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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):       Mathieu Carriere
+ *
+ *    Copyright (C) 2017 Inria
+ *
+ *    Modification(s):
+ *      - YYYY/MM Author: Description of the modification
+ */
+
+#ifndef DOC_COVER_COMPLEX_INTRO_COVER_COMPLEX_H_
+#define DOC_COVER_COMPLEX_INTRO_COVER_COMPLEX_H_
+
+namespace Gudhi {
+
+namespace cover_complex {
+
+/**  \defgroup cover_complex Cover complex
+ * 
+ * \author    Mathieu Carrière
+ * 
+ * @{
+ * 
+ * Visualizations of the simplicial complexes can be done with either
+ * neato (from <a target="_blank" href="http://www.graphviz.org/">graphviz</a>),
+ * <a target="_blank" href="http://www.geomview.org/">geomview</a>,
+ * <a target="_blank" href="https://github.com/MLWave/kepler-mapper">KeplerMapper</a>.
+ * Input point clouds are assumed to be \ref FileFormatsOFF "OFF files"
+ *
+ * \section covers Covers
+ *
+ * Nerves and Graph Induced Complexes require a cover C of the input point cloud P,
+ * that is a set of subsets of P whose union is P itself.
+ * Very often, this cover is obtained from the preimage of a family of intervals covering
+ * the image of some scalar-valued function f defined on P. This family is parameterized
+ * by its resolution, which can be either the number or the length of the intervals,
+ * and its gain, which is the overlap percentage between consecutive intervals (ordered by their first values).
+ *
+ * \section nerves Nerves
+ *
+ * \subsection nervedefinition Nerve definition
+ *
+ * Assume you are given a cover C of your point cloud P. Then, the Nerve of this cover
+ * is the simplicial complex that has one k-simplex per k-fold intersection of cover elements.
+ * See also <a target="_blank" href="https://en.wikipedia.org/wiki/Nerve_of_a_covering"> Wikipedia </a>.
+ *
+ * \image html "nerve.png" "Nerve of a double torus"
+ *
+ * \subsection nerveexample Example
+ *
+ * This example builds the Nerve of a point cloud sampled on a 3D human shape (human.off).
+ * The cover C comes from the preimages of intervals (10 intervals with gain 0.3)
+ * covering the height function (coordinate 2),
+ * which are then refined into their connected components using the triangulation of the .OFF file.
+ *
+ * \include Nerve_GIC/Nerve.cpp
+ *
+ * When launching:
+ *
+ * \code $> ./Nerve ../../data/points/human.off 2 10 0.3 -v
+ * \endcode
+ *
+ * the program output is:
+ *
+ * \include Nerve_GIC/Nerve.txt
+ *
+ * The program also writes a file ../../data/points/human_sc.txt. The first three lines in this file are the location
+ * of the input point cloud and the function used to compute the cover.
+ * The fourth line contains the number of vertices nv and edges ne of the Nerve.
+ * The next nv lines represent the vertices. Each line contains the vertex ID,
+ * the number of data points it contains, and their average color function value.
+ * Finally, the next ne lines represent the edges, characterized by the ID of their vertices.
+ *
+ * Using KeplerMapper, one can obtain the following visualization:
+ *
+ * \image html "nervevisu.jpg" "Visualization with KeplerMapper"
+ *
+ * \section gic Graph Induced Complexes (GIC)
+ *
+ * \subsection gicdefinition GIC definition
+ *
+ * Again, assume you are given a cover C of your point cloud P. Moreover, assume
+ * you are also given a graph G built on top of P. Then, for any clique in G
+ * whose nodes all belong to different elements of C, the GIC includes a corresponding
+ * simplex, whose dimension is the number of nodes in the clique minus one.
+ * See \cite Dey13 for more details.
+ *
+ * \image html "GIC.jpg" "GIC of a point cloud."
+ *
+ * \subsection gicexamplevor Example with cover from Voronoï
+ *
+ * This example builds the GIC of a point cloud sampled on a 3D human shape (human.off).
+ * We randomly subsampled 100 points in the point cloud, which act as seeds of
+ * a geodesic Voronoï diagram. Each cell of the diagram is then an element of C.
+ * The graph G (used to compute both the geodesics for Voronoï and the GIC)
+ * comes from the triangulation of the human shape. Note that the resulting simplicial complex is in dimension 3
+ * in this example.
+ *
+ * \include Nerve_GIC/VoronoiGIC.cpp
+ *
+ * When launching:
+ *
+ * \code $> ./VoronoiGIC ../../data/points/human.off 700 -v
+ * \endcode
+ *
+ * the program outputs SC.off. Using e.g.
+ *
+ * \code $> geomview ../../data/points/human_sc.off
+ * \endcode
+ *
+ * one can obtain the following visualization:
+ *
+ * \image html "gicvoronoivisu.jpg" "Visualization with Geomview"
+ *
+ * \subsection functionalGICdefinition Functional GIC
+ *
+ * If one restricts to the cliques in G whose nodes all belong to preimages of consecutive
+ * intervals (assuming the cover of the height function is minimal, i.e. no more than
+ * two intervals can intersect at a time), the GIC is of dimension one, i.e. a graph.
+ * We call this graph the functional GIC. See \cite Carriere16 for more details.
+ *
+ * \subsection functionalGICexample Example
+ *
+ * Functional GIC comes with automatic selection of the Rips threshold,
+ * the resolution and the gain of the function cover. See \cite Carriere17c for more details. In this example,
+ * we compute the functional GIC of a Klein bottle embedded in R^5,
+ * where the graph G comes from a Rips complex with automatic threshold,
+ * and the cover C comes from the preimages of intervals covering the first coordinate,
+ * with automatic resolution and gain. Note that automatic threshold, resolution and gain
+ * can be computed as well for the Nerve.
+ *
+ * \include Nerve_GIC/CoordGIC.cpp
+ *
+ * When launching:
+ *
+ * \code $> ./CoordGIC ../../data/points/KleinBottle5D.off 0 -v
+ * \endcode
+ *
+ * the program outputs SC.dot. Using e.g.
+ *
+ * \code $> neato SC.dot -Tpdf -o SC.pdf
+ * \endcode
+ *
+ * one can obtain the following visualization:
+ *
+ * \image html "coordGICvisu2.jpg" "Visualization with Neato"
+ *
+ * where nodes are colored by the filter function values and, for each node, the first number is its ID
+ * and the second is the number of data points that its contain.
+ *
+ * We also provide an example on a set of 72 pictures taken around the same object (lucky_cat.off).
+ * The function is now the first eigenfunction given by PCA, whose values
+ * are written in a file (lucky_cat_PCA1). Threshold, resolution and gain are automatically selected as before.
+ *
+ * \include Nerve_GIC/FuncGIC.cpp
+ *
+ * When launching:
+ *
+ * \code $> ./FuncGIC ../../data/points/COIL_database/lucky_cat.off ../../data/points/COIL_database/lucky_cat_PCA1 -v
+ * \endcode
+ *
+ * the program outputs again SC.dot which gives the following visualization after using neato:
+ *
+ * \image html "funcGICvisu.jpg" "Visualization with neato"
+ *
+ */
+/** @} */  // end defgroup cover_complex
+
+}  // namespace cover_complex
+
+}  // namespace Gudhi
+
+#endif  // DOC_COVER_COMPLEX_INTRO_COVER_COMPLEX_H_