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-rw-r--r--src/Persistent_cohomology/example/CMakeLists.txt68
-rw-r--r--src/Persistent_cohomology/example/README67
-rw-r--r--src/Persistent_cohomology/example/custom_persistence_sort.cpp125
-rw-r--r--src/Persistent_cohomology/example/persistence_from_file.cpp130
-rw-r--r--src/Persistent_cohomology/example/persistence_from_simple_simplex_tree.cpp163
-rw-r--r--src/Persistent_cohomology/example/plain_homology.cpp91
-rw-r--r--src/Persistent_cohomology/example/rips_multifield_persistence.cpp142
-rw-r--r--src/Persistent_cohomology/example/rips_persistence_step_by_step.cpp154
-rw-r--r--src/Persistent_cohomology/example/rips_persistence_via_boundary_matrix.cpp160
9 files changed, 1100 insertions, 0 deletions
diff --git a/src/Persistent_cohomology/example/CMakeLists.txt b/src/Persistent_cohomology/example/CMakeLists.txt
new file mode 100644
index 00000000..94ec13c5
--- /dev/null
+++ b/src/Persistent_cohomology/example/CMakeLists.txt
@@ -0,0 +1,68 @@
+project(Persistent_cohomology_examples)
+
+add_executable(plain_homology plain_homology.cpp)
+
+add_executable(persistence_from_simple_simplex_tree persistence_from_simple_simplex_tree.cpp)
+
+add_executable(rips_persistence_step_by_step rips_persistence_step_by_step.cpp)
+target_link_libraries(rips_persistence_step_by_step ${Boost_PROGRAM_OPTIONS_LIBRARY})
+
+add_executable(rips_persistence_via_boundary_matrix rips_persistence_via_boundary_matrix.cpp)
+target_link_libraries(rips_persistence_via_boundary_matrix ${Boost_PROGRAM_OPTIONS_LIBRARY})
+
+add_executable(persistence_from_file persistence_from_file.cpp)
+target_link_libraries(persistence_from_file ${Boost_PROGRAM_OPTIONS_LIBRARY})
+
+if (TBB_FOUND)
+ target_link_libraries(plain_homology ${TBB_LIBRARIES})
+ target_link_libraries(persistence_from_simple_simplex_tree ${TBB_LIBRARIES})
+ target_link_libraries(rips_persistence_step_by_step ${TBB_LIBRARIES})
+ target_link_libraries(rips_persistence_via_boundary_matrix ${TBB_LIBRARIES})
+ target_link_libraries(persistence_from_file ${TBB_LIBRARIES})
+endif()
+
+add_test(NAME Persistent_cohomology_example_plain_homology COMMAND $<TARGET_FILE:plain_homology>)
+add_test(NAME Persistent_cohomology_example_from_simple_simplex_tree COMMAND $<TARGET_FILE:persistence_from_simple_simplex_tree>
+ "1" "0")
+add_test(NAME Persistent_cohomology_example_from_rips_step_by_step_on_tore_3D COMMAND $<TARGET_FILE:rips_persistence_step_by_step>
+ "${CMAKE_SOURCE_DIR}/data/points/tore3D_1307.off" "-r" "0.25" "-m" "0.5" "-d" "3" "-p" "3")
+add_test(NAME Persistent_cohomology_example_via_boundary_matrix COMMAND $<TARGET_FILE:rips_persistence_via_boundary_matrix>
+ "${CMAKE_SOURCE_DIR}/data/points/Kl.off" "-r" "0.16" "-d" "3" "-p" "3" "-m" "100")
+add_test(NAME Persistent_cohomology_example_from_file_3_2_0 COMMAND $<TARGET_FILE:persistence_from_file>
+ "${CMAKE_SOURCE_DIR}/data/filtered_simplicial_complex/bunny_5000_complex.fsc" "-p" "2" "-m" "0")
+add_test(NAME Persistent_cohomology_example_from_file_3_3_100 COMMAND $<TARGET_FILE:persistence_from_file>
+ "${CMAKE_SOURCE_DIR}/data/filtered_simplicial_complex/bunny_5000_complex.fsc" "-p" "3" "-m" "100")
+
+install(TARGETS plain_homology DESTINATION bin)
+install(TARGETS persistence_from_simple_simplex_tree DESTINATION bin)
+install(TARGETS rips_persistence_step_by_step DESTINATION bin)
+install(TARGETS rips_persistence_via_boundary_matrix DESTINATION bin)
+install(TARGETS persistence_from_file DESTINATION bin)
+
+if(GMP_FOUND)
+ if(GMPXX_FOUND)
+ add_executable(rips_multifield_persistence rips_multifield_persistence.cpp )
+ target_link_libraries(rips_multifield_persistence
+ ${Boost_PROGRAM_OPTIONS_LIBRARY} ${GMPXX_LIBRARIES} ${GMP_LIBRARIES})
+ if (TBB_FOUND)
+ target_link_libraries(rips_multifield_persistence ${TBB_LIBRARIES})
+ endif(TBB_FOUND)
+ add_test(NAME Persistent_cohomology_example_multifield_2_71 COMMAND $<TARGET_FILE:rips_multifield_persistence>
+ "${CMAKE_SOURCE_DIR}/data/points/tore3D_1307.off" "-r" "0.25" "-m" "0.5" "-d" "3" "-p" "2" "-q" "71")
+ install(TARGETS rips_multifield_persistence DESTINATION bin)
+ endif(GMPXX_FOUND)
+endif(GMP_FOUND)
+
+if (NOT CGAL_WITH_EIGEN3_VERSION VERSION_LESS 4.11.0)
+
+ add_executable(custom_persistence_sort custom_persistence_sort.cpp)
+ target_link_libraries(custom_persistence_sort ${CGAL_LIBRARY})
+
+ if (TBB_FOUND)
+ target_link_libraries(custom_persistence_sort ${TBB_LIBRARIES})
+ endif(TBB_FOUND)
+ add_test(NAME Persistent_cohomology_example_custom_persistence_sort COMMAND $<TARGET_FILE:custom_persistence_sort>)
+
+ install(TARGETS custom_persistence_sort DESTINATION bin)
+
+endif (NOT CGAL_WITH_EIGEN3_VERSION VERSION_LESS 4.11.0)
diff --git a/src/Persistent_cohomology/example/README b/src/Persistent_cohomology/example/README
new file mode 100644
index 00000000..f39d9584
--- /dev/null
+++ b/src/Persistent_cohomology/example/README
@@ -0,0 +1,67 @@
+To build the examples, run in a Terminal:
+
+cd /path-to-examples/
+cmake .
