Čech complex

Čech complex

cech_persistence

This program computes the persistent homology with coefficient field Z/pZ of a Čech complex defined on a set of input points, using Euclidean distance.

Different versions of Delaunay-Čech 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.

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

cech_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-radius ] (default = inf) Maximal radius for the Čech complex construction.
  • -d [ --cpx-dimension ] (default = 1) Maximal dimension of the Čech complex we want to compute.
  • -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.
  • -e [ --exact ] for the exact computation version.
  • -f [ --fast ] for the fast computation version.

Beware: this program may use a lot of RAM and take a lot of time if max-radius is set to a large value.

Example 1 with Z/2Z coefficients

cech_persistence ../../data/points/tore3D_1307.off -r 0.25 -m 0.5 -d 3 -p 2

Example 2 with Z/3Z coefficients

cech_persistence ../../data/points/tore3D_1307.off -r 0.25 -m 0.5 -d 3 -p 3

dealaunay_cech_persistence

This program Computes the persistent homology with coefficient field Z/pZ of a Delaunay-Čech complex defined on a set of input points.

Different versions of Delaunay-Čech 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.

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

delaunay_cech_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-radius ] (default = inf) Maximal length of an edge for the Delaunay-Čech 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
  • -e [ --exact ] for the exact computation version.
  • -f [ --fast ] for the fast computation version.

Example 1 with Z/2Z coefficients

delaunay_cech_persistence ../../data/points/tore3D_1307.off -r 0.25 -m 0.5 -p 2

Example 2 with Z/3Z coefficients

delaunay_cech_persistence ../../data/points/tore3D_1307.off -r 0.25 -m 0.5 -p 3

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