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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): Vincent Rouvreau
*
* Copyright (C) 2018 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_CECH_COMPLEX_INTRO_CECH_COMPLEX_H_
#define DOC_CECH_COMPLEX_INTRO_CECH_COMPLEX_H_
namespace Gudhi {
namespace cech_complex {
/** \defgroup cech_complex Cech complex
*
* \author Clément Maria, Pawel Dlotko, Vincent Rouvreau
*
* @{
*
* \section cechdefinition Cech complex definition
*
* Cech_complex
* <a target="_blank" href="https://en.wikipedia.org/wiki/%C4%8Cech_cohomology">(Wikipedia)</a> is a
* proximity graph that allows to construct a
* <a target="_blank" href="https://en.wikipedia.org/wiki/Simplicial_complex">simplicial complex</a>
* from it.
* The input can be a point cloud with a given distance function.
*
* The filtration value of each edge is computed from a user-given distance function.
*
* All edges that have a filtration value strictly greater than a given threshold value are not inserted into
* the complex.
*
* When creating a simplicial complex from this proximity graph, Cech inserts the proximity graph into the data
* structure, and then expands the simplicial complex when required.
*
* Vertex name correspond to the index of the point in the given range (aka. the point cloud).
*
* \image html "cech_complex_representation.png" "Cech complex proximity graph representation"
*
* On this example, as edges (4,5), (4,6) and (5,6) are in the complex, simplex (4,5,6) is added with the filtration
* value set with \f$max(filtration(4,5), filtration(4,6), filtration(5,6))\f$.
* And so on for simplex (0,1,2,3).
*
* If the Cech_complex interfaces are not detailed enough for your need, please refer to
* <a href="_persistent_cohomology_2cech_persistence_step_by_step_8cpp-example.html">
* cech_persistence_step_by_step.cpp</a> example, where the graph construction over the Simplex_tree is more detailed.
*
* \section cechpointsdistance Point cloud and distance function
*
* \subsection cechpointscloudexample Example from a point cloud and a distance function
*
* This example builds the proximity graph from the given points, threshold value, and distance function.
* Then it creates a `Simplex_tree` with it.
*
* Then, it is asked to display information about the simplicial complex.
*
* \include Cech_complex/cech_complex_example_from_points.cpp
*
* When launching (Cech maximal distance between 2 points is 7.1, is expanded until dimension 2):
*
* \code $> ./Cech_complex_example_from_points
* \endcode
*
* the program output is:
*
* \include Cech_complex/cech_complex_example_from_points_for_doc.txt
*
*/
/** @} */ // end defgroup cech_complex
} // namespace cech_complex
} // namespace Gudhi
#endif // DOC_CECH_COMPLEX_INTRO_CECH_COMPLEX_H_
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