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-rw-r--r--src/Rips_complex/doc/Intro_rips_complex.h240
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diff --git a/src/Rips_complex/doc/COPYRIGHT b/src/Rips_complex/doc/COPYRIGHT
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+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/Rips_complex/doc/Intro_rips_complex.h b/src/Rips_complex/doc/Intro_rips_complex.h
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+/* 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): Clément Maria, Pawel Dlotko, Vincent Rouvreau
+ *
+ * Copyright (C) 2016 Inria
+ *
+ * Modification(s):
+ * - YYYY/MM Author: Description of the modification
+ */
+
+#ifndef DOC_RIPS_COMPLEX_INTRO_RIPS_COMPLEX_H_
+#define DOC_RIPS_COMPLEX_INTRO_RIPS_COMPLEX_H_
+
+namespace Gudhi {
+
+namespace rips_complex {
+
+/** \defgroup rips_complex Rips complex
+ *
+ * \author Clément Maria, Pawel Dlotko, Vincent Rouvreau, Marc Glisse
+ *
+ * @{
+ *
+ * \section ripsdefinition Rips complex definition
+ *
+ * The Vietoris-Rips complex
+ * <a target="_blank" href="https://en.wikipedia.org/wiki/Vietoris%E2%80%93Rips_complex">(Wikipedia)</a>
+ * is an abstract simplicial complex
+ * defined on a finite metric space, where each simplex corresponds to a subset
+ * of points whose diameter is smaller that some given threshold.
+ * Varying the threshold, we can also see the Rips complex as a filtration of
+ * the \f$(n-1)-\f$dimensional simplex, where the filtration value of each
+ * simplex is the diameter of the corresponding subset of points.
+ *
+ * This filtered complex is most often used as an approximation of the
+ * Čech complex. After rescaling (Rips using the length of the edges and Čech
+ * the half-length), they share the same 1-skeleton and are multiplicatively
+ * 2-interleaved or better. While it is slightly bigger, it is also much
+ * easier to compute.
+ *
+ * The number of simplices in the full Rips complex is exponential in the
+ * number of vertices, it is thus usually restricted, by excluding all the
+ * simplices with filtration value larger than some threshold, and keeping only
+ * the dim_max-skeleton. It may also be a good idea to start by making the
+ * point set sparser, for instance with
+ * `Gudhi::subsampling::sparsify_point_set()`, since small clusters of points
+ * have a disproportionate cost without affecting the persistence diagram much.
+ *
+ * In order to build this complex, the algorithm first builds the graph.
+ * The filtration value of each edge is computed from a user-given distance
+ * function, or directly read from the distance matrix.
+ * In a second step, this graph is inserted in a simplicial complex, which then
+ * gets expanded to a flag complex.
+ *
+ * The input can be given as a range of points and a distance function, or as a
+ * distance matrix.
+ *
+ * Vertex name correspond to the index of the point in the given range (aka. the point cloud).
+ *
+ * \image html "rips_complex_representation.png" "Rips-complex one skeleton 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 Rips_complex interfaces are not detailed enough for your need, please refer to
+ * <a href="_persistent_cohomology_2rips_persistence_step_by_step_8cpp-example.html">
+ * rips_persistence_step_by_step.cpp</a> example, where the constructions of the graph and
+ * the Simplex_tree are more detailed.
+ *
+ * \section sparserips Sparse Rips complex
+ *
+ * Even truncated in filtration value and dimension, the Rips complex remains
+ * quite large. However, it is possible to approximate it by a much smaller
+ * filtered simplicial complex (linear size, with constants that depend on
+ * &epsilon; and the doubling dimension of the space) that is
+ * \f$(1+O(\epsilon))-\f$interleaved with it (in particular, their persistence
+ * diagrams are at log-bottleneck distance at most \f$O(\epsilon)\f$).
+ *
+ * The sparse Rips filtration was introduced by Don Sheehy
+ * \cite sheehy13linear. We are using the version described in
+ * \cite buchet16efficient (except that we multiply all filtration values
+ * by 2, to match the usual Rips complex), for which \cite cavanna15geometric proves a
+ * \f$(1,\frac{1}{1-\epsilon})\f$-interleaving, although in practice the
+ * error is usually smaller.
+ * A more intuitive presentation of the idea is available in
+ * \cite cavanna15geometric, and in a video \cite cavanna15visualizing.
+ *
+ * The interface of `Sparse_rips_complex` is similar to the one for the usual
+ * `Rips_complex`, except that one has to specify the approximation factor.
+ * There is an option to limit the minimum and maximum filtration values, but
+ * they are not recommended: the way the approximation is done means that
+ * larger filtration values are much cheaper to handle than low filtration
+ * values, so the gain in ignoring the large ones is small, and
+ * `Gudhi::subsampling::sparsify_point_set()` is a more efficient way of
+ * ignoring small filtration values.
+ *
+ * Theoretical guarantees are only available for \f$\epsilon<1\f$. The
+ * construction accepts larger values of &epsilon;, and the size of the complex
+ * keeps decreasing, but there is no guarantee on the quality of the result.
