAN BMAJ= 1.0000E-03 BMIN= 1.0000E-03 B HISTORY AIPS CLEAN NITER= 1 PRODUCT=1 / NORMAL HISTORY /END FITS tape header "HISTORY" information HISTORY ---- IMLOD OUTNAME =' HISTORY IMLOD OUTSEQ = 0 INTAPE = 1 OUTDISK= 1 HISTORY IMLOD INFILE = 'T:M31MOD.M HISTORY ' IMLOD RELEASE = '15JAN90' HISTORY AIPS IMNAME='M31MOD ' IMCLASS='IMOD ' IMSEQ= 1 HISTORY AIPS USERNO= 400 / HISTORY AIPS CLEAN BMAJ= 1.0000E-03 BMIN= 1.0000E-03 BPA= 0. HISTORY AIPS CLEAN NITER= 1 PRODUCT=1 / NORMAL END 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sopt-2.0.0/python/000077500000000000000000000000001277570055300140325ustar00rootroot00000000000000sopt-2.0.0/python/CMakeLists.txt000066400000000000000000000001071277570055300165700ustar00rootroot00000000000000add_subdirectory(wavelets) if(tests) add_subdirectory(tests) endif() sopt-2.0.0/python/tests/000077500000000000000000000000001277570055300151745ustar00rootroot00000000000000sopt-2.0.0/python/tests/CMakeLists.txt000066400000000000000000000000261277570055300177320ustar00rootroot00000000000000add_pytest(test_*.py) sopt-2.0.0/python/tests/test_pywavelet.py000066400000000000000000000043611277570055300206310ustar00rootroot00000000000000"""test cython binding of wavelets""" def test_1D(): import sopt.wavelets as wv import numpy as np signal = np.random.rand(64) + 1e0 coefficient = wv.dwt(signal, "DB4", 1) inv_signal = wv.dwt(coefficient, "DB4", 1, inverse=True) np.testing.assert_allclose(signal, inv_signal, rtol=1e-8) def test_2D(): import sopt.wavelets as wv import numpy as np db_name = ["DB1", "DB2", "DB3", "DB4", "DB5", "DB6", "DB7"] for ncol in [1, 4, 32, 64]: signal = np.random.random((32, ncol)) for name in db_name: coefficient = wv.dwt(signal, name, 1) inv_signal = wv.dwt(coefficient, name, 1, inverse=True) np.testing.assert_allclose(signal, inv_signal) def test_complex(): import sopt.wavelets as wv import numpy as np s_real = np.random.random((64, 64)) + 1e0 s_img = np.random.random((64, 64)) + 1e0 signal = s_real + s_img*1j coefficient = wv.dwt(signal, "DB4", 1) inv_signal = wv.dwt(coefficient, "DB4", 1, inverse=True) np.testing.assert_allclose(signal, inv_signal) def test_1D_pywt(): """compare the result of 1D DB1 direct transform with pywt library""" import numpy as np import sopt.wavelets as wv import pywt input = np.random.random(128) + 1e0 coefficient_sopt = wv.dwt(input, "DB1", 1) cA_pywt, cD_pywt = pywt.dwt(input, "DB1") coefficient_pywt = np.concatenate( (cA_pywt, -1*cD_pywt)).reshape(coefficient_sopt.shape) np.testing.assert_allclose(coefficient_sopt, coefficient_pywt) def test_wrong_dims(): from pytest import raises import sopt.wavelets as wv import numpy as np signal = np.random.random((32, 32, 32)) with raises(ValueError): wv.dwt(signal, "DB1", 1) def test_wrong_type(): from pytest import raises import sopt.wavelets as wv import numpy as np signal = np.random.random((32, 32)) with raises(ValueError): wv.dwt(signal.astype("S1"), "DB1", 1) def test_noncontiguous(): import sopt.wavelets as wv import numpy as np signal = np.random.random((32, )) + 1e0 coeff_nc = wv.dwt(signal[::4], "DB1", 1) coeff_cont = wv.dwt(signal[::4].copy(), "DB1", 1) np.testing.assert_allclose(coeff_nc, coeff_cont) sopt-2.0.0/python/wavelets/000077500000000000000000000000001277570055300156645ustar00rootroot00000000000000sopt-2.0.0/python/wavelets/CMakeLists.txt000066400000000000000000000003221277570055300204210ustar00rootroot00000000000000include_directories(SYSTEM ${PYTHON_INCLUDE_DIR} ${NUMPY_INCLUDE_DIRS}) include_directories(${CMAKE_CURRENT_SOURCE_DIR}) add_python_module(sopt CPP TARGETNAME pywavelets LIBRARIES sopt FAKE_INIT wavelets.pyx) sopt-2.0.0/python/wavelets/python.wavelets.h000066400000000000000000000042171277570055300212130ustar00rootroot00000000000000#ifndef SOPT_TESTME #define SOPT_TESTME #include #include #include #include namespace sopt { namespace pyWavelets { template using Array = Eigen::Array; template using Map = Eigen::Map>; template using Vector = Eigen::Array; template using MapV = Eigen::Map>; //! \brief direct transform