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Cover complex

## Classes

class | Gudhi::cover_complex::Cover_complex< Point >
| Cover complex data structure. More...

## Detailed Description

Author
  Mathieu Carrière

Visualizations of the simplicial complexes can be done with either neato (from graphviz), geomview, KeplerMapper. Input point clouds are assumed to be OFF files

# Covers

Nerves and Graph Induced Complexes require a cover C of the input point cloud P, that is a set of subsets of P whose union is P itself. Very often, this cover is obtained from the preimage of a family of intervals covering the image of some scalar-valued function f defined on P. This family is parameterized by its resolution, which can be either the number or the length of the intervals, and its gain, which is the overlap percentage between consecutive intervals (ordered by their first values).

# Nerves

## Nerve definition

Assume you are given a cover C of your point cloud P. Then, the Nerve of this cover is the simplicial complex that has one k-simplex per k-fold intersection of cover elements. See also Wikipedia .

Nerve of a double torus

## Example

This example builds the Nerve of a point cloud sampled on a 3D human shape (human.off). The cover C comes from the preimages of intervals (10 intervals with gain 0.3) covering the height function (coordinate 2), which are then refined into their connected components using the triangulation of the .OFF file.

/* 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): Mathieu Carrière
*
* Copyright (C) 2017 Inria
*
* Modification(s):
* - YYYY/MM Author: Description of the modification
*/
#include <gudhi/GIC.h>
#include <string>
#include <vector>
void usage(int nbArgs, char *const progName) {
std::cerr << "Error: Number of arguments (" << nbArgs << ") is not correct\n";
std::cerr << "Usage: " << progName << " filename.off coordinate resolution gain [-v] \n";
std::cerr << " i.e.: " << progName << " ../../data/points/human.off 2 10 0.3 -v \n";
exit(-1); // ----- >>
}
int main(int argc, char **argv) {
if ((argc != 5) && (argc != 6)) usage(argc, argv[0]);
using Point = std::vector<float>;
std::string off_file_name(argv[1]);
int coord = atoi(argv[2]);
int resolution = atoi(argv[3]);
double gain = atof(argv[4]);
bool verb = 0;
if (argc == 6) verb = 1;
// --------------------------------
// Init of a Nerve from an OFF file
// --------------------------------
Gudhi::cover_complex::Cover_complex<Point> SC;
SC.set_verbose(verb);
bool check = SC.read_point_cloud(off_file_name);
if (!check) {
std::clog << "Incorrect OFF file." << std::endl;
} else {
SC.set_type("Nerve");
SC.set_color_from_coordinate(coord);
SC.set_function_from_coordinate(coord);
SC.set_graph_from_OFF();
SC.set_resolution_with_interval_number(resolution);
SC.set_gain(gain);
SC.set_cover_from_function();
SC.find_simplices();
SC.write_info();
Gudhi::Simplex_tree<> stree;
SC.create_complex(stree);
SC.compute_PD();
// ----------------------------------------------------------------------------
// Display information about the graph induced complex
// ----------------------------------------------------------------------------
if (verb) {
std::clog << "Nerve is of dimension " << stree.dimension() << " - " << stree.num_simplices() << " simplices - "
<< stree.num_vertices() << " vertices." << std::endl;
std::clog << "Iterator on Nerve simplices" << std::endl;
for (auto f_simplex : stree.filtration_simplex_range()) {
for (auto vertex : stree.simplex_vertex_range(f_simplex)) {
std::clog << vertex << " ";
}
std::clog << std::endl;
}
}
}
return 0;
}
Gudhi::Simplex_tree
Simplex Tree data structure for representing simplicial complexes.
Definition Simplex_tree.h:98
Gudhi::Simplex_tree::simplex_vertex_range
Simplex_vertex_range simplex_vertex_range(Simplex_handle sh) const
Returns a range over the vertices of a simplex.
Definition Simplex_tree.h:369
Gudhi::Simplex_tree::filtration_simplex_range
