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| -rw-r--r-- | src/Cech_complex/doc/COPYRIGHT | 12 | ||||
| -rw-r--r-- | src/Cech_complex/doc/Intro_cech_complex.h | 102 | ||||
| -rw-r--r-- | src/Cech_complex/doc/cech_complex_representation.ipe | 330 | ||||
| -rw-r--r-- | src/Cech_complex/doc/cech_complex_representation.png | bin | 0 -> 39938 bytes | |||
| -rw-r--r-- | src/Cech_complex/doc/cech_one_skeleton.ipe | 314 | ||||
| -rw-r--r-- | src/Cech_complex/doc/cech_one_skeleton.png | bin | 0 -> 24662 bytes |
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diff --git a/src/Cech_complex/doc/COPYRIGHT b/src/Cech_complex/doc/COPYRIGHT new file mode 100644 index 00000000..61f17f6d --- /dev/null +++ b/src/Cech_complex/doc/COPYRIGHT @@ -0,0 +1,12 @@ +The files of this directory are part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT. +See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details. + +Author(s): Vincent Rouvreau + +Copyright (C) 2015 Inria + +This gives everyone the freedoms to use openFrameworks in any context: +commercial or non-commercial, public or private, open or closed source. + +You should have received a copy of the MIT License along with this program. +If not, see https://opensource.org/licenses/MIT.
\ No newline at end of file diff --git a/src/Cech_complex/doc/Intro_cech_complex.h b/src/Cech_complex/doc/Intro_cech_complex.h new file mode 100644 index 00000000..90086de7 --- /dev/null +++ b/src/Cech_complex/doc/Intro_cech_complex.h @@ -0,0 +1,102 @@ +/* This file is part of the Gudhi Library - https://gudhi.inria.fr/ - which is released under MIT. + * See file LICENSE or go to https://gudhi.inria.fr/licensing/ for full license details. + * Author(s): Vincent Rouvreau + * + * Copyright (C) 2018 Inria + * + * Modification(s): + * - YYYY/MM Author: Description of the modification + */ + +#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_ diff --git a/src/Cech_complex/doc/cech_complex_representation.ipe b/src/Cech_complex/doc/cech_complex_representation.ipe new file mode 100644 index 00000000..377745a3 --- /dev/null +++ b/src/Cech_complex/doc/cech_complex_representation.ipe @@ -0,0 +1,330 @@ +<?xml version="1.0"?> +<!DOCTYPE ipe SYSTEM "ipe.dtd"> +<ipe version="70107" creator="Ipe 7.1.10"> +<info created="D:20150603143945" modified="D:20180305162524"/> +<ipestyle name="basic"> +<symbol name="arrow/arc(spx)"> +<path 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