diff options
Diffstat (limited to 'doc/Cech_complex/Intro_cech_complex.h')
-rw-r--r-- | doc/Cech_complex/Intro_cech_complex.h | 114 |
1 files changed, 0 insertions, 114 deletions
diff --git a/doc/Cech_complex/Intro_cech_complex.h b/doc/Cech_complex/Intro_cech_complex.h deleted file mode 100644 index 4483bcb9..00000000 --- a/doc/Cech_complex/Intro_cech_complex.h +++ /dev/null @@ -1,114 +0,0 @@ -/* 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 Čech complex - * - * \author Vincent Rouvreau - * - * @{ - * - * \section cechdefinition Čech complex definition - * - * Čech complex - * <a target="_blank" href="https://en.wikipedia.org/wiki/%C4%8Cech_cohomology">(Wikipedia)</a> is a - * <a target="_blank" href="https://en.wikipedia.org/wiki/Simplicial_complex">simplicial complex</a> constructed - * from a proximity graph. The set of all simplices is filtered by the radius of their minimal enclosing ball. - * - * The input shall be a point cloud in an Euclidean space. - * - * \remark For people only interested in the topology of the \ref cech_complex (for instance persistence), - * \ref alpha_complex is equivalent to the \ref cech_complex and much smaller if you do not bound the radii. - * \ref cech_complex can still make sense in higher dimension precisely because you can bound the radii. - * - * \subsection cechalgorithm Algorithm - * - * Cech_complex first builds a proximity graph from a point cloud. - * The filtration value of each edge of the `Gudhi::Proximity_graph` is computed from - * `Gudhi::Minimal_enclosing_ball_radius` function. - * - * All edges that have a filtration value strictly greater than a user given maximal radius value, \f$max\_radius\f$, - * are not inserted into the complex. - * - * Vertex name correspond to the index of the point in the given range (aka. the point cloud). - * - * \image html "cech_one_skeleton.png" "Čech complex proximity graph representation" - * - * When creating a simplicial complex from this proximity graph, Cech_complex inserts the proximity graph into the - * simplicial complex data structure, and then expands the simplicial complex when required. - * - * On this example, as edges \f$(x,y)\f$, \f$(y,z)\f$ and \f$(z,y)\f$ are in the complex, the minimal ball radius - * containing the points \f$(x,y,z)\f$ is computed. - * - * \f$(x,y,z)\f$ is inserted to the simplicial complex with the filtration value set with - * \f$mini\_ball\_radius(x,y,z))\f$ iff \f$mini\_ball\_radius(x,y,z)) \leq max\_radius\f$. - * - * And so on for higher dimensions. - * - * \image html "cech_complex_representation.png" "Čech complex expansion" - * - * The minimal ball radius computation is insured by - * <a target="_blank" href="https://people.inf.ethz.ch/gaertner/subdir/software/miniball.html"> - * the miniball software (V3.0)</a> - Smallest Enclosing Balls of Points - and distributed with GUDHI. - * Please refer to - * <a target="_blank" href="https://people.inf.ethz.ch/gaertner/subdir/texts/own_work/esa99_final.pdf"> - * the miniball software design description</a> for more information about this computation. - * - * This radius computation is the reason why the Cech_complex is taking much more time to be computed than the - * \ref rips_complex but it offers more topological guarantees. - * - * If the Cech_complex interfaces are not detailed enough for your need, please refer to - * <a href="_cech_complex_2cech_complex_step_by_step_8cpp-example.html"> - * cech_complex_step_by_step.cpp</a> example, where the graph construction over the Simplex_tree is more detailed. - * - * \subsection cechpointscloudexample Example from a point cloud - * - * This example builds the proximity graph from the given points, and maximal radius values. - * 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 (maximal enclosing ball radius is 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_ |