+make
+
+***********************************************************************************************************************
+Example of use of RIPS:
+
+Computation of the persistent homology with Z/2Z and Z/3Z coefficients simultaneously of the Rips complex
+on points sampling a 3D torus:
+
+./rips_multifield_persistence ../../data/points/tore3D_1307.off -r 0.25 -m 0.12 -d 3 -p 2 -q 3
+
+output:
+6 0 0 inf
+6 1 0.0983494 inf
+6 1 0.104347 inf
+6 2 0.138335 inf
+6 0 0 0.122545
+6 0 0 0.121171
+6 0 0 0.120964
+6 0 0 0.12057
+6 0 0 0.12047
+6 0 0 0.120414
+
+Every line is of this format: p1*...*pr dim b d
+where
+ p1*...*pr is the product of prime numbers pi such that the homology feature exists in homology with Z/piZ coefficients.
+ dim is the dimension of the homological feature,
+ b and d are respectively the birth and death of the feature and
+
+and the computation with all Z/pZ for 2 <= p <= 71 (20 first prime numbers):
+
+ ./rips_multifield_persistence ../../data/points/Kl.off -r 0.25 -m 0.5 -d 3 -p 2 -q 71
+
+output:
+557940830126698960967415390 0 0 inf
+557940830126698960967415390 1 0.0983494 inf
+557940830126698960967415390 1 0.104347 inf
+557940830126698960967415390 2 0.138335 inf
+557940830126698960967415390 0 0 0.122545
+557940830126698960967415390 0 0 0.121171
+557940830126698960967415390 0 0 0.120964
+557940830126698960967415390 0 0 0.12057
+557940830126698960967415390 0 0 0.12047
+557940830126698960967415390 0 0 0.120414
+
+***********************************************************************************************************************
+Example of use of PLAIN HOMOLOGY:
+
+This example computes the plain homology of the following simplicial complex without filtration values:
+ /* Complex to build. */
+ /* 1 3 */
+ /* o---o */
+ /* /X\ / */
+ /* o---o o */
+ /* 2 0 4 */
+
+./plain_homology
+
+output:
+2 0 0 inf
+2 0 0 inf
+2 1 0 inf
+
+Here we retrieve the 2 entities {0,1,2,3} and {4} (Betti numbers[0] = 2) and the hole in {0,1,3} (Betti numbers[1] = 1)
diff --git a/src/Persistent_cohomology/example/custom_persistence_sort.cpp b/src/Persistent_cohomology/example/custom_persistence_sort.cpp
new file mode 100644
index 00000000..be74cf50
--- /dev/null
+++ b/src/Persistent_cohomology/example/custom_persistence_sort.cpp
@@ -0,0 +1,125 @@
+/* This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+ * See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+ * Author(s): Vincent Rouvreau
+ *
+ * Copyright (C) 2014 Inria
+ *
+ * Modification(s):
+ * - YYYY/MM Author: Description of the modification
+ */
+
+#include <CGAL/Epick_d.h>
+#include <CGAL/point_generators_d.h>
+#include <CGAL/algorithm.h>
+#include <CGAL/assertions.h>
+
+#include <gudhi/Alpha_complex.h>
+#include <gudhi/Persistent_cohomology.h>
+// to construct a simplex_tree from alpha complex
+#include <gudhi/Simplex_tree.h>
+
+#include <iostream>
+#include <iterator>
+#include <vector>
+#include <fstream> // for std::ofstream
+#include <algorithm> // for std::sort
+
+
+using Kernel = CGAL::Epick_d< CGAL::Dimension_tag<3> >;
+using Point = Kernel::Point_d;
+using Alpha_complex = Gudhi::alpha_complex::Alpha_complex<Kernel>;
+using Simplex_tree = Gudhi::Simplex_tree<>;
+using Persistent_cohomology = Gudhi::persistent_cohomology::Persistent_cohomology< Simplex_tree,
+ Gudhi::persistent_cohomology::Field_Zp >;
+
+std::vector<Point> random_points() {
+ // Instanciate a random point generator
+ CGAL::Random rng(0);
+
+ // Generate "points_number" random points in a vector
+ std::vector<Point> points;
+
+ // Generates 1000 random 3D points on a sphere of radius 4.0
+ CGAL::Random_points_on_sphere_d<Point> rand_outside(3, 4.0, rng);
+ CGAL::cpp11::copy_n(rand_outside, 1000, std::back_inserter(points));
+ // Generates 2000 random 3D points in a sphere of radius 3.0
+ CGAL::Random_points_in_ball_d<Point> rand_inside(3, 3.0, rng);
+ CGAL::cpp11::copy_n(rand_inside, 2000, std::back_inserter(points));
+
+ return points;
+}
+
+/*
+ * Compare two intervals by dimension, then by length.
