Alpha complex user manual

Definition

Author:Vincent Rouvreau	Introduced in:GUDHI 2.0.0	Copyright:GPL v3
Requires:CGAL \(\geq\) 4.7.0	Eigen3

Alpha complex representation

Alpha_complex is a simplicial complex constructed from the finite cells of a Delaunay Triangulation. 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. This package requires having CGAL version 4.7 or higher (4.8.1 is advised for better performance).

Alpha complex user manual

Alpha complex reference manual

Alpha_complex is constructing a Simplex_tree using Delaunay Triangulation [1] from CGAL (the Computational Geometry Algorithms Library [2]).

Remarks

When Alpha_complex is constructed with an infinite value of \(\alpha\), the complex is a Delaunay complex.

Example from points

This example builds the Delaunay triangulation from the given points, and initializes the alpha complex with it:

import gudhi
alpha_complex = gudhi.AlphaComplex(points=[[1, 1], [7, 0], [4, 6], [9, 6], [0, 14], [2, 19], [9, 17]])

simplex_tree = alpha_complex.create_simplex_tree(max_alpha_square=60.0)
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 output is:

Alpha complex is of dimension 2 - 25 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
[2, 6] -> 36.50
[2, 3, 6] -> 36.50
[2, 4, 6] -> 37.24
[0, 4] -> 59.71
[0, 2, 4] -> 59.71

Algorithm

Data structure

In order to build the alpha complex, first, a Simplex tree is built from the cells of a Delaunay Triangulation. (The filtration value is set to NaN, which stands for unknown value):

Simplex tree structure construction example

Filtration value computation algorithm

for i : dimension \(\rightarrow\) 0 do

for all \(\sigma\) of dimension i

if filtration(\(\sigma\)) is NaN then

filtration(\(\sigma\)) = \(\alpha^2(\sigma)\)

end if

//propagate alpha filtration value

for all \(\tau\) face of \(\sigma\)

if filtration(\(\tau\)) is not NaN then

filtration(\(\tau\)) = filtration(\(\sigma\))

end if

end for

end for

end for

make_filtration_non_decreasing()

prune_above_filtration()

Dimension 2

From the example above, it means the algorithm looks into each triangle ([0,1,2], [0,2,4], [1,2,3], …), computes the filtration value of the triangle, and then propagates the filtration value as described here:

Filtration value propagation example

Dimension 1

Then, the algorithm looks into each edge ([0,1], [0,2], [1,2], …), computes the filtration value of the edge (in this case, propagation will have no effect).

Dimension 0

Finally, the algorithm looks into each vertex ([0], [1], [2], [3], [4], [5] and [6]) and sets the filtration value (0 in case of a vertex - propagation will have no effect).

Non decreasing filtration values

As the squared radii computed by CGAL are an approximation, it might happen that these alpha squared values do not quite define a proper filtration (i.e. non-decreasing with respect to inclusion). We fix that up by calling Simplex_tree::make_filtration_non_decreasing() (cf. C++ version).

Prune above given filtration value

The simplex tree is pruned from the given maximum alpha squared value (cf. Simplex_tree::prune_above_filtration() in the C++ version). In the following example, the value is given by the user as argument of the program.

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.

Then, it is asked to display information about the alpha complex:

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=59.0)
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:

Alpha complex is of dimension 2 - 23 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
[2, 6] -> 36.50
[2, 3, 6] -> 36.50
[2, 4, 6] -> 37.24

CGAL citations

[1]	Samuel Hornus, Olivier Devillers, and Clément Jamin. dD triangulations. In CGAL User and Reference Manual. CGAL Editorial Board, 4.7 edition, 2015. URL: http://doc.cgal.org/4.7/Manual/packages.html#PkgTriangulationsSummary.

[2]	The CGAL Project. CGAL User and Reference Manual. CGAL Editorial Board, 4.7 edition, 2015. URL: http://doc.cgal.org/4.7/Manual/packages.html.