You cannot select more than 25 topics
Topics must start with a letter or number, can include dashes ('-') and can be up to 35 characters long.
346 lines
13 KiB
Python
346 lines
13 KiB
Python
"""
|
|
Spherical Voronoi Code
|
|
|
|
.. versionadded:: 0.18.0
|
|
|
|
"""
|
|
#
|
|
# Copyright (C) Tyler Reddy, Ross Hemsley, Edd Edmondson,
|
|
# Nikolai Nowaczyk, Joe Pitt-Francis, 2015.
|
|
#
|
|
# Distributed under the same BSD license as SciPy.
|
|
#
|
|
|
|
import warnings
|
|
import numpy as np
|
|
import scipy
|
|
from . import _voronoi
|
|
from scipy.spatial import cKDTree
|
|
|
|
__all__ = ['SphericalVoronoi']
|
|
|
|
|
|
def calculate_solid_angles(R):
|
|
"""Calculates the solid angles of plane triangles. Implements the method of
|
|
Van Oosterom and Strackee [VanOosterom]_ with some modifications. Assumes
|
|
that input points have unit norm."""
|
|
# Original method uses a triple product `R1 . (R2 x R3)` for the numerator.
|
|
# This is equal to the determinant of the matrix [R1 R2 R3], which can be
|
|
# computed with better stability.
|
|
numerator = np.linalg.det(R)
|
|
denominator = 1 + (np.einsum('ij,ij->i', R[:, 0], R[:, 1]) +
|
|
np.einsum('ij,ij->i', R[:, 1], R[:, 2]) +
|
|
np.einsum('ij,ij->i', R[:, 2], R[:, 0]))
|
|
return np.abs(2 * np.arctan2(numerator, denominator))
|
|
|
|
|
|
class SphericalVoronoi:
|
|
""" Voronoi diagrams on the surface of a sphere.
|
|
|
|
.. versionadded:: 0.18.0
|
|
|
|
Parameters
|
|
----------
|
|
points : ndarray of floats, shape (npoints, ndim)
|
|
Coordinates of points from which to construct a spherical
|
|
Voronoi diagram.
|
|
radius : float, optional
|
|
Radius of the sphere (Default: 1)
|
|
center : ndarray of floats, shape (ndim,)
|
|
Center of sphere (Default: origin)
|
|
threshold : float
|
|
Threshold for detecting duplicate points and
|
|
mismatches between points and sphere parameters.
|
|
(Default: 1e-06)
|
|
|
|
Attributes
|
|
----------
|
|
points : double array of shape (npoints, ndim)
|
|
the points in `ndim` dimensions to generate the Voronoi diagram from
|
|
radius : double
|
|
radius of the sphere
|
|
center : double array of shape (ndim,)
|
|
center of the sphere
|
|
vertices : double array of shape (nvertices, ndim)
|
|
Voronoi vertices corresponding to points
|
|
regions : list of list of integers of shape (npoints, _ )
|
|
the n-th entry is a list consisting of the indices
|
|
of the vertices belonging to the n-th point in points
|
|
|
|
Methods
|
|
----------
|
|
calculate_areas
|
|
Calculates the areas of the Voronoi regions. For 2D point sets, the
|
|
regions are circular arcs. The sum of the areas is `2 * pi * radius`.
|
|
For 3D point sets, the regions are spherical polygons. The sum of the
|
|
areas is `4 * pi * radius**2`.
|
|
|
|
Raises
|
|
------
|
|
ValueError
|
|
If there are duplicates in `points`.
|
|
If the provided `radius` is not consistent with `points`.
|
|
|
|
Notes
|
|
-----
|
|
The spherical Voronoi diagram algorithm proceeds as follows. The Convex
|
|
Hull of the input points (generators) is calculated, and is equivalent to
|
|
their Delaunay triangulation on the surface of the sphere [Caroli]_.
|
|
The Convex Hull neighbour information is then used to
|
|
order the Voronoi region vertices around each generator. The latter
|
|
approach is substantially less sensitive to floating point issues than
|
|
angle-based methods of Voronoi region vertex sorting.
|
|
|
|
Empirical assessment of spherical Voronoi algorithm performance suggests
|
|
quadratic time complexity (loglinear is optimal, but algorithms are more
|
|
challenging to implement).
|
|
|
|
References
|
|
----------
|
|
.. [Caroli] Caroli et al. Robust and Efficient Delaunay triangulations of
|
|
points on or close to a sphere. Research Report RR-7004, 2009.
|
|
|
|
.. [VanOosterom] Van Oosterom and Strackee. The solid angle of a plane
|
|
triangle. IEEE Transactions on Biomedical Engineering,
|
|
2, 1983, pp 125--126.
|
|
|
|
See Also
|
|
--------
|
|
Voronoi : Conventional Voronoi diagrams in N dimensions.