wrapper called by cython //! \param[in] py_input: double or complex pointer of input data generated by cython code. //! \param[in] name: Daubechie wavelets coefficient. //! \param[in] level: wavelet transform level. //! \param[in] nrow/ncol: number of rows and columns of input data. In the case of a vector //! input, ncol = 1. //! \param[out] py_out: result data in either double or complex pointer. template void direct(T *py_input, T *py_output, std::string const &name, t_uint level, t_uint nrow, t_uint ncol) { auto const wavelets = sopt::wavelets::factory(name, level); if(nrow == 1 or ncol == 1) { wavelets.direct(MapV(py_output, nrow * ncol), MapV(py_input, nrow * ncol)); } else wavelets.direct(Map(py_output, nrow, ncol), Map(py_input, nrow, ncol)); }; //! \brief indirect transform wrapper called by cython //! \param[in] py_input: double or complex pointer of input data generated by cython code. //! \param[in] name: Daubechie wavelets coefficient. //! \param[in] level: wavelet transform level. //! \param[in] nrow/ncol: number of rows and columns of input data. In the case of a vector //! input, ncol = 1. //! \param[out] py_out: result data in either double or complex pointer. template void indirect(T *py_input, T *py_output, std::string const &name, t_uint level, t_uint nrow, t_uint ncol) { auto const wavelets = sopt::wavelets::factory(name, level); if(nrow == 1 or ncol == 1) wavelets.indirect(MapV(py_input, nrow * ncol), MapV(py_output, nrow * ncol)); else wavelets.indirect(Map(py_input, nrow, ncol), Map(py_output, nrow, ncol)); }; } } #endif sopt-2.0.0/python/wavelets/wavelets.pyx000066400000000000000000000060411277570055300202610ustar00rootroot00000000000000import numpy as np from cython cimport view from libcpp.string cimport string cdef extern from "python.wavelets.h" namespace "sopt::pyWavelets": ctypedef unsigned t_uint void direct[T](T * signal, T* out, const string & name, t_uint level, t_uint nrow, t_uint ncol) except + void indirect[T](T * signal, T* out, const string & name, t_uint level, t_uint nrow, t_uint ncol) except + cdef _dwt(input, name, level, inverse=False): if input.ndim > 2: raise ValueError("Expect 1D or 2D arrays") nrow, ncol = input.shape if input.ndim == 2 else (input.size, 1) output = np.zeros(input.shape, dtype=input.dtype) cdef: long input_data = input.ctypes.data long output_data = output.ctypes.data string cname = name.encode("UTF-8") if inverse: if input.dtype == "float64": indirect[double]( input_data, output_data, cname, level, nrow, ncol ) elif input.dtype == "complex128": indirect[complex]( input_data, output_data, cname, level, nrow, ncol ) else: if input.dtype == "float64": direct[double]( input_data, output_data, cname, level, nrow, ncol ) elif input.dtype == "complex128": direct[complex]( input_data, output_data, cname, level, nrow, ncol ) return output def dwt(input, name, level, inverse=False): """ direct/inverse Daubechies wavelet transform Parameters: ------------ inputs: numpy array 1D/2D numerical input signal name: string Daubechies wavelets coefficients, e.g. "DB1" through "DB38" level: int Wavelets transform level inverse: bool True - indirect transform False - direct transform Returns ------------ numpy array Approximation matrix in the same size as input Notes ------------ * Input will be converted to "float64" or "complex128" automatically * To avoid extra copies, please use those types exclusively * Size of signal must be a multiple of 2^levels Examples ----------- signal = np.random.random((64,64)) coefficient = wv.dwt(signal, "DB4", 2) # direct transform recover = wv.dwt(coefficient, "DB4", 2, inverse = True) # inverse transform """ from numpy import iscomplex, isreal, require, all is_complex = all(iscomplex(input)) if (not is_complex) and not all(isreal(input)): raise ValueError("Incorrect array type") dtype = "complex128" if is_complex else "float64" normalized_input = require(input, requirements=['C'], dtype=dtype) return _dwt(normalized_input, name, level, inverse=inverse).astype(input.dtype)