Filtration_simplex_range const & filtration_simplex_range(Indexing_tag=Indexing_tag()) const
Returns a range over the simplices of the simplicial complex, in the order of the filtration.
Definition Simplex_tree.h:358
Gudhi::Simplex_tree::num_vertices
size_t num_vertices() const
Returns the number of vertices in the complex.
Definition Simplex_tree.h:778
Gudhi::Simplex_tree::num_simplices
size_t num_simplices() const
Returns the number of simplices in the simplex_tree.
Definition Simplex_tree.h:791
Gudhi::cover_complex::Cover_complex
Cover complex data structure.
Definition GIC.h:99
Gudhi::cover_complex::Cover_complex::set_resolution_with_interval_number
void set_resolution_with_interval_number(int reso)
Sets a number of intervals from a value stored in memory.
Definition GIC.h:589
Gudhi::cover_complex::Cover_complex::compute_PD
Persistence_diagram compute_PD()
Computes the extended persistence diagram of the complex.
Definition GIC.h:1165
Gudhi::cover_complex::Cover_complex::create_complex
void create_complex(SimplicialComplex &complex)
Creates the simplicial complex.
Definition GIC.h:1325
Gudhi::cover_complex::Cover_complex::set_type
void set_type(const std::string &t)
Specifies whether the type of the output simplicial complex.
Definition GIC.h:188
Gudhi::cover_complex::Cover_complex::find_simplices
void find_simplices()
Computes the simplices of the simplicial complex.
Definition GIC.h:1339
Gudhi::cover_complex::Cover_complex::set_cover_from_function
void set_cover_from_function()
Creates a cover C from the preimages of the function f.
Definition GIC.h:601
Gudhi::cover_complex::Cover_complex::write_info
void write_info()
Creates a .txt file called SC.txt describing the 1-skeleton, which can then be plotted with e....
Definition GIC.h:1078
Gudhi::cover_complex::Cover_complex::set_gain
void set_gain(double g=0.3)
Sets a gain from a value stored in memory (default value 0.3).
Definition GIC.h:595
Gudhi::cover_complex::Cover_complex::set_function_from_coordinate
void set_function_from_coordinate(int k)
Creates the function f from the k-th coordinate of the point cloud P.
Definition GIC.h:504
Gudhi::cover_complex::Cover_complex::set_color_from_coordinate
void set_color_from_coordinate(int k=0)
Computes the function used to color the nodes of the simplicial complex from the k-th coordinate.
Definition GIC.h:1005
Gudhi::cover_complex::Cover_complex::set_graph_from_OFF
void set_graph_from_OFF()
Creates a graph G from the triangulation given by the input .OFF file.
Definition GIC.h:344
Gudhi::cover_complex::Cover_complex::read_point_cloud
bool read_point_cloud(const std::string &off_file_name)
Reads and stores the input point cloud from .(n)OFF file.
Definition GIC.h:244
Gudhi::cover_complex::Cover_complex::set_verbose
void set_verbose(bool verb=false)
Specifies whether the program should display information or not.
Definition GIC.h:196

When launching:

$> ./Nerve ../../data/points/human.off 2 10 0.3 -v

the program output is:

Min function value = -0.979672 and Max function value = 0.816414
Interval 0 = [-0.979672, -0.761576]
Interval 1 = [-0.838551, -0.581967]
Interval 2 = [-0.658942, -0.402359]
Interval 3 = [-0.479334, -0.22275]
Interval 4 = [-0.299725, -0.0431415]
Interval 5 = [-0.120117, 0.136467]
Interval 6 = [0.059492, 0.316076]
Interval 7 = [0.239101, 0.495684]
Interval 8 = [0.418709, 0.675293]
Interval 9 = [0.598318, 0.816414]
Computing preimages...
Computing connected components...
.txt generated. It can be visualized with e.g. python KeplerMapperVisuFromTxtFile.py and firefox.
5 interval(s) in dimension 0:
[-0.909111, 0.00817529]
[-0.171433, 0.367392]
[-0.171433, 0.367392]
[-0.909111, 0.745853]
0 interval(s) in dimension 1:
Nerve is of dimension 1 - 41 simplices - 21 vertices.
Iterator on Nerve simplices
1
0
4
4 0
2
2 1
8
8 2
5
5 4
9
9 8
13
13 5
14
14 9
19
19 13
25
32
20
32 20
33
33 25
26
26 14
26 19
42
42 26
34
34 33
27
27 20
35
35 27
35 34
42 35
44
44 35
54
54 44

