Alpha complex

alpha_complex_persistence

This program computes the persistent homology with coefficient field Z/pZ of the dD alpha complex built from a dD point cloud.

Different versions of Alpha complex computation are available:

fast: right combinatorics, values can be arbitrarily bad
safe (default): values can have a relative error at most 1e-5
exact: true values rounded to double.

Default Alpha complex filtrations computation are square of the circumradius of the simplex. If you are interested in the circumradius of the simplex as filtration values, pass the ‘–squared-filtrations off’ (or ‘-s off’) option.

Alpha complex can be, or not, weighted (requires a file containing weights values).

The output diagram contains one bar per line, written with the convention:

   p dim birth death

where dim is the dimension of the homological feature, birth and death are respectively the birth and death of the feature, and p is the characteristic of the field Z/pZ used for homology coefficients (p must be a prime number).

Usage

   alpha_complex_persistence [options] <input OFF file>

where <input OFF file> is the path to the input point cloud in nOFF ASCII format.

Allowed options

-h [ --help ] Produce help message.
-o [ --output-file ] Name of file in which the persistence diagram is written. Default print in standard output.
-r [ --max-alpha-square-value ] (default = inf) Maximal alpha square value for the Alpha complex construction.
-p [ --field-charac ] (default = 11) Characteristic p of the coefficient field Z/pZ for computing homology.
-m [ --min-persistence ] (default = 0) Minimal lifetime of homology feature to be recorded. Enter a negative value to see zero length intervals.
-w [ --weight-file ] is the path to the file containing the weights of the points (one value per line). Default version is not weighted.
-e [ --exact ] for the exact computation version.
-f [ --fast ] for the fast computation version.
-s [ --squared-filtrations ] to activate square filtration computations (default is ‘on’, can be ‘off’).

Example

   alpha_complex_persistence -p 2 -m 0.45 ../../data/points/tore3D_300.off

N.B.:

Filtration values are alpha square values.
Weights values are explained on CGAL dD Triangulations and Regular triangulation documentation. In this case, the filtration value of each simplex is computed as the power distance of the smallest power sphere passing through all of its vertices. Weighted Alpha complex can have negative filtration values. This is the reason why ‘-s off’ or ‘–square-root-filtrations off’ is ignored in this case (filtration values would be NaN in this case).

alpha_complex_3d_persistence

This program computes the persistent homology with coefficient field Z/pZ of the 3D Alpha complex built from a 3D point cloud.

Different versions of 3D Alpha complex computation are available:

fast: right combinatorics, values can be arbitrarily bad
safe (default): values can have a relative error at most 1e-5
exact: true values rounded to double.

3D Alpha complex can be, or not, weighted (requires a file containing weights values) and/or periodic (requires a file describing the periodic domain).