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Diffstat (limited to 'src/Alpha_complex/doc/Intro_alpha_complex.h')
-rw-r--r-- | src/Alpha_complex/doc/Intro_alpha_complex.h | 103 |
1 files changed, 64 insertions, 39 deletions
diff --git a/src/Alpha_complex/doc/Intro_alpha_complex.h b/src/Alpha_complex/doc/Intro_alpha_complex.h index 7a375c9f..ab7c4794 100644 --- a/src/Alpha_complex/doc/Intro_alpha_complex.h +++ b/src/Alpha_complex/doc/Intro_alpha_complex.h @@ -29,34 +29,34 @@ namespace Gudhi { 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. @@ -65,30 +65,30 @@ namespace alpha_complex { * \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 $> ./Alpha_complex_example_from_points * \endcode * * the program output is: - * + * * \include Alpha_complex/alphaoffreader_for_doc_60.txt - * + * * \section createcomplexalgorithm Create complex algorithm - * + * * \subsection datastructure Data structure - * + * * In order to create the simplicial complex, first, it is built from the cells of the Delaunay Triangulation. * The filtration values are set to NaN, which stands for unknown value. - * + * * In example, : * \image html "alpha_complex_doc.png" "Simplicial complex structure construction example" * @@ -106,7 +106,7 @@ namespace alpha_complex { * \quad\quad\quad\quad \text{filtration(} \tau \text{) = min( filtration(} \tau \text{), filtration(} \sigma * \text{) )}\\ * \quad\quad\quad \textbf{else}\\ - * \quad\quad\quad\quad \textbf{if } \textbf{if } \tau \text{ is not Gabriel for } \sigma \textbf{ then}\\ + * \quad\quad\quad\quad \textbf{if } \tau \text{ is not Gabriel for } \sigma \textbf{ then}\\ * \quad\quad\quad\quad\quad \text{filtration(} \tau \text{) = filtration(} \sigma \text{)}\\ * \quad\quad\quad\quad \textbf{end if}\\ * \quad\quad\quad \textbf{end if}\\ @@ -118,53 +118,78 @@ namespace alpha_complex { * \f$ * * \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 |