Periodic cubical complex reference manual

class gudhi.PeriodicCubicalComplex

Bases: object

The PeriodicCubicalComplex is an example of a structured complex useful in computational mathematics (specially rigorous numerics) and image analysis.

__init__()

PeriodicCubicalComplex constructor from dimensions and top_dimensional_cells or from a Perseus-style file name.

Parameters

• dimensions (list of int) – A list of number of top dimensional cells.

• top_dimensional_cells (list of double) – A list of cells filtration values.

• periodic_dimensions (list of boolean) – A list of top dimensional cells periodicity value.

Or

Parameters

• top_dimensional_cells (anything convertible to a numpy ndarray) – A multidimensional array of cells filtration values.

• periodic_dimensions (list of boolean) – A list of top dimensional cells periodicity value.

Or

Parameters

perseus_file (string) – A Perseus-style file name.

betti_numbers()

This function returns the Betti numbers of the complex.

Returns

list of int – The Betti numbers ([B0, B1, …, Bn]).

Note

betti_numbers function requires persistence function to be launched first.

Note

betti_numbers function always returns [1, 0, 0, …] as infinity filtration cubes are not removed from the complex.

dimension()

This function returns the dimension of the complex.

Returns

int – the complex dimension.

num_simplices()

This function returns the number of all cubes in the complex.

Returns

int – the number of all cubes in the complex.

persistence()

This function returns the persistence of the complex.

Parameters

• homology_coeff_field (int.) – The homology coefficient field. Must be a prime number

• min_persistence (float.) – The minimum persistence value to take into account (strictly greater than min_persistence). Default value is 0.0. Sets min_persistence to -1.0 to see all values.

Returns

list of pairs(dimension, pair(birth, death)) – the persistence of the complex.

persistence_intervals_in_dimension()

This function returns the persistence intervals of the complex in a specific dimension.

Parameters

dimension (int.) – The specific dimension.

Returns

The persistence intervals.

Return type

numpy array of dimension 2

Note

intervals_in_dim function requires persistence function to be launched first.

persistent_betti_numbers()

This function returns the persistent Betti numbers of the complex.

Parameters

• from_value (float.) – The persistence birth limit to be added in the numbers (persistent birth <= from_value).

• to_value (float.) – The persistence death limit to be added in the numbers (persistent death > to_value).

Returns

list of int – The persistent Betti numbers ([B0, B1, …, Bn]).

Note

persistent_betti_numbers function requires persistence function to be launched first.