The program also writes a file ../../data/points/human_sc.txt. The first three lines in this file are the location of the input point cloud and the function used to compute the cover. The fourth line contains the number of vertices nv and edges ne of the Nerve. The next nv lines represent the vertices. Each line contains the vertex ID, the number of data points it contains, and their average color function value. Finally, the next ne lines represent the edges, characterized by the ID of their vertices.

Using KeplerMapper, one can obtain the following visualization:

Visualization with KeplerMapper

# Graph Induced Complexes (GIC)

## GIC definition

Again, assume you are given a cover C of your point cloud P. Moreover, assume you are also given a graph G built on top of P. Then, for any clique in G whose nodes all belong to different elements of C, the GIC includes a corresponding simplex, whose dimension is the number of nodes in the clique minus one. See [Dey13] for more details.

GIC of a point cloud.

## Example with cover from Voronoï

This example builds the GIC of a point cloud sampled on a 3D human shape (human.off). We randomly subsampled 100 points in the point cloud, which act as seeds of a geodesic Voronoï diagram. Each cell of the diagram is then an element of C. The graph G (used to compute both the geodesics for Voronoï and the GIC) comes from the triangulation of the human shape. Note that the resulting simplicial complex is in dimension 3 in this example.

/* 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): Mathieu Carrière
*
* Copyright (C) 2017 Inria
*
* Modification(s):
* - YYYY/MM Author: Description of the modification
*/
#include <gudhi/GIC.h>
#include <string>
#include <vector>
void usage(int nbArgs, char *const progName) {
std::cerr << "Error: Number of arguments (" << nbArgs << ") is not correct\n";
std::cerr << "Usage: " << progName << " filename.off N [-v] \n";
std::cerr << " i.e.: " << progName << " ../../data/points/human.off 100 -v \n";
exit(-1); // ----- >>
}
int main(int argc, char **argv) {
if ((argc != 3) && (argc != 4)) usage(argc, argv[0]);
using Point = std::vector<float>;
std::string off_file_name(argv[1]);
int m = atoi(argv[2]);
bool verb = 0;
if (argc == 4) verb = 1;
// ----------------------------------------------------------------------------
// Init of a graph induced complex from an OFF file
// ----------------------------------------------------------------------------
Gudhi::cover_complex::Cover_complex<Point> GIC;
GIC.set_verbose(verb);
bool check = GIC.read_point_cloud(off_file_name);
if (!check) {
std::clog << "Incorrect OFF file." << std::endl;
} else {
GIC.set_type("GIC");
GIC.set_color_from_coordinate();
GIC.set_graph_from_OFF();
GIC.set_cover_from_Voronoi(Gudhi::Euclidean_distance(), m);
GIC.find_simplices();
GIC.plot_OFF();
Gudhi::Simplex_tree<> stree;
GIC.create_complex(stree);
// ----------------------------------------------------------------------------
// Display information about the graph induced complex
// ----------------------------------------------------------------------------
if (verb) {
std::clog << "Graph induced complex is of dimension " << stree.dimension() << " - " << stree.num_simplices()
<< " simplices - " << stree.num_vertices() << " vertices." << std::endl;
std::clog << "Iterator on graph induced complex simplices" << std::endl;
for (auto f_simplex : stree.filtration_simplex_range()) {
for (auto vertex : stree.simplex_vertex_range(f_simplex)) {
std::clog << vertex << " ";
}
std::clog << std::endl;
}
}
}
return 0;
}
Gudhi::Euclidean_distance
Compute the Euclidean distance between two Points given by a range of coordinates....
Definition distance_functions.h:32
Gudhi::cover_complex::Cover_complex::set_cover_from_Voronoi
void set_cover_from_Voronoi(Distance distance, int m=100)
Creates the cover C from the Voronoï cells of a subsampling of the point cloud.
Definition GIC.h:895
Gudhi::cover_complex::Cover_complex::plot_OFF
void plot_OFF()
Creates a .off file called SC.off for 3D visualization, which contains the 2-skeleton of the GIC....
Definition GIC.h:1117

When launching:

$> ./VoronoiGIC ../../data/points/human.off 700 -v

the program outputs SC.off. Using e.g.

