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Diffstat (limited to 'src/cython/doc/source/witness_complex_sum.rst')
| -rw-r--r-- | src/cython/doc/source/witness_complex_sum.rst | 26 |
1 files changed, 15 insertions, 11 deletions
diff --git a/src/cython/doc/source/witness_complex_sum.rst b/src/cython/doc/source/witness_complex_sum.rst index a91ff01b..0d65d420 100644 --- a/src/cython/doc/source/witness_complex_sum.rst +++ b/src/cython/doc/source/witness_complex_sum.rst @@ -3,19 +3,23 @@ ===================================== ===================================== ===================================== +---------------------------------------------+----------------------------------------------------------------------+ -| .. image:: | Alpha_complex is a simplicial complex constructed from the finite | -| img/Witness_complex_representation.png | cells of a Delaunay Triangulation. | +| .. image:: | Witness complex :math:`Wit(W,L)` is a simplicial complex defined on | +| img/Witness_complex_representation.png | two sets of points in :math:`\mathbb{R}^D`:Wit(W,L)` is a simplicial | +| | complex defined on two sets of points in :math:`\mathbb{R}^D`: | | | | -| | 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. | +| | * :math:`W` set of **witnesses** and | +| | * :math:`L \subseteq W` set of **landmarks**. | | | | -| | This package requires having CGAL version 4.7 or higher (4.8.1 is | -| | advised for better perfomances). | +| | The simplices are based on landmarks and a simplex belongs to the | +| | witness complex if and only if it is witnessed, that is: | +| | | +| | :math:`\sigma \subset L` is witnessed if there exists a point | +| | :math:`w \in W` such that w is closer to the vertices of | +| | :math:`\sigma` than other points in :math:`L` and all of its faces | +| | are witnessed as well. | +| | | +| | The data structure is described in | +| | :cite:`boissonnatmariasimplextreealgorithmica`. | +---------------------------------------------+----------------------------------------------------------------------+ | :doc:`witness_complex_user` | :doc:`witness_complex_ref` | +---------------------------------------------+----------------------------------------------------------------------+ |