+ */
+struct cmp_intervals_by_dim_then_length {
+ explicit cmp_intervals_by_dim_then_length(Simplex_tree * sc)
+ : sc_(sc) { }
+
+ template<typename Persistent_interval>
+ bool operator()(const Persistent_interval & p1, const Persistent_interval & p2) {
+ if (sc_->dimension(get < 0 > (p1)) == sc_->dimension(get < 0 > (p2)))
+ return (sc_->filtration(get < 1 > (p1)) - sc_->filtration(get < 0 > (p1))
+ > sc_->filtration(get < 1 > (p2)) - sc_->filtration(get < 0 > (p2)));
+ else
+ return (sc_->dimension(get < 0 > (p1)) > sc_->dimension(get < 0 > (p2)));
+ }
+ Simplex_tree* sc_;
+};
+
+int main(int argc, char **argv) {
+ std::vector<Point> points = random_points();
+
+ std::cout << "Points size=" << points.size() << std::endl;
+ // Alpha complex persistence computation from generated points
+ Alpha_complex alpha_complex_from_points(points);
+ std::cout << "alpha_complex_from_points" << std::endl;
+
+ Simplex_tree simplex;
+ std::cout << "simplex" << std::endl;
+ if (alpha_complex_from_points.create_complex(simplex, 0.6)) {
+ std::cout << "simplex" << std::endl;
+ // ----------------------------------------------------------------------------
+ // Display information about the alpha complex
+ // ----------------------------------------------------------------------------
+ std::cout << "Simplicial complex is of dimension " << simplex.dimension() <<
+ " - " << simplex.num_simplices() << " simplices - " <<
+ simplex.num_vertices() << " vertices." << std::endl;
+
+ // Sort the simplices in the order of the filtration
+ simplex.initialize_filtration();
+
+ std::cout << "Simplex_tree dim: " << simplex.dimension() << std::endl;
+
+ Persistent_cohomology pcoh(simplex);
+
+ // initializes the coefficient field for homology - Z/3Z
+ pcoh.init_coefficients(3);
+ pcoh.compute_persistent_cohomology(0.2);
+
+ // Custom sort and output persistence
+ cmp_intervals_by_dim_then_length cmp(&simplex);
+ auto persistent_pairs = pcoh.get_persistent_pairs();
+ std::sort(std::begin(persistent_pairs), std::end(persistent_pairs), cmp);
+ for (auto pair : persistent_pairs) {
+ std::cout << simplex.dimension(get<0>(pair)) << " "
+ << simplex.filtration(get<0>(pair)) << " "
+ << simplex.filtration(get<1>(pair)) << std::endl;
+ }
+
+ // Persistent Betti numbers
+ std::cout << "The persistent Betti numbers in interval [0.40, 0.41] are : ";
+ for (int dim = 0; dim < simplex.dimension(); dim++)
+ std::cout << "b" << dim << " = " << pcoh.persistent_betti_number(dim, 0.40, 0.41) << " ; ";
+ std::cout << std::endl;
+
+ // Betti numbers
+ std::vector<int> betti_numbers = pcoh.betti_numbers();
+ std::cout << "The Betti numbers are : ";
+ for (std::size_t i = 0; i < betti_numbers.size(); i++)
+ std::cout << "b" << i << " = " << betti_numbers[i] << " ; ";
+ std::cout << std::endl;
+ }
+ return 0;
+}
+
diff --git a/src/Persistent_cohomology/example/persistence_from_file.cpp b/src/Persistent_cohomology/example/persistence_from_file.cpp
new file mode 100644
index 00000000..d169cc63
--- /dev/null
+++ b/src/Persistent_cohomology/example/persistence_from_file.cpp
@@ -0,0 +1,130 @@
+/* This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+ * See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+ * Author(s): Vincent Rouvreau
+ *
+ * Copyright (C) 2014 Inria
+ *
+ * Modification(s):
+ * - YYYY/MM Author: Description of the modification
+ */
+
+#include <gudhi/reader_utils.h>
+#include <gudhi/graph_simplicial_complex.h>
+#include <gudhi/distance_functions.h>
+#include <gudhi/Simplex_tree.h>
+#include <gudhi/Persistent_cohomology.h>
+
+#include <boost/program_options.hpp>
+
+#include <string>
+
+using namespace Gudhi;
+using namespace Gudhi::persistent_cohomology;
+
+typedef double Filtration_value;
+
+void program_options(int argc, char * argv[]
+ , std::string & simplex_tree_file
+ , std::string & output_file
+ , int & p
+ , Filtration_value & min_persistence);
+
+int main(int argc, char * argv[]) {
+ std::string simplex_tree_file;
+ std::string output_file;
+ int p;
+ Filtration_value min_persistence;
+
+ program_options(argc, argv, simplex_tree_file, output_file, p, min_persistence);
+
+ std::cout << "Simplex_tree from file=" << simplex_tree_file.c_str() << " - output_file=" << output_file.c_str()
+ << std::endl;
+ std::cout << " - p=" << p << " - min_persistence=" << min_persistence << std::endl;
+
+ // Read the list of simplices from a file.
+ Simplex_tree<> simplex_tree;
+
+ std::ifstream simplex_tree_stream(simplex_tree_file);
+ simplex_tree_stream >> simplex_tree;
+
+ std::cout << "The complex contains " << simplex_tree.num_simplices() << " simplices" << std::endl;
+ std::cout << " - dimension " << simplex_tree.dimension() << std::endl;
+
+ /*
+ std::cout << std::endl << std::endl << "Iterator on Simplices in the filtration, with [filtration value]:" << std::endl;
+ for( auto f_simplex : simplex_tree.filtration_simplex_range() )
+ { std::cout << " " << "[" << simplex_tree.filtration(f_simplex) << "] ";
+ for( auto vertex : simplex_tree.simplex_vertex_range(f_simplex) )
+ { std::cout << vertex << " "; }
+ std::cout << std::endl;
+ }*/
+
+ // Sort the simplices in the order of the filtration
+ simplex_tree.initialize_filtration();
+
+ // Compute the persistence diagram of the complex
+ Persistent_cohomology< Simplex_tree<>, Field_Zp > pcoh(simplex_tree);
+ // initializes the coefficient field for homology
+ pcoh.init_coefficients(p);
+
+ pcoh.compute_persistent_cohomology(min_persistence);
+
+ // Output the diagram in output_file
+ if (output_file.empty()) {
+ pcoh.output_diagram();
+ } else {
+ std::ofstream out(output_file);
+ pcoh.output_diagram(out);
+ out.close();
+ }
+
+ return 0;
+}
+
+void program_options(int argc, char * argv[]
+ , std::string & simplex_tree_file
+ , std::string & output_file
+ , int & p
+ , Filtration_value & min_persistence) {
+ namespace po = boost::program_options;
+ po::options_description hidden("Hidden options");
+ hidden.add_options()
+ ("input-file", po::value<std::string>(&simplex_tree_file),
+ "Name of file containing a simplex set. Format is one simplex per line (cf. reader_utils.h - read_simplex): Dim1 X11 X12 ... X1d Fil1 ");
+
+ po::options_description visible("Allowed options", 100);
+ visible.add_options()
+ ("help,h", "produce help message")
+ ("output-file,o", po::value<std::string>(&output_file)->default_value(std::string()),
+ "Name of file in which the persistence diagram is written. Default print in std::cout")
+ ("field-charac,p", po::value<int>(&p)->default_value(11),
+ "Characteristic p of the coefficient field Z/pZ for computing homology.")
