CubicalPersistence¶

class gtda.homology.CubicalPersistence(homology_dimensions=0, 1, coeff=2, periodic_dimensions=None, infinity_values=None, reduced_homology=True, n_jobs=None)[source]

Persistence diagrams resulting from filtered cubical complexes.

Given a greyscale image, information about the appearance and disappearance of topological features (technically, homology classes) of various dimensions and at different scales is summarised in the corresponding persistence diagram.

Important note:
• Persistence diagrams produced by this class must be interpreted with care due to the presence of padding triples which carry no information. See transform for additional information.

Parameters
• homology_dimensions (list or tuple, optional, default: (0, 1)) – Dimensions (non-negative integers) of the topological features to be detected.

• coeff (int prime, optional, default: 2) – Compute homology with coefficients in the prime field $$\mathbb{F}_p = \{ 0, \ldots, p - 1 \}$$ where $$p$$ equals coeff.

• periodic_dimensions (boolean ndarray of shape (n_dimensions,) or None, optional, default: None) – Periodicity of the boundaries along each of the axes, where n_dimensions is the dimension of the images of the collection. The boolean in the dth position expresses whether the boundaries along the dth axis are periodic. The default None is equivalent to passing numpy.zeros((n_dimensions,), dtype=np.bool), i.e. none of the boundaries are periodic.

• infinity_values (float or None, default: None) – Which death value to assign to features which are still alive at filtration value numpy.inf. None assigns the maximum pixel values within all images passed to fit.

• reduced_homology (bool, optional, default: True) – If True, the earliest-born triple in homology dimension 0 which has infinite death is discarded from each diagram computed in transform.

• n_jobs (int or None, optional, default: None) – The number of jobs to use for the computation. None means 1 unless in a joblib.parallel_backend context. -1 means using all processors.

periodic_dimensions_

Effective periodicity of the boundaries along each of the axes. Set in fit.

Type

boolean ndarray of shape (n_dimensions,)

infinity_values_

Effective death value to assign to features which have infinite persistence. Set in fit.

Type

float

images.HeightFiltration, images.RadialFiltration, images.DilationFiltration, images.ErosionFiltration, images.SignedDistanceFiltration

Notes

GUDHI is used as a C++ backend for computing cubical persistent homology 1. Python bindings were modified for performance.

References

1

P. Dlotko, “Cubical complex”, 2015; GUDHI User and Reference Manual.

__init__(homology_dimensions=0, 1, coeff=2, periodic_dimensions=None, infinity_values=None, reduced_homology=True, n_jobs=None)[source]

Initialize self. See help(type(self)) for accurate signature.

fit(X, y=None)[source]

Do nothing and return the estimator unchanged.

This method is here to implement the usual scikit-learn API and hence work in pipelines.

Parameters
• X (ndarray of shape (n_samples, n_pixels_1, .., n_pixels_d)) – Input data. Array of d-dimensional images.

• y (None) – There is no need of a target in a transformer, yet the pipeline API requires this parameter.

Returns

self

Return type

object

fit_transform(X, y=None, **fit_params)

Fit to data, then transform it.

Fits transformer to X and y with optional parameters fit_params and returns a transformed version of X.

Parameters
• X ({array-like, sparse matrix, dataframe} of shape (n_samples, n_features)) –

• y (ndarray of shape (n_samples,), default=None) – Target values.

• **fit_params (dict) – Additional fit parameters.

Returns

X_new – Transformed array.

Return type

ndarray array of shape (n_samples, n_features_new)

fit_transform_plot(X, y=None, sample=0, **plot_params)

Fit to data, then apply transform_plot.

Parameters
• X (ndarray of shape (n_samples, ..)) – Input data.

• y (ndarray of shape (n_samples,) or None) – Target values for supervised problems.

• sample (int) – Sample to be plotted.

• **plot_params – Optional plotting parameters.

Returns

Xt – Transformed one-sample slice from the input.

Return type

ndarray of shape (1, ..)

get_params(deep=True)

Get parameters for this estimator.

Parameters

deep (bool, default=True) – If True, will return the parameters for this estimator and contained subobjects that are estimators.

Returns

params – Parameter names mapped to their values.

Return type

mapping of string to any

static plot(Xt, sample=0, homology_dimensions=None, plotly_params=None)[source]

Plot a sample from a collection of persistence diagrams, with homology in multiple dimensions.

Parameters
• Xt (ndarray of shape (n_samples, n_features, 3)) – Collection of persistence diagrams, such as returned by transform.

• sample (int, optional, default: 0) – Index of the sample in Xt to be plotted.

• homology_dimensions (list, tuple or None, optional, default: None) – Which homology dimensions to include in the plot. None means plotting all dimensions present in Xt[sample].

• plotly_params (dict or None, optional, default: None) – Custom parameters to configure the plotly figure. Allowed keys are "traces" and "layout", and the corresponding values should be dictionaries containing keyword arguments as would be fed to the update_traces and update_layout methods of plotly.graph_objects.Figure.

Returns

fig – Plotly figure.

Return type

plotly.graph_objects.Figure object

set_params(**params)

Set the parameters of this estimator.

The method works on simple estimators as well as on nested objects (such as pipelines). The latter have parameters of the form <component>__<parameter> so that it’s possible to update each component of a nested object.

Parameters

**params (dict) – Estimator parameters.

Returns

self – Estimator instance.

Return type

object

transform(X, y=None)[source]

For each image in X, compute the relevant persistence diagram as an array of triples [b, d, q]. Each triple represents a persistent topological feature in dimension q (belonging to homology_dimensions) which is born at b and dies at d. Only triples in which b < d are meaningful. Triples in which b and d are equal (“diagonal elements”) may be artificially introduced during the computation for padding purposes, since the number of non-trivial persistent topological features is typically not constant across samples. They carry no information and hence should be effectively ignored by any further computation.

Parameters
• X (ndarray of shape (n_samples, n_pixels_1, .., n_pixels_d)) – Input data. Array of d-dimensional images.

• y (None) – There is no need of a target in a transformer, yet the pipeline API requires this parameter.

Returns

Xt – Array of persistence diagrams computed from the feature arrays or distance matrices in X. n_features equals $$\sum_q n_q$$, where $$n_q$$ is the maximum number of topological features in dimension $$q$$ across all samples in X.

Return type

ndarray of shape (n_samples, n_features, 3)

transform_plot(X, sample=0, **plot_params)

Take a one-sample slice from the input collection and transform it. Before returning the transformed object, plot the transformed sample.

Parameters
• X (ndarray of shape (n_samples, ..)) – Input data.

• sample (int) – Sample to be plotted.

• **plot_params – Optional plotting parameters.

Returns

Xt – Transformed one-sample slice from the input.

Return type

ndarray of shape (1, ..)