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Diffstat (limited to 'src/Nerve_GIC/doc/Intro_graph_induced_complex.h')
-rw-r--r-- | src/Nerve_GIC/doc/Intro_graph_induced_complex.h | 25 |
1 files changed, 21 insertions, 4 deletions
diff --git a/src/Nerve_GIC/doc/Intro_graph_induced_complex.h b/src/Nerve_GIC/doc/Intro_graph_induced_complex.h index 8c31b262..f30d51a4 100644 --- a/src/Nerve_GIC/doc/Intro_graph_induced_complex.h +++ b/src/Nerve_GIC/doc/Intro_graph_induced_complex.h @@ -33,23 +33,31 @@ namespace graph_induced_complex { * * @{ * + * \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, that is a set of subsets of P - * whose union is P itself. Then, the Nerve of this cover + * 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. - * All intervals have the resolution (either the length or the number of the intervals) - * and gain (overlap percentage). * * \include Nerve_GIC/Nerve.cpp * @@ -62,6 +70,13 @@ namespace graph_induced_complex { * * \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 @@ -73,6 +88,8 @@ namespace graph_induced_complex { * 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 * * This example builds the GIC of a point cloud sampled on a 3D human shape (human.off). |