+ * Note that while the number of edges decreases when &epsilon; increases, the
+ * number of higher-dimensional simplices may not be monotonous when
+ * \f$\frac12\leq\epsilon\leq 1\f$.
+ *
+ * \section ripspointsdistance Point cloud and distance function
+ *
+ * \subsection ripspointscloudexample Example from a point cloud and a distance function
+ *
+ * This example builds the one skeleton 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 Rips_complex/example_one_skeleton_rips_from_points.cpp
+ *
+ * When launching (Rips maximal distance between 2 points is 12.0, is expanded
+ * until dimension 1 - one skeleton graph in other words):
+ *
+ * \code $> ./Rips_complex_example_one_skeleton_from_points
+ * \endcode
+ *
+ * the program output is:
+ *
+ * \include Rips_complex/one_skeleton_rips_for_doc.txt
+ *
+ * \subsection ripsoffexample Example from OFF file
+ *
+ * This example builds the Rips_complex from the given points in an OFF file, threshold value, and distance
+ * function.
+ * Then it creates a `Simplex_tree` with it.
+ *
+ *
+ * Then, it is asked to display information about the Rips complex.
+ *
+ * \include Rips_complex/example_rips_complex_from_off_file.cpp
+ *
+ * When launching:
+ *
+ * \code $> ./Rips_complex_example_from_off ../../data/points/alphacomplexdoc.off 12.0 3
+ * \endcode
+ *
+ * the program output is:
+ *
+ * \include Rips_complex/full_skeleton_rips_for_doc.txt
+ *
+ *
+ * \subsection sparseripspointscloudexample Example of a sparse Rips from a point cloud
+ *
+ * This example builds the full sparse Rips of a set of 2D Euclidean points, then prints some minimal
+ * information about the complex.
+ *
+ * \include Rips_complex/example_sparse_rips.cpp
+ *
+ * When launching:
+ *
+ * \code $> ./Rips_complex_example_sparse
+ * \endcode
+ *
+ * the program output may be (the exact output varies from one run to the next):
+ *
+ * \code Sparse Rips complex is of dimension 2 - 19 simplices - 7 vertices.
+ * \endcode
+ *
+ *
+ *
+ * \section ripsdistancematrix Distance matrix
+ *
+ * \subsection ripsdistancematrixexample Example from a distance matrix
+ *
+ * This example builds the one skeleton graph from the given distance matrix and threshold value.
+ * Then it creates a `Simplex_tree` with it.
+ *
+ * Then, it is asked to display information about the simplicial complex.
+ *
+ * \include Rips_complex/example_one_skeleton_rips_from_distance_matrix.cpp
+ *
+ * When launching (Rips maximal distance between 2 points is 1.0, is expanded until dimension 1 - one skeleton graph
+ * with other words):
+ *
+ * \code $> ./Rips_complex_example_one_skeleton_from_distance_matrix
+ * \endcode
+ *
+ * the program output is:
+ *
+ * \include Rips_complex/one_skeleton_rips_for_doc.txt
+ *
+ * \subsection ripscsvdistanceexample Example from a distance matrix read in a csv file
+ *
+ * This example builds the one skeleton graph from the given distance matrix read in a csv file and threshold value.
+ * Then it creates a `Simplex_tree` with it.
+ *
+ *
+ * Then, it is asked to display information about the Rips complex.
+ *
+ * \include Rips_complex/example_rips_complex_from_csv_distance_matrix_file.cpp
+ *
+ * When launching:
+ *
+ * \code $> ./Rips_complex_example_from_csv_distance_matrix ../../data/distance_matrix/full_square_distance_matrix.csv 1.0 3
+ * \endcode
+ *
+ * the program output is:
+ *
+ * \include Rips_complex/full_skeleton_rips_for_doc.txt
+ *
+ *
+ * \section ripscorrelationematrix Correlation matrix
+ *
+ * Analogously to the case of distance matrix, Rips complexes can be also constructed based on correlation matrix.
+ * Given a correlation matrix M, comportment-wise 1-M is a distance matrix.
+ * This example builds the one skeleton graph from the given corelation matrix and threshold value.
+ * Then it creates a `Simplex_tree` with it.
+ *
+ * Then, it is asked to display information about the simplicial complex.
+ *
+ * \include Rips_complex/example_one_skeleton_rips_from_correlation_matrix.cpp
+ *
+ * When launching:
+ *
+ * \code $> ./example_one_skeleton_from_correlation_matrix
+ * \endcode
+ *
+ * the program output is:
+ *
+ * \include Rips_complex/one_skeleton_rips_from_correlation_matrix_for_doc.txt
+ *
+ * All the other constructions discussed for Rips complex for distance matrix can be also performed for Rips complexes
+ * construction from correlation matrices.
+ *
+ * @warning As persistence diagrams points will be under the diagonal, bottleneck distance and persistence graphical
+ * tool will not work properly, this is a known issue.
+ *
+ */
+/** @} */ // end defgroup rips_complex
+
+} // namespace rips_complex
+
+} // namespace Gudhi
+
+#endif // DOC_RIPS_COMPLEX_INTRO_RIPS_COMPLEX_H_
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