+ ("min-persistence,m", po::value<Filtration_value>(&min_persistence),
+ "Minimal lifetime of homology feature to be recorded. Default is 0");
+
+ po::positional_options_description pos;
+ pos.add("input-file", 1);
+
+ po::options_description all;
+ all.add(visible).add(hidden);
+
+ po::variables_map vm;
+ po::store(po::command_line_parser(argc, argv).
+ options(all).positional(pos).run(), vm);
+ po::notify(vm);
+
+ if (vm.count("help") || !vm.count("input-file")) {
+ std::cout << std::endl;
+ std::cout << "Compute the persistent homology with coefficient field Z/pZ \n";
+ std::cout << "of a Rips complex defined on a set of input points.\n \n";
+ std::cout << "The output diagram contains one bar per line, written with the convention: \n";
+ std::cout << " p dim b d \n";
+ std::cout << "where dim is the dimension of the homological feature,\n";
+ std::cout << "b and d are respectively the birth and death of the feature and \n";
+ std::cout << "p is the characteristic of the field Z/pZ used for homology coefficients." << std::endl << std::endl;
+
+ std::cout << "Usage: " << argv[0] << " [options] input-file" << std::endl << std::endl;
+ std::cout << visible << std::endl;
+ exit(-1);
+ }
+}
diff --git a/src/Persistent_cohomology/example/persistence_from_simple_simplex_tree.cpp b/src/Persistent_cohomology/example/persistence_from_simple_simplex_tree.cpp
new file mode 100644
index 00000000..3c91662f
--- /dev/null
+++ b/src/Persistent_cohomology/example/persistence_from_simple_simplex_tree.cpp
@@ -0,0 +1,163 @@
+/* This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+ * See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+ * Author(s): Vincent Rouvreau
+ *
+ * Copyright (C) 2014 Inria
+ *
+ * Modification(s):
+ * - YYYY/MM Author: Description of the modification
+ */
+
+#include <gudhi/graph_simplicial_complex.h>
+#include <gudhi/Simplex_tree.h>
+#include <gudhi/Persistent_cohomology.h>
+
+#include <iostream>
+#include <ctime>
+#include <utility>
+#include <vector>
+
+// Types definition
+using Simplex_tree = Gudhi::Simplex_tree<>;
+using Filtration_value = Simplex_tree::Filtration_value;
+using Field_Zp = Gudhi::persistent_cohomology::Field_Zp;
+using Persistent_cohomology = Gudhi::persistent_cohomology::Persistent_cohomology<Simplex_tree, Field_Zp >;
+using typeVectorVertex = std::vector< Simplex_tree::Vertex_handle >;
+
+void usage(char * const progName) {
+ std::cerr << "Usage: " << progName << " coeff_field_characteristic[integer > 0] min_persistence[float >= -1.0]\n";
+ exit(-1);
+}
+
+int main(int argc, char * const argv[]) {
+ // program args management
+ if (argc != 3) {
+ std::cerr << "Error: Number of arguments (" << argc << ") is not correct\n";
+ usage(argv[0]);
+ }
+
+ int coeff_field_characteristic = 0;
+ int returnedScanValue = sscanf(argv[1], "%d", &coeff_field_characteristic);
+ if ((returnedScanValue == EOF) || (coeff_field_characteristic <= 0)) {
+ std::cerr << "Error: " << argv[1] << " is not correct\n";
+ usage(argv[0]);
+ }
+
+ Filtration_value min_persistence = 0.0;
+ returnedScanValue = sscanf(argv[2], "%lf", &min_persistence);
+ if ((returnedScanValue == EOF) || (min_persistence < -1.0)) {
+ std::cerr << "Error: " << argv[2] << " is not correct\n";
+ usage(argv[0]);
+ }
+
+ // TEST OF INSERTION
+ std::cout << "********************************************************************" << std::endl;
+ std::cout << "TEST OF INSERTION" << std::endl;
+ Simplex_tree st;
+
+ // ++ FIRST
+ std::cout << " - INSERT (0,1,2)" << std::endl;
+ typeVectorVertex SimplexVector = {0, 1, 2};
+ st.insert_simplex_and_subfaces(SimplexVector, 0.3);
+
+ // ++ SECOND
+ std::cout << " - INSERT 3" << std::endl;
+ SimplexVector = {3};
+ st.insert_simplex_and_subfaces(SimplexVector, 0.1);
+
+ // ++ THIRD
+ std::cout << " - INSERT (0,3)" << std::endl;
+ SimplexVector = {0, 3};
+ st.insert_simplex_and_subfaces(SimplexVector, 0.2);
+
+ // ++ FOURTH
+ std::cout << " - INSERT (0,1) (already inserted)" << std::endl;
+ SimplexVector = {0, 1};
+ st.insert_simplex_and_subfaces(SimplexVector, 0.2);
+
+ // ++ FIFTH
+ std::cout << " - INSERT (3,4,5)" << std::endl;
+ SimplexVector = {3, 4, 5};
+ st.insert_simplex_and_subfaces(SimplexVector, 0.3);
+
+ // ++ SIXTH
+ std::cout << " - INSERT (0,1,6,7)" << std::endl;
+ SimplexVector = {0, 1, 6, 7};
+ st.insert_simplex_and_subfaces(SimplexVector, 0.4);
+
+ // ++ SEVENTH
+ std::cout << " - INSERT (4,5,8,9)" << std::endl;
+ SimplexVector = {4, 5, 8, 9};
+ st.insert_simplex_and_subfaces(SimplexVector, 0.4);
+
+ // ++ EIGHTH
+ std::cout << " - INSERT (9,10,11)" << std::endl;
+ SimplexVector = {9, 10, 11};
+ st.insert_simplex_and_subfaces(SimplexVector, 0.3);
+
+ // ++ NINETH
+ std::cout << " - INSERT (2,10,12)" << std::endl;
+ SimplexVector = {2, 10, 12};
+ st.insert_simplex_and_subfaces(SimplexVector, 0.3);
+
+ // ++ TENTH
+ std::cout << " - INSERT (11,6)" << std::endl;
+ SimplexVector = {6, 11};
