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diff --git a/src/Alpha_complex/doc/Intro_alpha_complex.h b/src/Alpha_complex/doc/Intro_alpha_complex.h new file mode 100644 index 00000000..b075d1fc --- /dev/null +++ b/src/Alpha_complex/doc/Intro_alpha_complex.h @@ -0,0 +1,190 @@ +/* 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) 2015 Inria + * + * Modification(s): + * - YYYY/MM Author: Description of the modification + */ + +#ifndef DOC_ALPHA_COMPLEX_INTRO_ALPHA_COMPLEX_H_ +#define DOC_ALPHA_COMPLEX_INTRO_ALPHA_COMPLEX_H_ + +// needs namespace for Doxygen to link on classes +namespace Gudhi { +// needs namespace for Doxygen to link on classes +namespace alpha_complex { + +/** \defgroup alpha_complex Alpha complex + * + * \author Vincent Rouvreau + * + * @{ + * + * \section definition Definition + * + * Alpha_complex is a <a target="_blank" href="https://en.wikipedia.org/wiki/Simplicial_complex">simplicial complex</a> + * constructed from the finite cells of a Delaunay Triangulation. + * + * The filtration value of each simplex is computed as the square of the circumradius of the simplex if the + * circumsphere is empty (the simplex is then said to be Gabriel), and as the minimum of the filtration + * values of the codimension 1 cofaces that make it not Gabriel otherwise. + * + * All simplices that have a filtration value strictly greater than a given alpha squared value are not inserted into + * the complex. + * + * \image html "alpha_complex_representation.png" "Alpha-complex representation" + * + * Alpha_complex is constructing a <a target="_blank" + * href="http://doc.cgal.org/latest/Triangulation/index.html#Chapter_Triangulations">Delaunay Triangulation</a> + * \cite cgal:hdj-t-15b from <a target="_blank" href="http://www.cgal.org/">CGAL</a> (the Computational Geometry + * Algorithms Library \cite cgal:eb-15b) and is able to create a `SimplicialComplexForAlpha`. + * + * The complex is a template class requiring an Epick_d <a target="_blank" + * href="http://doc.cgal.org/latest/Kernel_d/index.html#Chapter_dD_Geometry_Kernel">dD Geometry Kernel</a> + * \cite cgal:s-gkd-15b from CGAL as template parameter. + * + * \remark + * - When the simplicial complex is constructed with an infinite value of alpha, the complex is a Delaunay + * complex. + * - For people only interested in the topology of the \ref alpha_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. + * + * \section pointsexample Example from points + * + * This example builds the Delaunay triangulation from the given points in a 2D static kernel, and creates a + * `Simplex_tree` with it. + * + * Then, it is asked to display information about the simplicial complex. + * + * \include Alpha_complex/Alpha_complex_from_points.cpp + * + * When launching: + * + * \code $> ./Alpha_complex_example_from_points + * \endcode + * + * the program output is: + * + * \include Alpha_complex/alphaoffreader_for_doc_60.txt + * + * \section createcomplexalgorithm Create complex algorithm + * + * \subsection datastructure Data structure + * + * In order to create the simplicial complex, first, it is built from the cells of the Delaunay Triangulation. + * The filtration values are set to NaN, which stands for unknown value. + * + * In example, : + * \image html "alpha_complex_doc.png" "Simplicial complex structure construction example" + * + * \subsection filtrationcomputation Filtration value computation algorithm + * <br> + * \f$ + * \textbf{for } \text{i : dimension } \rightarrow 0 \textbf{ do}\\ + * \quad \textbf{for all } \sigma \text{ of dimension i}\\ + * \quad\quad \textbf{if } \text{filtration(} \sigma ) \text{ is NaN} \textbf{ then}\\ + * \quad\quad\quad \text{filtration(} \sigma ) = \alpha^2( \sigma )\\ + * \quad\quad \textbf{end if}\\ + * \quad\quad \textbf{for all } \tau \text{ face of } \sigma \textbf{ do}\quad\quad + * \textit{// propagate alpha filtration value}\\ + * \quad\quad\quad \textbf{if } \text{filtration(} \tau ) \text{ is not NaN} \textbf{ then}\\ + * \quad\quad\quad\quad \text{filtration(} \tau \text{) = min( filtration(} \tau \text{), filtration(} \sigma + * \text{) )}\\ + * \quad\quad\quad \textbf{else}\\ + * \quad\quad\quad\quad \textbf{if } \tau \text{ is not Gabriel for } \sigma \textbf{ then}\\ + * \quad\quad\quad\quad\quad \text{filtration(} \tau \text{) = filtration(} \sigma \text{)}\\ + * \quad\quad\quad\quad \textbf{end if}\\ + * \quad\quad\quad \textbf{end if}\\ + * \quad\quad \textbf{end for}\\ + * \quad \textbf{end for}\\ + * \textbf{end for}\\ + * \text{make_filtration_non_decreasing()}\\ + * \text{prune_above_filtration()}\\ + * \f$ + * + * \subsubsection dimension2 Dimension 2 + * + * From the example above, it means the algorithm looks into each triangle ([0,1,2], [0,2,4], [1,2,3], ...), + * computes the filtration value of the triangle, and then propagates the filtration value as described + * here : + * \image html "alpha_complex_doc_420.png" "Filtration value propagation example" + * + * \subsubsection dimension1 Dimension 1 + * + * Then, the algorithm looks into each edge ([0,1], [0,2], [1,2], ...), + * computes the filtration value of the edge (in this case, propagation will have no effect). + * + * \subsubsection dimension0 Dimension 0 + * + * Finally, the algorithm looks into each vertex ([0], [1], [2], [3], [4], [5] and [6]) and + * sets the filtration value (0 in case of a vertex - propagation will have no effect). + * + * \subsubsection nondecreasing Non decreasing filtration values + * + * As the squared radii computed by CGAL are an approximation, it might happen that these alpha squared values do not + * quite define a proper filtration (i.e. non-decreasing with respect to inclusion). + * We fix that up by calling `SimplicialComplexForAlpha::make_filtration_non_decreasing()`. + * + * \subsubsection pruneabove Prune above given filtration value + * + * The simplex tree is pruned from the given maximum alpha squared value (cf. + * `SimplicialComplexForAlpha::prune_above_filtration()`). + * In the following example, the value is given by the user as argument of the program. + * + * + * \section offexample Example from OFF file + * + * This example builds the Delaunay triangulation in a dynamic kernel, and initializes the alpha complex with it. + * + * + * Then, it is asked to display information about the alpha complex. + * + * \include Alpha_complex/Alpha_complex_from_off.cpp + * + * When launching: + * + * \code $> ./Alpha_complex_example_from_off ../../data/points/alphacomplexdoc.off 32.0 + * \endcode + * + * the program output is: + * + * \include Alpha_complex/alphaoffreader_for_doc_32.txt + * + * + * \section weighted3dexample 3d specific example + * + * A specific module for Alpha complex is available in 3d (cf. Alpha_complex_3d) and allows to construct standard, + * weighted, periodic or weighted and periodic versions of alpha complexes. Alpha values computation can be + * Gudhi::alpha_complex::complexity::FAST, Gudhi::alpha_complex::complexity::SAFE (default value) or + * Gudhi::alpha_complex::complexity::EXACT. + * + * This example builds the CGAL 3d weighted alpha shapes from a small molecule, and initializes the alpha complex with + * it. This example is taken from <a href="https://doc.cgal.org/latest/Alpha_shapes_3/index.html#title13">CGAL 3d + * weighted alpha shapes</a>. + * + * Then, it is asked to display information about the alpha complex. + * + * \include Alpha_complex/Weighted_alpha_complex_3d_from_points.cpp + * + * When launching: + * + * \code $> ./Alpha_complex_example_weighted_3d_from_points + * \endcode + * + * the program output is: + * + * \include Alpha_complex/weightedalpha3dfrompoints_for_doc.txt + * + */ +/** @} */ // end defgroup alpha_complex + +} // namespace alpha_complex + +namespace alphacomplex = alpha_complex; + +} // namespace Gudhi + +#endif // DOC_ALPHA_COMPLEX_INTRO_ALPHA_COMPLEX_H_ |