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Diffstat (limited to 'src/python/doc/alpha_complex_user.rst')
| -rw-r--r-- | src/python/doc/alpha_complex_user.rst | 71 |
1 files changed, 31 insertions, 40 deletions
diff --git a/src/python/doc/alpha_complex_user.rst b/src/python/doc/alpha_complex_user.rst index d49f45b4..fffcb3db 100644 --- a/src/python/doc/alpha_complex_user.rst +++ b/src/python/doc/alpha_complex_user.rst @@ -9,15 +9,33 @@ Definition .. include:: alpha_complex_sum.inc -`AlphaComplex` is constructing a :doc:`SimplexTree <simplex_tree_ref>` using +:doc:`AlphaComplex <alpha_complex_ref>` is constructing a :doc:`SimplexTree <simplex_tree_ref>` using `Delaunay Triangulation <http://doc.cgal.org/latest/Triangulation/index.html#Chapter_Triangulations>`_ :cite:`cgal:hdj-t-19b` from the `Computational Geometry Algorithms Library <http://www.cgal.org/>`_ :cite:`cgal:eb-19b`. 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. +* The vertices in the output simplex tree are not guaranteed to match the order of the input points. One can use + :func:`~gudhi.AlphaComplex.get_point` to get the initial point back. Example from points ------------------- @@ -160,49 +178,22 @@ In the following example, a threshold of :math:`\alpha^2 = 32.0` is used. Example from OFF file ^^^^^^^^^^^^^^^^^^^^^ -This example builds the Delaunay triangulation from the points given by an OFF file, and initializes the alpha complex -with it. +This example builds the alpha complex from 300 random points on a 2-torus. +Then, it computes the persistence diagram and displays it: -Then, it is asked to display information about the alpha complex: - -.. testcode:: +.. plot:: + :include-source: + import matplotlib.pyplot as plt import gudhi alpha_complex = gudhi.AlphaComplex(off_file=gudhi.__root_source_dir__ + \ - '/data/points/alphacomplexdoc.off') - simplex_tree = alpha_complex.create_simplex_tree(max_alpha_square=32.0) + '/data/points/tore3D_300.off') + simplex_tree = alpha_complex.create_simplex_tree() result_str = 'Alpha complex is of dimension ' + repr(simplex_tree.dimension()) + ' - ' + \ repr(simplex_tree.num_simplices()) + ' simplices - ' + \ repr(simplex_tree.num_vertices()) + ' vertices.' print(result_str) - fmt = '%s -> %.2f' - for filtered_value in simplex_tree.get_filtration(): - print(fmt % tuple(filtered_value)) - -the program output is: - -.. testoutput:: - - Alpha complex is of dimension 2 - 20 simplices - 7 vertices. - [0] -> 0.00 - [1] -> 0.00 - [2] -> 0.00 - [3] -> 0.00 - [4] -> 0.00 - [5] -> 0.00 - [6] -> 0.00 - [2, 3] -> 6.25 - [4, 5] -> 7.25 - [0, 2] -> 8.50 - [0, 1] -> 9.25 - [1, 3] -> 10.00 - [1, 2] -> 11.25 - [1, 2, 3] -> 12.50 - [0, 1, 2] -> 13.00 - [5, 6] -> 13.25 - [2, 4] -> 20.00 - [4, 6] -> 22.74 - [4, 5, 6] -> 22.74 - [3, 6] -> 30.25 - + diag = simplex_tree.persistence() + gudhi.plot_persistence_diagram(diag) + plt.show() |