+ st.insert_simplex_and_subfaces(SimplexVector, 0.2);
+
+ // ++ ELEVENTH
+ std::cout << " - INSERT (13,14,15)" << std::endl;
+ SimplexVector = {13, 14, 15};
+ st.insert_simplex_and_subfaces(SimplexVector, 0.25);
+
+ /* Inserted simplex: */
+ /* 1 6 */
+ /* o---o */
+ /* /X\7/ 4 2 */
+ /* o---o---o---o o */
+ /* 2 0 3\X/8\ 10 /X\ */
+ /* o---o---o---o */
+ /* 5 9\X/ 12 */
+ /* o---o */
+ /* 11 6 */
+ /* In other words: */
+ /* A facet [2,1,0] */
+ /* An edge [0,3] */
+ /* A facet [3,4,5] */
+ /* A cell [0,1,6,7] */
+ /* A cell [4,5,8,9] */
+ /* A facet [9,10,11] */
+ /* An edge [11,6] */
+ /* An edge [10,12,2] */
+
+
+ std::cout << "The complex contains " << st.num_simplices() << " simplices - " << st.num_vertices() << " vertices "
+ << std::endl;
+ std::cout << " - dimension " << st.dimension() << std::endl;
+ std::cout << std::endl << std::endl << "Iterator on Simplices in the filtration, with [filtration value]:"
+ << std::endl;
+ std::cout << "**************************************************************" << std::endl;
+ std::cout << "strict graph G { " << std::endl;
+
+ for (auto f_simplex : st.filtration_simplex_range()) {
+ std::cout << " " << "[" << st.filtration(f_simplex) << "] ";
+ for (auto vertex : st.simplex_vertex_range(f_simplex)) {
+ std::cout << static_cast<int>(vertex) << " -- ";
+ }
+ std::cout << ";" << std::endl;
+ }
+
+ std::cout << "}" << std::endl;
+ std::cout << "**************************************************************" << std::endl;
+
+ // Compute the persistence diagram of the complex
+ Persistent_cohomology pcoh(st);
+ // initializes the coefficient field for homology
+ pcoh.init_coefficients(coeff_field_characteristic);
+
+ pcoh.compute_persistent_cohomology(min_persistence);
+
+ // Output the diagram in filediag
+ pcoh.output_diagram();
+ return 0;
+}
diff --git a/src/Persistent_cohomology/example/plain_homology.cpp b/src/Persistent_cohomology/example/plain_homology.cpp
new file mode 100644
index 00000000..84333e46
--- /dev/null
+++ b/src/Persistent_cohomology/example/plain_homology.cpp
@@ -0,0 +1,91 @@
+/* This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+ * See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+ * Author(s): Marc Glisse
+ *
+ * Copyright (C) 2015 Inria
+ *
+ * Modification(s):
+ * - YYYY/MM Author: Description of the modification
+ */
+
+#include <gudhi/Simplex_tree.h>
+#include <gudhi/Persistent_cohomology.h>
+
+#include <iostream>
+#include <vector>
+#include <cstdint> // for std::uint8_t
+
+/* We could perfectly well use the default Simplex_tree<> (which uses
+ * Simplex_tree_options_full_featured), the following simply demonstrates
+ * how to save on storage by not storing a filtration value. */
+
+struct MyOptions : Gudhi::Simplex_tree_options_full_featured {
+ // Implicitly use 0 as filtration value for all simplices
+ static const bool store_filtration = false;
+ // The persistence algorithm needs this
+ static const bool store_key = true;
+ // I have few vertices
+ typedef short Vertex_handle;
+ // Maximum number of simplices to compute persistence is 2^8 - 1 = 255. One is reserved for null_key
+ typedef std::uint8_t Simplex_key;
+};
+
+using ST = Gudhi::Simplex_tree<MyOptions>;
+using Field_Zp = Gudhi::persistent_cohomology::Field_Zp;
+using Persistent_cohomology = Gudhi::persistent_cohomology::Persistent_cohomology<ST, Field_Zp>;
+
+int main() {
+ ST st;
+
+ /* Complex to build. */
+ /* 1 3 5 */
+ /* o---o---o */
+ /* / \ / */
+ /* o---o o */
+ /* 2 0 4 */
+
+ const short edge01[] = {0, 1};
+ const short edge02[] = {0, 2};
+ const short edge12[] = {1, 2};
+ const short edge03[] = {0, 3};
+ const short edge13[] = {1, 3};
+ const short edge35[] = {3, 5};
+ const short vertex4[] = {4};
+ st.insert_simplex_and_subfaces(edge01);
+ st.insert_simplex_and_subfaces(edge02);
+ st.insert_simplex_and_subfaces(edge12);
+ st.insert_simplex_and_subfaces(edge03);
+ st.insert_simplex(edge13);
+ st.insert_simplex_and_subfaces(edge35);
+ st.insert_simplex(vertex4);
+
+ // Sort the simplices in the order of the filtration
+ st.initialize_filtration();
+
+ // Class for homology computation
+ // By default, since the complex has dimension 1, only 0-dimensional homology would be computed.
+ // Here we also want persistent homology to be computed for the maximal dimension in the complex (persistence_dim_max = true)
+ Persistent_cohomology pcoh(st, true);
+
+ // Initialize the coefficient field Z/2Z for homology
+ pcoh.init_coefficients(2);
+
+ // Compute the persistence diagram of the complex
+ pcoh.compute_persistent_cohomology();
+
+ // Print the result. The format is, on each line: 2 dim 0 inf
+ // where 2 represents the field, dim the dimension of the feature.
+ // 2 0 0 inf
+ // 2 0 0 inf
+ // 2 1 0 inf
+ // 2 1 0 inf
+ // means that in Z/2Z-homology, the Betti numbers are b0=2 and b1=2.
+ pcoh.output_diagram();
+
+ // Print the Betti numbers are b0=2 and b1=2.
