1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
16
17
18
19
20
21
22
23
24
25
26
27
28
29
30
31
32
33
34
35
36
37
38
39
40
41
42
43
44
45
46
47
48
49
50
51
52
53
54
55
56
57
58
59
60
61
62
63
64
65
66
67
68
69
70
71
72
73
74
75
76
77
78
79
80
81
82
83
84
85
86
87
88
89
90
91
92
93
94
95
96
97
98
99
100
101
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_
|
