Source code for gtda.homology.cubical

"""Persistent homology on grids."""
# License: GNU AGPLv3

from numbers import Real

import numpy as np
from joblib import Parallel, delayed
from sklearn.base import BaseEstimator, TransformerMixin
from sklearn.utils.validation import check_array, check_is_fitted

from ._utils import _pad_diagram
from ..base import PlotterMixin
from ..externals.python import CubicalComplex, PeriodicCubicalComplex
from ..plotting import plot_diagram
from ..utils.intervals import Interval
from ..utils.validation import validate_params


[docs]class CubicalPersistence(BaseEstimator, TransformerMixin, PlotterMixin): """:ref:`Persistence diagrams <persistence_diagram>` resulting from :ref:`filtered cubical complexes <cubical_complex>`. Given a :ref:`greyscale image <cubical_chains_and_cubical_homology>`, information about the appearance and disappearance of topological features (technically, :ref:`homology classes <homology_and_cohomology>`) of various dimensions and at different scales is summarised in the corresponding persistence diagram. 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 :math:`\\mathbb{F}_p = \\{ 0, \\ldots, p - 1 \\}` where :math:`p` equals `coeff`. periodic_dimensions : boolean ndarray of shape (n_dimensions,) or None, \ optional, default: ``None`` Periodicity of the boundaries along each of the axis, where ``n_dimensions`` is the dimension of the images of the collection. The boolean in the `d`th position expresses whether the boundaries along the `d`th 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 `np.inf`. ``None`` assigns the maximum pixel values within all images passed to :meth:`fit`. n_jobs : int or None, optional, default: ``None`` The number of jobs to use for the computation. ``None`` means 1 unless in a :obj:`joblib.parallel_backend` context. ``-1`` means using all processors. Attributes ---------- periodic_dimensions_ : boolean ndarray of shape (n_dimensions,) Effective periodicity of the boundaries along each of the axis. Set in :meth:`fit`. infinity_values_ : float Effective death value to assign to features which have infinite persistence. Set in :meth:`fit`. See also -------- VietorisRipsPersistence, SparseRipsPersistence, EuclideanCechPersistence Notes ----- `GUDHI <https://github.com/GUDHI/gudhi-devel>`_ is used as a C++ backend for computing cubical persistent homology. Python bindings were modified for performance. Persistence diagrams produced by this class must be interpreted with care due to the presence of padding triples which carry no information. See :meth:`transform` for additional information. References ---------- [1] P. Dlotko, "Cubical complex", 2015; `GUDHI User and Reference Manual \ <http://gudhi.gforge.inria.fr/doc/latest/group__cubical__complex.html>`_. """ _hyperparameters = { 'homology_dimensions': { 'type': (list, tuple), 'of': { 'type': int, 'in': Interval(0, np.inf, closed='left')}}, 'coeff': {'type': int, 'in': Interval(2, np.inf, closed='left')}, 'periodic_dimensions': { 'type': (np.ndarray, type(None)), 'of': {'type': np.bool_}}, 'infinity_values': {'type': (Real, type(None))}}
[docs] def __init__(self, homology_dimensions=(0, 1), coeff=2, periodic_dimensions=None, infinity_values=None, n_jobs=None): self.homology_dimensions = homology_dimensions self.coeff = coeff self.periodic_dimensions = periodic_dimensions self.infinity_values = infinity_values self.n_jobs = n_jobs
def _gudhi_diagram(self, X): cubical_complex = self._filtration( dimensions=X.shape, top_dimensional_cells=X.flatten(order="F"), **self._filtration_kwargs) Xdgms = cubical_complex.persistence(homology_coeff_field=self.coeff, min_persistence=0) # Separate diagrams by homology dimensions Xdgms = {dim: np.array([Xdgms[i][1] for i in range(len(Xdgms)) if Xdgms[i][0] == dim]).reshape((-1, 2)) for dim in self.homology_dimensions} # Add dimension as the third elements of each (b, d) tuple Xdgms = {dim: np.hstack([Xdgms[dim], dim * np.ones((Xdgms[dim].shape[0], 1), dtype=Xdgms[dim].dtype)]) for dim in self._homology_dimensions} return Xdgms
[docs] def fit(self, X, y=None): """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 : object """ X = check_array(X, allow_nd=True) validate_params( self.get_params(), self._hyperparameters, exclude=['n_jobs']) self._filtration_kwargs = {} if self.periodic_dimensions is None or \ np.sum(self.periodic_dimensions) == 0: self._filtration = CubicalComplex self.periodic_dimensions_ = np.zeros(len(X) - 1, dtype=np.bool) else: self._filtration = PeriodicCubicalComplex self.periodic_dimensions_ = np.array(self.periodic_dimensions, dtype=np.bool) self._filtration_kwargs['periodic_dimensions'] = \ self.periodic_dimensions_ if self.infinity_values is None: self.infinity_values_ = np.max(X) else: self.infinity_values_ = self.infinity_values self._homology_dimensions = sorted(self.homology_dimensions) self._max_homology_dimension = self._homology_dimensions[-1] return self
[docs] def transform(self, X, y=None): """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 : ndarray of shape (n_samples, n_features, 3) Array of persistence diagrams computed from the feature arrays or distance matrices in `X`. ``n_features`` equals :math:`\\sum_q n_q`, where :math:`n_q` is the maximum number of topological features in dimension :math:`q` across all samples in `X`. """ check_is_fitted(self) Xt = check_array(X, allow_nd=True) Xt = Parallel(n_jobs=self.n_jobs)( delayed(self._gudhi_diagram)(x) for x in Xt) max_n_points = { dim: max(1, np.max([x[dim].shape[0] for x in Xt])) for dim in self.homology_dimensions} min_values = { dim: min([np.min(x[dim][:, 0]) if x[dim].size else np.inf for x in Xt]) for dim in self.homology_dimensions} min_values = { dim: min_values[dim] if min_values[dim] != np.inf else 0 for dim in self.homology_dimensions} Xt = Parallel(n_jobs=self.n_jobs)(delayed(_pad_diagram)( x, self._homology_dimensions, max_n_points, min_values) for x in Xt) Xt = np.stack(Xt) Xt = np.nan_to_num(Xt, posinf=self.infinity_values_) return Xt
[docs] @staticmethod def plot(Xt, sample=0, homology_dimensions=None): """Plot a sample from a collection of persistence diagrams, with homology in multiple dimensions. Parameters ---------- Xt : ndarray of shape (n_samples, n_points, 3) Collection of persistence diagrams, such as returned by :meth:`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]``. """ return plot_diagram( Xt[sample], homology_dimensions=homology_dimensions)