+ std::cout << std::endl;
+ std::cout << "The Betti numbers are : ";
+ for (int i = 0; i < 3; i++)
+ std::cout << "b" << i << " = " << pcoh.betti_number(i) << " ; ";
+ std::cout << std::endl;
+}
diff --git a/src/Persistent_cohomology/example/rips_multifield_persistence.cpp b/src/Persistent_cohomology/example/rips_multifield_persistence.cpp
new file mode 100644
index 00000000..9eb5ccfc
--- /dev/null
+++ b/src/Persistent_cohomology/example/rips_multifield_persistence.cpp
@@ -0,0 +1,142 @@
+/* This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+ * See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+ * Author(s): Clément Maria
+ *
+ * Copyright (C) 2014 Inria
+ *
+ * Modification(s):
+ * - YYYY/MM Author: Description of the modification
+ */
+
+#include <gudhi/Rips_complex.h>
+#include <gudhi/distance_functions.h>
+#include <gudhi/Simplex_tree.h>
+#include <gudhi/Persistent_cohomology.h>
+#include <gudhi/Persistent_cohomology/Multi_field.h>
+#include <gudhi/Points_off_io.h>
+
+#include <boost/program_options.hpp>
+
+#include <string>
+#include <vector>
+
+// Types definition
+using Simplex_tree = Gudhi::Simplex_tree<Gudhi::Simplex_tree_options_fast_persistence>;
+using Filtration_value = Simplex_tree::Filtration_value;
+using Rips_complex = Gudhi::rips_complex::Rips_complex<Filtration_value>;
+using Multi_field = Gudhi::persistent_cohomology::Multi_field;
+using Persistent_cohomology = Gudhi::persistent_cohomology::Persistent_cohomology<Simplex_tree, Multi_field >;
+using Point = std::vector<double>;
+using Points_off_reader = Gudhi::Points_off_reader<Point>;
+
+void program_options(int argc, char * argv[]
+ , std::string & off_file_points
+ , std::string & filediag
+ , Filtration_value & threshold
+ , int & dim_max
+ , int & min_p
+ , int & max_p
+ , Filtration_value & min_persistence);
+
+int main(int argc, char * argv[]) {
+ std::string off_file_points;
+ std::string filediag;
+ Filtration_value threshold;
+ int dim_max;
+ int min_p;
+ int max_p;
+ Filtration_value min_persistence;
+
+ program_options(argc, argv, off_file_points, filediag, threshold, dim_max, min_p, max_p, min_persistence);
+
+ Points_off_reader off_reader(off_file_points);
+ Rips_complex rips_complex_from_file(off_reader.get_point_cloud(), threshold, Gudhi::Euclidean_distance());
+
+ // Construct the Rips complex in a Simplex Tree
+ Simplex_tree simplex_tree;
+
+ rips_complex_from_file.create_complex(simplex_tree, dim_max);
+ std::cout << "The complex contains " << simplex_tree.num_simplices() << " simplices \n";
+ std::cout << " and has dimension " << simplex_tree.dimension() << " \n";
+
+ // Sort the simplices in the order of the filtration
+ simplex_tree.initialize_filtration();
+
+ // Compute the persistence diagram of the complex
+ Persistent_cohomology pcoh(simplex_tree);
+ // initializes the coefficient field for homology
+ pcoh.init_coefficients(min_p, max_p);
+
+ pcoh.compute_persistent_cohomology(min_persistence);
+
+ // Output the diagram in filediag
+ if (filediag.empty()) {
+ pcoh.output_diagram();
+ } else {
+ std::ofstream out(filediag);
+ pcoh.output_diagram(out);
+ out.close();
+ }
+
+ return 0;
+}
+
+void program_options(int argc, char * argv[]
+ , std::string & off_file_points
+ , std::string & filediag
+ , Filtration_value & threshold
+ , int & dim_max
+ , int & min_p
+ , int & max_p
+ , Filtration_value & min_persistence) {
+ namespace po = boost::program_options;
+ po::options_description hidden("Hidden options");
+ hidden.add_options()
+ ("input-file", po::value<std::string>(&off_file_points),
+ "Name of an OFF file containing a point set.\n");
+
+ po::options_description visible("Allowed options");
+ visible.add_options()
+ ("help,h", "produce help message")
+ ("output-file,o", po::value<std::string>(&filediag)->default_value(std::string()),
+ "Name of file in which the persistence diagram is written. Default print in std::cout")
+ ("max-edge-length,r", po::value<Filtration_value>(&threshold)->default_value(0),
+ "Maximal length of an edge for the Rips complex construction.")
+ ("cpx-dimension,d", po::value<int>(&dim_max)->default_value(1),
+ "Maximal dimension of the Rips complex we want to compute.")
+ ("min-field-charac,p", po::value<int>(&min_p)->default_value(2),
+ "Minimal characteristic p of the coefficient field Z/pZ.")
+ ("max-field-charac,q", po::value<int>(&max_p)->default_value(1223),
+ "Minimial characteristic q of the coefficient field Z/pZ.")
+ ("min-persistence,m", po::value<Filtration_value>(&min_persistence),
+ "Minimal lifetime of homology feature to be recorded. Default is 0");
+
+ po::positional_options_description pos;
+ pos.add("input-file", 1);
+
+ po::options_description all;
+ all.add(visible).add(hidden);
+
+ po::variables_map vm;
+ po::store(po::command_line_parser(argc, argv).
+ options(all).positional(pos).run(), vm);
+ po::notify(vm);
+
+ if (vm.count("help") || !vm.count("input-file")) {
+ std::cout << std::endl;
+ std::cout << "Compute the persistent homology with various coefficient fields \n";
+ std::cout << "of a Rips complex defined on a set of input points. The coefficient \n";
+ std::cout << "fields are all the Z/rZ for a prime number r contained in the \n";
+ std::cout << "specified range [p,q]\n \n";
+ std::cout << "The output diagram contains one bar per line, written with the convention: \n";
+ std::cout << " p1*...*pr dim b d \n";
+ std::cout << "where dim is the dimension of the homological feature,\n";
+ std::cout << "b and d are respectively the birth and death of the feature and \n";
+ std::cout << "p1*...*pr is the product of prime numbers pi such that the homology \n";
+ std::cout << "feature exists in homology with Z/piZ coefficients." << std::endl << std::endl;
+
+ std::cout << "Usage: " << argv[0] << " [options] input-file" << std::endl << std::endl;
+ std::cout << visible << std::endl;
+ exit(-1);
+ }
+}
diff --git a/src/Persistent_cohomology/example/rips_persistence_step_by_step.cpp b/src/Persistent_cohomology/example/rips_persistence_step_by_step.cpp
new file mode 100644
index 00000000..02db05ec
--- /dev/null
+++ b/src/Persistent_cohomology/example/rips_persistence_step_by_step.cpp
@@ -0,0 +1,154 @@
+/* This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+ * See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+ * Author(s): Clément Maria
+ *
+ * Copyright (C) 2014 Inria
+ *
+ * Modification(s):
+ * - YYYY/MM Author: Description of the modification
+ */
+
+#include <gudhi/graph_simplicial_complex.h>
+#include <gudhi/distance_functions.h>
+#include <gudhi/Simplex_tree.h>
+#include <gudhi/Persistent_cohomology.h>
+#include <gudhi/Points_off_io.h>
+
+#include <boost/program_options.hpp>
+
+#include <string>
+#include <vector>
+#include <limits> // infinity
+#include <utility> // for pair
+#include <map>
+
+// ----------------------------------------------------------------------------
+// rips_persistence_step_by_step is an example of each step that is required to
+// build a Rips over a Simplex_tree. Please refer to rips_persistence to see
+// how to do the same thing with the Rips_complex wrapper for less detailed
+// steps.