$> geomview ../../data/points/human_sc.off

one can obtain the following visualization:

Visualization with Geomview

## Functional GIC

If one restricts to the cliques in G whose nodes all belong to preimages of consecutive intervals (assuming the cover of the height function is minimal, i.e. no more than two intervals can intersect at a time), the GIC is of dimension one, i.e. a graph. We call this graph the functional GIC. See [Carriere16] for more details.

## Example

Functional GIC comes with automatic selection of the Rips threshold, the resolution and the gain of the function cover. See [Carriere17c] for more details. In this example, we compute the functional GIC of a Klein bottle embedded in R^5, where the graph G comes from a Rips complex with automatic threshold, and the cover C comes from the preimages of intervals covering the first coordinate, with automatic resolution and gain. Note that automatic threshold, resolution and gain can be computed as well for the Nerve.

/* 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): Mathieu Carrière
*
* Copyright (C) 2017 Inria
*
* Modification(s):
* - YYYY/MM Author: Description of the modification
*/
#include <gudhi/GIC.h>
#include <string>
#include <vector>
void usage(int nbArgs, char *const progName) {
std::cerr << "Error: Number of arguments (" << nbArgs << ") is not correct\n";
std::cerr << "Usage: " << progName << " filename.off coordinate [-v] \n";
std::cerr << " i.e.: " << progName << " ../../data/points/human.off 2 -v \n";
exit(-1); // ----- >>
}
int main(int argc, char **argv) {
if ((argc != 3) && (argc != 4)) usage(argc, argv[0]);
using Point = std::vector<float>;
std::string off_file_name(argv[1]);
int coord = atoi(argv[2]);
bool verb = 0;
if (argc == 4) verb = 1;
// -----------------------------------------
// Init of a functional GIC from an OFF file
// -----------------------------------------
Gudhi::cover_complex::Cover_complex<Point> GIC;
GIC.set_verbose(verb);
bool check = GIC.read_point_cloud(off_file_name);
if (!check) {
std::clog << "Incorrect OFF file." << std::endl;
} else {
GIC.set_type("GIC");
GIC.set_color_from_coordinate(coord);
GIC.set_function_from_coordinate(coord);
GIC.set_graph_from_automatic_rips(Gudhi::Euclidean_distance());
GIC.set_automatic_resolution();
GIC.set_gain();
GIC.set_cover_from_function();
GIC.find_simplices();
GIC.compute_distribution(10);
GIC.compute_p_value();
GIC.plot_DOT();
Gudhi::Simplex_tree<> stree;
GIC.create_complex(stree);
// --------------------------------------------
// Display information about the functional GIC
// --------------------------------------------
if (verb) {
std::clog << "Coordinate GIC is of dimension " << stree.dimension() << " - " << stree.num_simplices()
<< " simplices - " << stree.num_vertices() << " vertices." << std::endl;
std::clog << "Iterator on coordinate GIC simplices" << std::endl;
for (auto f_simplex : stree.filtration_simplex_range()) {
for (auto vertex : stree.simplex_vertex_range(f_simplex)) {
std::clog << vertex << " ";
}
std::clog << std::endl;
}
}
}
return 0;
}
Gudhi::cover_complex::Cover_complex::set_automatic_resolution
double set_automatic_resolution()
Computes the optimal length of intervals (i.e. the smallest interval length avoiding discretization a...
Definition GIC.h:542
Gudhi::cover_complex::Cover_complex::plot_DOT
void plot_DOT()
Creates a .dot file called SC.dot for neato (part of the graphviz package) once the simplicial comple...
Definition GIC.h:1034
Gudhi::cover_complex::Cover_complex::set_graph_from_automatic_rips
double set_graph_from_automatic_rips(Distance distance, int N=100)
Creates a graph G from a Rips complex whose threshold value is automatically tuned with subsampling—s...
Definition GIC.h:431
Gudhi::cover_complex::Cover_complex::compute_distribution
void compute_distribution(unsigned int N=100)
Computes bootstrapped distances distribution.
Definition GIC.h:1228
Gudhi::cover_complex::Cover_complex::compute_p_value
double compute_p_value()
Computes the p-value, i.e. the opposite of the confidence level of the largest bottleneck distance pr...
Definition GIC.h:1306

