/* 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): Clément Maria, Pawel Dlotko, Vincent Rouvreau * * Copyright (C) 2016 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 . */ #ifndef DOC_GRAPH_INDUCED_COMPLEX_INTRO_GRAPH_INDUCED_COMPLEX_H_ #define DOC_GRAPH_INDUCED_COMPLEX_INTRO_GRAPH_INDUCED_COMPLEX_H_ namespace Gudhi { namespace graph_induced_complex { /** \defgroup graph_induced_complex Graph induced complex * * \author Mathieu Carrière * * @{ * * Visualizations of the simplicial complexes require neato, python and firefox!! * * \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 Wikipedia . * * \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 first three lines are requirements for visualization with Kepler-Mapper. * 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. * * * \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 this article * for more details. * * \image html "gic_complex.png" "GIC of a point cloud." * * \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 output is: * * \include Nerve_GIC/GIC.txt * * \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. * * \include Nerve_GIC/GICvoronoi.cpp * * When launching: * * \code $> ./GICvoronoi ../../../data/points/human.off 100 --v * \endcode * * the program output is: * * \include Nerve_GIC/GICvoronoi.txt * * \subsection mapperdeltadefinition Mapper Delta * * 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 Mapper Delta, since it is related to the usual Mapper. See * this article for more details. * * \subsection mapperdeltaexample Example * * Mapper Delta comes with optimal selection for the Rips threshold, * the resolution and the gain of the function cover. In this example, * we compute the Mapper Delta of a point cloud sampled on a 3D human shape (human.off), * where the graph G comes from a Rips complex with optimal threshold, * and the cover C comes from the preimages of intervals covering the height function (coordinate 2), * with optimal resolution and gain. Note that optimal threshold, resolution and gain * also exist for the Nerve of this cover. * * \include Nerve_GIC/MapperDeltaCoord.cpp * * When launching: * * \code $> ./MapperDeltaCoord ../../../data/points/human.off 2 --v * \endcode * * the program output is: * * \include MapperDeltaCoord.txt * * 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/MapperDeltaFunc.cpp * * When launching: * * \code $> ./MapperDeltaFunc ../../../data/points/COIL_database/lucky_cat.off ../../../data/points/COIL_database/lucky_cat_PCA1 --v * \endcode * * the program output is: * * \include MapperDeltaFunc.txt * * \copyright GNU General Public License v3. * \verbatim Contact: gudhi-users@lists.gforge.inria.fr \endverbatim */ /** @} */ // end defgroup graph_induced_complex } // namespace graph_induced_complex } // namespace Gudhi #endif // DOC_GRAPH_INDUCED_COMPLEX_INTRO_GRAPH_INDUCED_COMPLEX_H_