+// ----------------------------------------------------------------------------
+
+// Types definition
+using Simplex_tree = Gudhi::Simplex_tree<Gudhi::Simplex_tree_options_fast_persistence>;
+using Vertex_handle = Simplex_tree::Vertex_handle;
+using Filtration_value = Simplex_tree::Filtration_value;
+using Proximity_graph = Gudhi::Proximity_graph<Simplex_tree>;
+
+using Field_Zp = Gudhi::persistent_cohomology::Field_Zp;
+using Persistent_cohomology = Gudhi::persistent_cohomology::Persistent_cohomology<Simplex_tree, Field_Zp >;
+using Point = std::vector<double>;
+using Points_off_reader = Gudhi::Points_off_reader<Point>;
+
+void program_options(int argc, char * argv[]
+ , std::string & off_file_points
+ , std::string & filediag
+ , Filtration_value & threshold
+ , int & dim_max
+ , int & p
+ , Filtration_value & min_persistence);
+
+int main(int argc, char * argv[]) {
+ std::string off_file_points;
+ std::string filediag;
+ Filtration_value threshold;
+ int dim_max;
+ int p;
+ Filtration_value min_persistence;
+
+ program_options(argc, argv, off_file_points, filediag, threshold, dim_max, p, min_persistence);
+
+ // Extract the points from the file filepoints
+ Points_off_reader off_reader(off_file_points);
+
+ // Compute the proximity graph of the points
+ Proximity_graph prox_graph = Gudhi::compute_proximity_graph<Simplex_tree>(off_reader.get_point_cloud(),
+ threshold,
+ Gudhi::Euclidean_distance());
+
+ // Construct the Rips complex in a Simplex Tree
+ Simplex_tree st;
+ // insert the proximity graph in the simplex tree
+ st.insert_graph(prox_graph);
+ // expand the graph until dimension dim_max
+ st.expansion(dim_max);
+
+ std::cout << "The complex contains " << st.num_simplices() << " simplices \n";
+ std::cout << " and has dimension " << st.dimension() << " \n";
+
+ // Sort the simplices in the order of the filtration
+ st.initialize_filtration();
+
+ // Compute the persistence diagram of the complex
+ Persistent_cohomology pcoh(st);
+ // initializes the coefficient field for homology
+ pcoh.init_coefficients(p);
+
+ pcoh.compute_persistent_cohomology(min_persistence);
+
+ // Output the diagram in filediag
+ if (filediag.empty()) {
+ pcoh.output_diagram();
+ } else {
+ std::ofstream out(filediag);
+ pcoh.output_diagram(out);
+ out.close();
+ }
+
+ return 0;
+}
+
+void program_options(int argc, char * argv[]
+ , std::string & off_file_points
+ , std::string & filediag
+ , Filtration_value & threshold
+ , int & dim_max
+ , int & p
+ , Filtration_value & min_persistence) {
+ namespace po = boost::program_options;
+ po::options_description hidden("Hidden options");
+ hidden.add_options()
+ ("input-file", po::value<std::string>(&off_file_points),
+ "Name of an OFF file containing a point set.\n");
+
+ po::options_description visible("Allowed options", 100);
+ visible.add_options()
+ ("help,h", "produce help message")
+ ("output-file,o", po::value<std::string>(&filediag)->default_value(std::string()),
+ "Name of file in which the persistence diagram is written. Default print in std::cout")
+ ("max-edge-length,r",
+ po::value<Filtration_value>(&threshold)->default_value(std::numeric_limits<Filtration_value>::infinity()),
+ "Maximal length of an edge for the Rips complex construction.")
+ ("cpx-dimension,d", po::value<int>(&dim_max)->default_value(1),
+ "Maximal dimension of the Rips complex we want to compute.")
+ ("field-charac,p", po::value<int>(&p)->default_value(11),
+ "Characteristic p of the coefficient field Z/pZ for computing homology.")
+ ("min-persistence,m", po::value<Filtration_value>(&min_persistence),
+ "Minimal lifetime of homology feature to be recorded. Default is 0. Enter a negative value to see zero length intervals");
+
+ po::positional_options_description pos;
+ pos.add("input-file", 1);
+
+ po::options_description all;
+ all.add(visible).add(hidden);
+
+ po::variables_map vm;
+ po::store(po::command_line_parser(argc, argv).
+ options(all).positional(pos).run(), vm);
+ po::notify(vm);
+
+ if (vm.count("help") || !vm.count("input-file")) {
+ std::cout << std::endl;
+ std::cout << "Compute the persistent homology with coefficient field Z/pZ \n";
+ std::cout << "of a Rips complex defined on a set of input points.\n \n";
+ std::cout << "The output diagram contains one bar per line, written with the convention: \n";
+ std::cout << " p dim b d \n";
+ std::cout << "where dim is the dimension of the homological feature,\n";
+ std::cout << "b and d are respectively the birth and death of the feature and \n";
+ std::cout << "p is the characteristic of the field Z/pZ used for homology coefficients." << std::endl << std::endl;
+
+ std::cout << "Usage: " << argv[0] << " [options] input-file" << std::endl << std::endl;
+ std::cout << visible << std::endl;
+ exit(-1);
+ }
+}
diff --git a/src/Persistent_cohomology/example/rips_persistence_via_boundary_matrix.cpp b/src/Persistent_cohomology/example/rips_persistence_via_boundary_matrix.cpp
new file mode 100644
index 00000000..37fa5e93
--- /dev/null
+++ b/src/Persistent_cohomology/example/rips_persistence_via_boundary_matrix.cpp
@@ -0,0 +1,160 @@
+/* This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT.
