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. 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.

Example

alpha_complex_persistence -r 32 -p 2 -m 0.45 ../../data/points/tore3D_300.off

N.B.:

  • Filtration values are alpha square values.

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. 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_3d_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.
  • -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.

Example

alpha_complex_3d_persistence ../../data/points/tore3D_300.off -p 2 -m 0.45

N.B.:

  • alpha_complex_3d_persistence only accepts OFF files in dimension 3.
  • Filtration values are alpha square values.

exact_alpha_complex_3d_persistence

Same as alpha_complex_3d_persistence, but using exact computation. It is slower, but it is necessary when points are on a grid for instance.

weighted_alpha_complex_3d_persistence

Same as alpha_complex_3d_persistence, but using weighted points.

Usage

weighted_alpha_complex_3d_persistence [options] <input OFF file> <weights input file>

where

  • <input OFF file> is the path to the input point cloud in nOFF ASCII format.
  • <input weights file> is the path to the file containing the weights of the points (one value per line).

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.
  • -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.

Example

weighted_alpha_complex_3d_persistence ../../data/points/tore3D_300.off ../../data/points/tore3D_300.weights -p 2 -m 0.45

N.B.:

periodic_alpha_complex_3d_persistence

Same as alpha_complex_3d_persistence, but using periodic alpha shape 3d. Refer to the CGAL’s 3D Periodic Triangulations User Manual for more details.

Usage

periodic_alpha_complex_3d_persistence [options] <input OFF file> <cuboid file>

where

  • <input OFF file> is the path to the input point cloud in nOFF ASCII format.
  • <cuboid file> is the path to the file describing the periodic domain. It must be in the format described here.

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.
  • -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

Example

periodic_alpha_complex_3d_persistence ../../data/points/grid_10_10_10_in_0_1.off ../../data/points/iso_cuboid_3_in_0_1.txt -p 3 -m 1.0

N.B.:

  • Cuboid file must be in the format described here.
  • Filtration values are alpha square values.