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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):       Clément Maria, Pawel Dlotko, Vincent Rouvreau
+ *
+ *    Copyright (C) 2016  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_GRAPH_INDUCED_COMPLEX_INTRO_GRAPH_INDUCED_COMPLEX_H_
+#define DOC_GRAPH_INDUCED_COMPLEX_INTRO_GRAPH_INDUCED_COMPLEX_H_
+
+namespace Gudhi {
+
+namespace graph_induced_complex {
+
+/**  \defgroup graph_induced_complex Graph induced complex
+ * 
+ * \author    Mathieu Carrière
+ * 
+ * @{
+ * 
+ * Visualizations of the simplicial complexes require neato, python and firefox!!
+ *
+ * \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 first three lines are requirements for visualization with Kepler-Mapper.
+ * 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.
+ *
+ *
+ * \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 <a target="_blank" href="https://arxiv.org/abs/1304.0662"> this article </a>
+ * for more details.
+ *
+ * \image html "gic_complex.png" "GIC of a point cloud."
+ *
+ * \subsection gicexample Example with cover from function
+ *
+ * This example builds the GIC of a point cloud sampled on a 3D human shape (human.off).
+ * The cover C comes from the preimages of intervals (with length 0.075 and gain 0)
+ * covering the height function (coordinate 2),
+ * and the graph G comes from a Rips complex built with threshold 0.075.
+ * Note that if the gain is too big, the number of cliques increases a lot,
+ * which make the computation time much larger.
+ *
+ * \include Nerve_GIC/GIC.cpp
+ *
+ * When launching:
+ *
+ * \code $> ./GIC ../../../data/points/human.off 0.075 2 0.075 0 --v
+ * \endcode
+ *
+ * the program output is:
+ *
+ * \include Nerve_GIC/GIC.txt
+ *
+ * \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.
+ *
+ * \include Nerve_GIC/GICvoronoi.cpp
+ *
+ * When launching:
+ *
+ * \code $> ./GICvoronoi ../../../data/points/human.off 100 --v
+ * \endcode
+ *
+ * the program output is:
+ *
+ * \include Nerve_GIC/GICvoronoi.txt
+ *
+ * \subsection mapperdeltadefinition Mapper Delta
+ *
+ * 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 Mapper Delta, since it is related to the usual Mapper. See
+ * <a target="_blank" href="https://arxiv.org/abs/1511.05823"> this article </a> for more details.
+ *
+ * \subsection mapperdeltaexample Example
+ *
+ * Mapper Delta comes with optimal selection for the Rips threshold,
+ * the resolution and the gain of the function cover. In this example,
+ * we compute the Mapper Delta of a point cloud sampled on a 3D human shape (human.off),
+ * where the graph G comes from a Rips complex with optimal threshold,
+ * and the cover C comes from the preimages of intervals covering the height function (coordinate 2),
+ * with optimal resolution and gain. Note that optimal threshold, resolution and gain
+ * also exist for the Nerve of this cover.
+ *
+ * \include Nerve_GIC/MapperDeltaCoord.cpp
+ *
+ * When launching:
+ *
+ * \code $> ./MapperDeltaCoord ../../../data/points/human.off 2 --v
+ * \endcode
+ *
+ * the program output is:
+ *
+ * \include MapperDeltaCoord.txt
+ *
+ * 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/MapperDeltaFunc.cpp
+ *
+ * When launching:
+ *
+ * \code $> ./MapperDeltaFunc ../../../data/points/COIL_database/lucky_cat.off ../../../data/points/COIL_database/lucky_cat_PCA1 --v
+ * \endcode
+ *
+ * the program output is:
+ *
+ * \include MapperDeltaFunc.txt
+ *
+ * \copyright GNU General Public License v3.                         
+ * \verbatim  Contact: gudhi-users@lists.gforge.inria.fr \endverbatim
+ */
+/** @} */  // end defgroup graph_induced_complex
+
+}  // namespace graph_induced_complex
+
+}  // namespace Gudhi
+
+#endif  // DOC_GRAPH_INDUCED_COMPLEX_INTRO_GRAPH_INDUCED_COMPLEX_H_