+ * See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details.
+ * Author(s): Clément Maria, Marc Glisse
+ *
+ * Copyright (C) 2014 Inria
+ *
+ * Modification(s):
+ * - YYYY/MM Author: Description of the modification
+ */
+
+#include <gudhi/Simplex_tree.h>
+#include <gudhi/Persistent_cohomology.h>
+#include <gudhi/Rips_complex.h>
+#include <gudhi/Hasse_complex.h>
+#include <gudhi/Points_off_io.h>
+#include <gudhi/distance_functions.h>
+
+#include <boost/program_options.hpp>
+
+#ifdef GUDHI_USE_TBB
+#include <tbb/task_scheduler_init.h>
+#endif
+
+#include <string>
+#include <vector>
+
+////////////////////////////////////////////////////////////////
+// //
+// WARNING: persistence computation itself is not parallel, //
+// and this uses more memory than rips_persistence. //
+// //
+////////////////////////////////////////////////////////////////
+
+// Types definition
+using Simplex_tree = Gudhi::Simplex_tree<>;
+using Filtration_value = Simplex_tree::Filtration_value;
+using Rips_complex = Gudhi::rips_complex::Rips_complex<Filtration_value>;
+using Field_Zp = Gudhi::persistent_cohomology::Field_Zp;
+using Point = std::vector<double>;
+using Points_off_reader = Gudhi::Points_off_reader<Point>;
+
+void program_options(int argc, char * argv[]
+ , std::string & off_file_points
+ , std::string & filediag
+ , Filtration_value & threshold
+ , int & dim_max
+ , int & p
+ , Filtration_value & min_persistence);
+
+int main(int argc, char * argv[]) {
+ std::string off_file_points;
+ std::string filediag;
+ Filtration_value threshold;
+ int dim_max;
+ int p;
+ Filtration_value min_persistence;
+
+ program_options(argc, argv, off_file_points, filediag, threshold, dim_max, p, min_persistence);
+
+ Points_off_reader off_reader(off_file_points);
+ Rips_complex rips_complex_from_file(off_reader.get_point_cloud(), threshold, Gudhi::Euclidean_distance());
+
+ // Construct the Rips complex in a Simplex Tree
+ Simplex_tree& st = *new Simplex_tree;
+ rips_complex_from_file.create_complex(st, dim_max);
+
+ std::cout << "The complex contains " << st.num_simplices() << " simplices \n";
+ std::cout << " and has dimension " << st.dimension() << " \n";
+
+#ifdef GUDHI_USE_TBB
+ // Unnecessary, but clarifies which operations are parallel.
+ tbb::task_scheduler_init ts;
+#endif
+
+ // Sort the simplices in the order of the filtration
+ st.initialize_filtration();
+ int count = 0;
+ for (auto sh : st.filtration_simplex_range())
+ st.assign_key(sh, count++);
+
+ // Convert to a more convenient representation.
+ Gudhi::Hasse_complex<> hcpx(st);
+
+#ifdef GUDHI_USE_TBB
+ ts.terminate();
+#endif
+
+ // Free some space.
+ delete &st;
+
+ // Compute the persistence diagram of the complex
+ Gudhi::persistent_cohomology::Persistent_cohomology< Gudhi::Hasse_complex<>, Field_Zp > pcoh(hcpx);
+ // initializes the coefficient field for homology
+ pcoh.init_coefficients(p);
+
+ pcoh.compute_persistent_cohomology(min_persistence);
+
+ // Output the diagram in filediag
+ if (filediag.empty()) {
+ pcoh.output_diagram();
+ } else {
+ std::ofstream out(filediag);
+ pcoh.output_diagram(out);
+ out.close();
+ }
+}
+
+void program_options(int argc, char * argv[]
+ , std::string & off_file_points
+ , std::string & filediag
+ , Filtration_value & threshold
+ , int & dim_max
+ , int & p
+ , Filtration_value & min_persistence) {
+ namespace po = boost::program_options;
+ po::options_description hidden("Hidden options");
+ hidden.add_options()
+ ("input-file", po::value<std::string>(&off_file_points),
+ "Name of file containing a point set. Format is one point per line: X1 ... Xd ");
+
+ po::options_description visible("Allowed options", 100);
+ visible.add_options()
+ ("help,h", "produce help message")
+ ("output-file,o", po::value<std::string>(&filediag)->default_value(std::string()),
+ "Name of file in which the persistence diagram is written. Default print in std::cout")
+ ("max-edge-length,r", po::value<Filtration_value>(&threshold)->default_value(0),
+ "Maximal length of an edge for the Rips complex construction.")
+ ("cpx-dimension,d", po::value<int>(&dim_max)->default_value(1),
+ "Maximal dimension of the Rips complex we want to compute.")
+ ("field-charac,p", po::value<int>(&p)->default_value(11),
+ "Characteristic p of the coefficient field Z/pZ for computing homology.")
+ ("min-persistence,m", po::value<Filtration_value>(&min_persistence),
+ "Minimal lifetime of homology feature to be recorded. Default is 0. Enter a negative value to see zero length intervals");
+
+ po::positional_options_description pos;
+ pos.add("input-file", 1);
+
+ po::options_description all;
+ all.add(visible).add(hidden);
+
+ po::variables_map vm;
+ po::store(po::command_line_parser(argc, argv).
+ options(all).positional(pos).run(), vm);
+ po::notify(vm);
+
+ if (vm.count("help") || !vm.count("input-file")) {
+ std::cout << std::endl;
+ std::cout << "Compute the persistent homology with coefficient field Z/pZ \n";
+ std::cout << "of a Rips complex defined on a set of input points.\n \n";
+ std::cout << "The output diagram contains one bar per line, written with the convention: \n";
+ std::cout << " p dim b d \n";
+ std::cout << "where dim is the dimension of the homological feature,\n";
+ std::cout << "b and d are respectively the birth and death of the feature and \n";
+ std::cout << "p is the characteristic of the field Z/pZ used for homology coefficients." << std::endl << std::endl;
+
+ std::cout << "Usage: " << argv[0] << " [options] input-file" << std::endl << std::endl;
+ std::cout << visible << std::endl;
+ exit(-1);
+ }
+}