When launching:

$> ./CoordGIC ../../data/points/KleinBottle5D.off 0 -v

the program outputs SC.dot. Using e.g.

$> neato SC.dot -Tpdf -o SC.pdf

one can obtain the following visualization:

Visualization with Neato

where nodes are colored by the filter function values and, for each node, the first number is its ID and the second is the number of data points that its contain.

We also provide an example on a set of 72 pictures taken around the same object (lucky_cat.off). The function is now the first eigenfunction given by PCA, whose values are written in a file (lucky_cat_PCA1). Threshold, resolution and gain are automatically selected as before.

/* 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): Mathieu Carrière
*
* Copyright (C) 2017 Inria
*
* Modification(s):
* - YYYY/MM Author: Description of the modification
*/
#include <gudhi/GIC.h>
#include <string>
#include <vector>
void usage(int nbArgs, char *const progName) {
std::cerr << "Error: Number of arguments (" << nbArgs << ") is not correct\n";
std::cerr << "Usage: " << progName << " filename.off function [-v] \n";
std::cerr << " i.e.: " << progName << " ../../data/points/COIL_database/lucky_cat.off "
"../../data/points/COIL_database/lucky_cat_PCA1 -v \n";
exit(-1); // ----- >>
}
int main(int argc, char **argv) {
if ((argc != 3) && (argc != 4)) usage(argc, argv[0]);
using Point = std::vector<float>;
std::string off_file_name(argv[1]);
std::string func_file_name = argv[2];
bool verb = 0;
if (argc == 4) verb = 1;
// -----------------------------------------
// Init of a functional GIC from an OFF file
// -----------------------------------------
Gudhi::cover_complex::Cover_complex<Point> GIC;
GIC.set_verbose(verb);
bool check = GIC.read_point_cloud(off_file_name);
if (!check) {
std::clog << "Incorrect OFF file." << std::endl;
} else {
GIC.set_type("GIC");
GIC.set_color_from_file(func_file_name);
GIC.set_function_from_file(func_file_name);
GIC.set_graph_from_automatic_rips(Gudhi::Euclidean_distance());
GIC.set_automatic_resolution();
GIC.set_gain();
GIC.set_cover_from_function();
GIC.find_simplices();
GIC.plot_DOT();
Gudhi::Simplex_tree<> stree;
GIC.create_complex(stree);
// --------------------------------------------
// Display information about the functional GIC
// --------------------------------------------
if (verb) {
std::clog << "Functional GIC is of dimension " << stree.dimension() << " - " << stree.num_simplices()
<< " simplices - " << stree.num_vertices() << " vertices." << std::endl;
std::clog << "Iterator on functional GIC simplices" << std::endl;
for (auto f_simplex : stree.filtration_simplex_range()) {
for (auto vertex : stree.simplex_vertex_range(f_simplex)) {
std::clog << vertex << " ";
}
std::clog << std::endl;
}
}
}
return 0;
}
Gudhi::cover_complex::Cover_complex::set_function_from_file
void set_function_from_file(const std::string &func_file_name)
Creates the function f from a file containing the function values.
Definition GIC.h:485
Gudhi::cover_complex::Cover_complex::set_color_from_file
void set_color_from_file(const std::string &color_file_name)
Computes the function used to color the nodes of the simplicial complex from a file containing the fu...
Definition GIC.h:987

When launching:

$> ./FuncGIC ../../data/points/COIL_database/lucky_cat.off ../../data/points/COIL_database/lucky_cat_PCA1 -v

the program outputs again SC.dot which gives the following visualization after using neato:

Visualization with neato