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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): Vincent Rouvreau
*
* Copyright (C) 2015 INRIA Saclay (France)
*
* 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 INTRO_ALPHA_COMPLEX_H_
#define INTRO_ALPHA_COMPLEX_H_
// needs namespace for Doxygen to link on classes
namespace Gudhi {
// needs namespace for Doxygen to link on classes
namespace alphacomplex {
/** \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 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.
*
* All simplices that have a filtration value strictly greater than a given alpha square value are not inserted into
* the simplex.
*
* \image html "alpha_complex_representation.png" "Alpha simplicial complex representation"
*
* 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).
*
* 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_points.cpp
*
* When launching:
*
* \code $> ./alphapoints 60.0
* \endcode
*
* the program output is:
*
* \include alphaoffreader_for_doc_60.txt
*
* \section algorithm Algorithm
*
* \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"
*
* \subsection filtrationcomputation Filtration value computation algorithm
*
* \f{algorithm}{
* \caption{Filtration value computation algorithm}\label{alpha}
* \begin{algorithmic}
* \For{i : dimension $\rightarrow$ 0}
* \ForAll{$\sigma$ of dimension i}
* \If {filtration($\sigma$) is NaN}
* \State filtration($\sigma$) = $\alpha^2(\sigma)$
* \EndIf
* \ForAll{$\tau$ face of $\sigma$} \Comment{propagate alpha filtration value}
* \If {filtration($\tau$) is not NaN}
* \State filtration($\tau$) = min (filtration($\tau$), filtration($\sigma$))
* \Else
* \If {$\tau$ is not Gabriel for $\sigma$}
* \State filtration($\tau$) = filtration($\sigma$)
* \EndIf
* \EndIf
* \EndFor
* \EndFor
* \EndFor
* \State make\_filtration\_non\_decreasing()
* \State prune\_above\_filtration()
* \end{algorithmic}
* \f}
*
* \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"
*
* \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 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
*
* \copyright GNU General Public License v3.
* \verbatim Contact: gudhi-users@lists.gforge.inria.fr \endverbatim
*/
/** @} */ // end defgroup alpha_complex
} // namespace alphacomplex
} // namespace Gudhi
#endif // INTRO_ALPHA_COMPLEX_H_
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