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cubical.py
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cubical.py
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"""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
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
--------
images.HeightFiltration, images.RadialFiltration, \
images.DilationFiltration, images.ErosionFiltration, \
images.SignedDistanceFiltration
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))}
}
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}
return Xdgms
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
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
@staticmethod
def plot(Xt, sample=0, homology_dimensions=None, plotly_params=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]``.
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
:meth:`update_traces` and :meth:`update_layout` methods of
:class:`plotly.graph_objects.Figure`.
Returns
-------
fig : :class:`plotly.graph_objects.Figure` object
Plotly figure.
"""
return plot_diagram(
Xt[sample], homology_dimensions=homology_dimensions,
plotly_params=plotly_params
)