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/* 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 <http://www.gnu.org/licenses/>.
*/
#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 <a target="_blank" href="https://en.wikipedia.org/wiki/Nerve_of_a_covering"> Wikipedia </a>.
*
* \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
* <a target="_blank" href="https://arxiv.org/abs/1511.05823"> this article </a>).
*
* \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_
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