diff options
Diffstat (limited to 'src/Nerve_GIC/doc/Intro_graph_induced_complex.h')
-rw-r--r-- | src/Nerve_GIC/doc/Intro_graph_induced_complex.h | 216 |
1 files changed, 216 insertions, 0 deletions
diff --git a/src/Nerve_GIC/doc/Intro_graph_induced_complex.h b/src/Nerve_GIC/doc/Intro_graph_induced_complex.h new file mode 100644 index 00000000..3a0d8154 --- /dev/null +++ b/src/Nerve_GIC/doc/Intro_graph_induced_complex.h @@ -0,0 +1,216 @@ +/* This file is part of the Gudhi Library. The Gudhi library + * (Geometric Understanding in Higher Dimensions) is a generic C++ + * library for computational topology. + * + * Author(s): Mathieu Carriere + * + * Copyright (C) 2017 INRIA + * + * 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 3 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. + * + * You should have received a copy of the GNU General Public License + * along with this program. If not, see <http://www.gnu.org/licenses/>. + */ + +#ifndef DOC_COVER_COMPLEX_INTRO_COVER_COMPLEX_H_ +#define DOC_COVER_COMPLEX_INTRO_COVER_COMPLEX_H_ + +namespace Gudhi { + +namespace cover_complex { + +/** \defgroup cover_complex Cover complex + * + * \author Mathieu Carrière + * + * @{ + * + * Visualizations of the simplicial complexes can be done with either + * neato (from <a target="_blank" href="http://www.graphviz.org/">graphviz</a>), + * <a target="_blank" href="http://www.geomview.org/">geomview</a>, + * <a target="_blank" href="https://github.com/MLWave/kepler-mapper">KeplerMapper</a>. + * Input point clouds are assumed to be + * <a target="_blank" href="http://www.geomview.org/docs/html/OFF.html">OFF files</a>. + * + * \section covers 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). + * + * \section nerves Nerves + * + * \subsection nervedefinition 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 <a target="_blank" href="https://en.wikipedia.org/wiki/Nerve_of_a_covering"> Wikipedia </a>. + * + * \image html "nerve.png" "Nerve of a double torus" + * + * \subsection nerveexample 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. + * + * \include Nerve_GIC/Nerve.cpp + * + * When launching: + * + * \code $> ./Nerve ../../../../data/points/human.off 2 10 0.3 --v + * \endcode + * + * the program output is: + * + * \include Nerve_GIC/Nerve.txt + * + * The program also writes a file 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: + * + * \image html "nervevisu.jpg" "Visualization with KeplerMapper" + * + * \section gic Graph Induced Complexes (GIC) + * + * \subsection gicdefinition 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 \cite Dey13 for more details. + * + * \image html "GIC.jpg" "GIC of a point cloud." + * + * \subsection gicexamplevor 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. + * + * \include Nerve_GIC/VoronoiGIC.cpp + * + * When launching: + * + * \code $> ./VoronoiGIC ../../../../data/points/human.off 700 --v + * \endcode + * + * the program outputs SC.off. Using e.g. + * + * \code $> geomview SC.off + * \endcode + * + * one can obtain the following visualization: + * + * \image html "gicvoronoivisu.jpg" "Visualization with Geomview" + * + * \subsection functionalGICdefinition 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 \cite Carriere16 for more details. + * + * \subsection functionalGICexample Example + * + * Functional GIC comes with automatic selection of the Rips threshold, + * the resolution and the gain of the function cover. See \cite 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. + * + * \include Nerve_GIC/CoordGIC.cpp + * + * When launching: + * + * \code $> ./CoordGIC ../../../../data/points/KleinBottle5D.off 0 --v + * \endcode + * + * the program outputs SC.dot. Using e.g. + * + * \code $> neato SC.dot -Tpdf -o SC.pdf + * \endcode + * + * one can obtain the following visualization: + * + * \image html "coordGICvisu2.jpg" "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. + * + * \include Nerve_GIC/FuncGIC.cpp + * + * When launching: + * + * \code $> ./FuncGIC ../../data/points/COIL_database/lucky_cat.off ../../data/points/COIL_database/lucky_cat_PCA1 --v + * \endcode + * + * the program outputs again SC.dot which gives the following visualization after using neato: + * + * \image html "funcGICvisu.jpg" "Visualization with neato" + * + * \copyright GNU General Public License v3. + * \verbatim Contact: gudhi-users@lists.gforge.inria.fr \endverbatim + */ +/** @} */ // end defgroup cover_complex + +} // namespace cover_complex + +} // namespace Gudhi + +#endif // DOC_COVER_COMPLEX_INTRO_COVER_COMPLEX_H_ + + +/* * \subsection gicexample Example with cover from function + * + * This example builds the GIC of a point cloud sampled on a 3D human shape (human.off). + * The cover C comes from the preimages of intervals (with length 0.075 and gain 0) + * covering the height function (coordinate 2), + * and the graph G comes from a Rips complex built with threshold 0.075. + * Note that if the gain is too big, the number of cliques increases a lot, + * which make the computation time much larger. + * + * \include Nerve_GIC/GIC.cpp + * + * When launching: + * + * \code $> ./GIC ../../../../data/points/human.off 0.075 2 0.075 0 --v + * \endcode + * + * the program outputs SC.txt, which can be visualized with python and firefox as before: + * + * \image html "gicvisu.jpg" "Visualization with KeplerMapper" + * */ + + +/* * Using e.g. + * + * \code $> python KeplerMapperVisuFromTxtFile.py && firefox SC.html + * \endcode */ |