From aa67dab1eebe3cdba573741857051005ba72cc3b Mon Sep 17 00:00:00 2001 From: mcarrier Date: Mon, 8 May 2017 16:56:25 +0000 Subject: git-svn-id: svn+ssh://scm.gforge.inria.fr/svnroot/gudhi/branches/Nerve_GIC@2406 636b058d-ea47-450e-bf9e-a15bfbe3eedb Former-commit-id: 2d857904595833667d469db97c746bbd8696eac4 --- src/Nerve_GIC/doc/Intro_graph_induced_complex.h | 130 ++++++++++++++++++++++++ 1 file changed, 130 insertions(+) create mode 100644 src/Nerve_GIC/doc/Intro_graph_induced_complex.h (limited to 'src/Nerve_GIC/doc') 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..0b51e345 --- /dev/null +++ b/src/Nerve_GIC/doc/Intro_graph_induced_complex.h @@ -0,0 +1,130 @@ +/* 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 + * + * @{ + * + * \section complexes Graph induced complexes (GIC) and Nerves + * + * GIC and Nerves are simplicial complexes built on top of a point cloud P. + * + * \subsection nervedefinition Nerve definition + * + * Assume you are given a cover C of your point cloud P, that is a set of subsets of P + * whose union is P itself. 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 . + * + * \subsection nerveexample Example + * + * This example builds the Nerve of a point cloud sampled on a 3D human shape. + * The cover C comes from the preimages of intervals covering the height function. + * All intervals have the resolution (either the length or the number of the intervals) + * and gain (overlap percentage). + * + * \include + * + * When launching: + * + * \code $> + * \endcode + * + * the program output is: + * + * \include + * + * \section 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. + * + * \subsection gicexample Example + * + * This example builds the GIC of a point cloud sampled on a 3D human shape. + * The cover C comes from the preimages of intervals covering the height function, + * and the graph G comes from a Rips complex built with a threshold parameter. + * Note that if the gain is too big, the number of cliques increases a lot, + * which make the computation time much larger. + * + * \include + * + * When launching: + * + * \code $> + * \endcode + * + * the program output is: + * + * \include + * + * \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 ). + * + * \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, + * 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, + * with optimal resolution and gain. Note that optimal threshold, resolution and gain + * also exist for the Nerve of this cover. + * + * \include + * + * When launching: + * + * \code $> + * \endcode + * + * the program output is: + * + * \include + * + * + * \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_ -- cgit v1.2.3