# 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 compute_persistence() function to be launched first.

Note

This function always returns the Betti numbers of a torus as infinity filtration cubes are not removed from the complex.

cofaces_of_persistence_pairs()

A persistence interval is described by a pair of cells, one that creates the feature and one that kills it. The filtration values of those 2 cells give coordinates for a point in a persistence diagram, or a bar in a barcode. Structurally, in the cubical complexes provided here, the filtration value of any cell is the minimum of the filtration values of the maximal cells that contain it. Connecting persistence diagram coordinates to the corresponding value in the input (i.e. the filtration values of the top-dimensional cells) is useful for differentiation purposes.

This function returns a list of pairs of top-dimensional cells corresponding to the persistence birth and death cells of the filtration. The cells are represented by their indices in the input list of top-dimensional cells (and not their indices in the internal datastructure that includes non-maximal cells). Note that when two adjacent top-dimensional cells have the same filtration value, we arbitrarily return one of the two when calling the function on one of their common faces.

Returns

The top-dimensional cells/cofaces of the positive and negative cells, together with the corresponding homological dimension, in two lists of numpy arrays of integers. The first list contains the regular persistence pairs, grouped by dimension. It contains numpy arrays of shape [number_of_persistence_points, 2]. The indices of the arrays in the list correspond to the homological dimensions, and the integers of each row in each array correspond to: (index of positive top-dimensional cell, index of negative top-dimensional cell). The second list contains the essential features, grouped by dimension. It contains numpy arrays of shape [number_of_persistence_points, 1]. The indices of the arrays in the list correspond to the homological dimensions, and the integers of each row in each array correspond to: (index of positive top-dimensional cell).

compute_persistence()

This function computes the persistence of the complex, so it can be accessed through persistent_betti_numbers(), persistence_intervals_in_dimension(), etc. This function is equivalent to persistence() when you do not want the list persistence() returns.

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

Nothing.

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 computes and 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 compute_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 compute_persistence() function to be launched first.