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author | Vincent Rouvreau <10407034+VincentRouvreau@users.noreply.github.com> | 2020-06-06 14:44:14 +0200 |
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committer | GitHub <noreply@github.com> | 2020-06-06 14:44:14 +0200 |
commit | 434cc66fbf832ca492e5f0710c8afc67d681637a (patch) | |
tree | 08cc98efa6a60aeaa59cf6407d7c4e722dbb212b /src/python/doc/alpha_complex_user.rst | |
parent | d1df63c72af9a239b5dad44d0f8545d24a4a292b (diff) | |
parent | 19dc69e6edaac076da5138fe091be367a3652057 (diff) |
Merge pull request #326 from VincentRouvreau/alpha_complex_python_fast_and_exact
Alpha complex python enhancement
Diffstat (limited to 'src/python/doc/alpha_complex_user.rst')
-rw-r--r-- | src/python/doc/alpha_complex_user.rst | 20 |
1 files changed, 18 insertions, 2 deletions
diff --git a/src/python/doc/alpha_complex_user.rst b/src/python/doc/alpha_complex_user.rst index e31194a7..097470c1 100644 --- a/src/python/doc/alpha_complex_user.rst +++ b/src/python/doc/alpha_complex_user.rst @@ -16,8 +16,24 @@ Definition Remarks ^^^^^^^ -When an :math:`\alpha`-complex is constructed with an infinite value of :math:`\alpha^2`, -the complex is a Delaunay complex (with special filtration values). +* When an :math:`\alpha`-complex is constructed with an infinite value of :math:`\alpha^2`, the complex is a Delaunay + complex (with special filtration values). The Delaunay complex without filtration values is also available by + passing :code:`default_filtration_value = True` to :func:`~gudhi.AlphaComplex.create_simplex_tree`. +* For people only interested in the topology of the Alpha complex (for instance persistence), Alpha complex is + equivalent to the `Čech complex <https://gudhi.inria.fr/doc/latest/group__cech__complex.html>`_ and much smaller if + you do not bound the radii. `Čech complex <https://gudhi.inria.fr/doc/latest/group__cech__complex.html>`_ can still + make sense in higher dimension precisely because you can bound the radii. +* Using the default :code:`precision = 'safe'` makes the construction safe. + If you pass :code:`precision = 'exact'` to :func:`~gudhi.AlphaComplex.__init__`, the filtration values are the exact + ones converted to float. This can be very slow. + If you pass :code:`precision = 'safe'` (the default), the filtration values are only + guaranteed to have a small multiplicative error compared to the exact value. + A drawback, when computing persistence, is that an empty exact interval [10^12,10^12] may become a + non-empty approximate interval [10^12,10^12+10^6]. + Using :code:`precision = 'fast'` makes the computations slightly faster, and the combinatorics are still exact, but + the computation of filtration values can exceptionally be arbitrarily bad. In all cases, we still guarantee that the + output is a valid filtration (faces have a filtration value no larger than their cofaces). +* For performances reasons, it is advised to use Alpha_complex with `CGAL <installation.html#cgal>`_ :math:`\geq` 5.0.0. For performances reasons, it is advised to use CGAL :math:`\geq` 5.0.0. |