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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 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_