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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 | 91 |
1 files changed, 67 insertions, 24 deletions
diff --git a/src/Alpha_complex/doc/Intro_alpha_complex.h b/src/Alpha_complex/doc/Intro_alpha_complex.h index 1fb8fdee..685a4c2f 100644 --- a/src/Alpha_complex/doc/Intro_alpha_complex.h +++ b/src/Alpha_complex/doc/Intro_alpha_complex.h @@ -36,48 +36,58 @@ namespace alphacomplex { * * \section definition Definition * - * Alpha_complex is a Simplex_tree constructed from each finite cell of a Delaunay Triangulation. + * Alpha_complex is a <a target="_blank" href="https://en.wikipedia.org/wiki/Simplicial_complex">simplicial complex</a> + * constructed from each finite cell of a Delaunay Triangulation. * * The filtration value of each simplex is computed from the alpha square value of the simplex if it is Gabriel or * from the alpha value of the simplex coface that makes the simplex not Gabriel. * - * Please refer to \cite AlphaShapesDefinition for a more complete alpha complex definition. + * All simplices that have a filtration value strictly greater than a given alpha square value are not inserted into + * the simplex. * - * Alpha complex are interesting because it looks like an \ref alpha-shape "Alpha shape" as described in - * \cite AlphaShapesIntroduction (an alpha complex concept vulgarization). + * \image html "alpha_complex_representation.png" "Alpha simplicial complex representation" * - * \section example Example + * Alpha_complex is constructing a `Simplex_tree` using <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). * - * This example loads points from an OFF file, builds the Delaunay triangulation from the points, and finally - * initialize the alpha complex with it. + * The complex is a template class requiring a <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. + * + * \section pointsexample Example from points + * + * This example builds the Delaunay triangulation from the given points in a 2D static kernel, and initializes the + * alpha complex with it. * * Then, it is asked to display information about the alpha complex. * - * \include Alpha_complex_from_off.cpp + * \include Alpha_complex_from_points.cpp * * When launching: * - * \code $> ./alphaoffreader ../../data/points/alphacomplexdoc.off 60.0 + * \code $> ./alphapoints 60.0 * \endcode * * the program output is: * - * \include alphaoffreader_for_doc.txt + * \include alphaoffreader_for_doc_60.txt * * \section algorithm Algorithm * - * <b>Data structure</b> + * \subsection datastructure Data structure * * In order to build the alpha complex, first, a Simplex tree is build from the cells of a Delaunay Triangulation. * (The filtration value is set to NaN, which stands for unknown value): * \image html "alpha_complex_doc.png" "Simplex tree structure construction example" * - * <b>Filtration value computation algorithm</b> - * + * \subsection filtrationcomputation Filtration value computation algorithm + * * \f{algorithm}{ * \caption{Filtration value computation algorithm}\label{alpha} * \begin{algorithmic} - * \For{i : dimension $\rightarrow$ 1} + * \For{i : dimension $\rightarrow$ 0} * \ForAll{$\sigma$ of dimension i} * \If {filtration($\sigma$) is NaN} * \State filtration($\sigma$) = $\alpha^2(\sigma)$ @@ -93,25 +103,58 @@ namespace alphacomplex { * \EndFor * \EndFor * \EndFor + * \State make\_filtration\_non\_decreasing() + * \State prune\_above\_filtration() * \end{algorithmic} * \f} * - * From the example above, it means the algorithm will look into each triangle ([1,2,3], [2,3,4], [1,3,5], ...), - * will compute the filtration value of the triangle, and then will propagate the filtration value as described + * \subsubsection dimension2 Dimension 2 + * + * From the example above, it means the algorithm looks into each triangle ([1,2,3], [2,3,4], [1,3,5], ...), + * computes the filtration value of the triangle, and then propagates the filtration value as described * here : * \image html "alpha_complex_doc_135.png" "Filtration value propagation example" - * Then, the algorithm will look into each edge ([1,2], [2,3], [1,3], ...), - * will compute the filtration value of the edge (in this case, propagation will have no effect). * - * Finally, the algorithm will look into each vertex ([1], [2], [3], [4], [5], [6] and [7]), - * will set the filtration value (0 in case of a vertex - propagation will have no effect). + * \subsubsection dimension1 Dimension 1 + * + * Then, the algorithm looks into each edge ([1,2], [2,3], [1,3], ...), + * 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 ([1], [2], [3], [4], [5], [6] and [7]) and + * sets the filtration value (0 in case of a vertex - propagation will have no effect). + * + * \subsubsection nondecreasing Non decreasing filtration values + * + * As Alpha square value computed from CGAL is an approximation, we have to make filtration non decreasing while + * increasing the dimension for our simplicial complex to be valid (cf. + * `Simplex_tree::make_filtration_non_decreasing()`). + * + * \subsubsection pruneabove Prune above given filtration value + * + * The simplex tree is pruned from the given maximum alpha square value (cf. `Simplex_tree::prune_above_filtration()`). + * In this example, the value is given by the user as argument of the program. * - * \section alpha-shape Alpha shape * - * In the example above, the alpha shape of \f$\alpha^2_{63} < \alpha^2 < \alpha^2_{62}\f$ is the alpha complex where the - * \f$\alpha^2_{63} <\f$ filtration value \f$< \alpha^2_{62}\f$ as described in \cite AlphaShapesIntroduction + * \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_from_off.cpp + * + * When launching: + * + * \code $> ./alphaoffreader ../../data/points/alphacomplexdoc.off 32.0 + * \endcode + * + * the program output is: + * + * \include alphaoffreader_for_doc_32.txt * - * \image html "alpha_complex_doc_alpha_shape.png" "Alpha shape example" * \copyright GNU General Public License v3. * \verbatim Contact: gudhi-users@lists.gforge.inria.fr \endverbatim */ |