|
|
|
|
Examples
|
|
--------
|
|
Do some imports and take some points on a cube:
|
|
|
|
>>> import matplotlib.pyplot as plt
|
|
>>> from scipy.spatial import SphericalVoronoi, geometric_slerp
|
|
>>> from mpl_toolkits.mplot3d import proj3d
|
|
>>> # set input data
|
|
>>> points = np.array([[0, 0, 1], [0, 0, -1], [1, 0, 0],
|
|
... [0, 1, 0], [0, -1, 0], [-1, 0, 0], ])
|
|
|
|
Calculate the spherical Voronoi diagram:
|
|
|
|
>>> radius = 1
|
|
>>> center = np.array([0, 0, 0])
|
|
>>> sv = SphericalVoronoi(points, radius, center)
|
|
|
|
Generate plot:
|
|
|
|
>>> # sort vertices (optional, helpful for plotting)
|
|
>>> sv.sort_vertices_of_regions()
|
|
>>> t_vals = np.linspace(0, 1, 2000)
|
|
>>> fig = plt.figure()
|
|
>>> ax = fig.add_subplot(111, projection='3d')
|
|
>>> # plot the unit sphere for reference (optional)
|
|
>>> u = np.linspace(0, 2 * np.pi, 100)
|
|
>>> v = np.linspace(0, np.pi, 100)
|
|
>>> x = np.outer(np.cos(u), np.sin(v))
|
|
>>> y = np.outer(np.sin(u), np.sin(v))
|
|
>>> z = np.outer(np.ones(np.size(u)), np.cos(v))
|
|
>>> ax.plot_surface(x, y, z, color='y', alpha=0.1)
|
|
>>> # plot generator points
|
|
>>> ax.scatter(points[:, 0], points[:, 1], points[:, 2], c='b')
|
|
>>> # plot Voronoi vertices
|
|
>>> ax.scatter(sv.vertices[:, 0], sv.vertices[:, 1], sv.vertices[:, 2],
|
|
... c='g')
|
|
>>> # indicate Voronoi regions (as Euclidean polygons)
|
|
>>> for region in sv.regions:
|
|
... n = len(region)
|
|
... for i in range(n):
|
|
... start = sv.vertices[region][i]
|
|
... end = sv.vertices[region][(i + 1) % n]
|
|
... result = geometric_slerp(start, end, t_vals)
|
|
... ax.plot(result[..., 0],
|
|
... result[..., 1],
|
|
... result[..., 2],
|
|
... c='k')
|
|
>>> ax.azim = 10
|
|
>>> ax.elev = 40
|
|
>>> _ = ax.set_xticks([])
|
|
>>> _ = ax.set_yticks([])
|
|
>>> _ = ax.set_zticks([])
|
|
>>> fig.set_size_inches(4, 4)
|
|
>>> plt.show()
|
|
|
|
"""
|
|
def __init__(self, points, radius=1, center=None, threshold=1e-06):
|
|
|
|
if radius is None:
|
|
radius = 1.
|
|
warnings.warn('`radius` is `None`. '
|
|
'This will raise an error in a future version. '
|
|
'Please provide a floating point number '
|
|
'(i.e. `radius=1`).',
|
|
DeprecationWarning)
|
|
|
|
self.radius = float(radius)
|
|
self.points = np.array(points).astype(np.double)
|
|
self._dim = len(points[0])
|
|
if center is None:
|
|
self.center = np.zeros(self._dim)
|
|
else:
|
|
self.center = np.array(center, dtype=float)
|
|
|
|
# test degenerate input
|
|
self._rank = np.linalg.matrix_rank(self.points - self.points[0],
|
|
tol=threshold * self.radius)
|
|
if self._rank < self._dim:
|
|
raise ValueError("Rank of input points must be at least {0}".format(self._dim))
|
|
|
|
if cKDTree(self.points).query_pairs(threshold * self.radius):
|
|
raise ValueError("Duplicate generators present.")
|
|
|
|
radii = np.linalg.norm(self.points - self.center, axis=1)
|
|
max_discrepancy = np.abs(radii - self.radius).max()
|
|
if max_discrepancy >= threshold * self.radius:
|
|
raise ValueError("Radius inconsistent with generators.")
|
|
|
|
self._calc_vertices_regions()
|
|
|
|
def _calc_vertices_regions(self):
|
|
"""
|
|
Calculates the Voronoi vertices and regions of the generators stored
|
|
in self.points. The vertices will be stored in self.vertices and the
|
|
regions in self.regions.
|
|
|
|
This algorithm was discussed at PyData London 2015 by
|
|
Tyler Reddy, Ross Hemsley and Nikolai Nowaczyk
|
|
"""
|
|
# get Convex Hull
|
|
conv = scipy.spatial.ConvexHull(self.points)
|
|
# get circumcenters of Convex Hull triangles from facet equations
|
|
# for 3D input circumcenters will have shape: (2N-4, 3)
|
|
self.vertices = self.radius * conv.equations[:, :-1] + self.center
|
|
self._simplices = conv.simplices
|
|
# calculate regions from triangulation
|
|
# for 3D input simplex_indices will have shape: (2N-4,)
|
|
simplex_indices = np.arange(len(self._simplices))
|
|
# for 3D input tri_indices will have shape: (6N-12,)
|
|
tri_indices = np.column_stack([simplex_indices] * self._dim).ravel()
|
|
# for 3D input point_indices will have shape: (6N-12,)
|
|
point_indices = self._simplices.ravel()
|
|
# for 3D input indices will have shape: (6N-12,)
|
|
indices = np.argsort(point_indices, kind='mergesort')
|
|
# for 3D input flattened_groups will have shape: (6N-12,)
|
|
flattened_groups = tri_indices[indices].astype(np.intp)
|
|
# intervals will have shape: (N+1,)
|
|
intervals = np.cumsum(np.bincount(point_indices + 1))
|
|
# split flattened groups to get nested list of unsorted regions
|
|
groups = [list(flattened_groups[intervals[i]:intervals[i + 1]])
|
|
for i in range(len(intervals) - 1)]
|
|
self.regions = groups
|
|
|
|
def sort_vertices_of_regions(self):
|
|
"""Sort indices of the vertices to be (counter-)clockwise ordered.
|
|
|
|
Raises
|
|
------
|
|
TypeError
|
|
If the points are not three-dimensional.
|
|
|
|
Notes
|
|
-----
|
|
For each region in regions, it sorts the indices of the Voronoi
|
|
vertices such that the resulting points are in a clockwise or
|
|
counterclockwise order around the generator point.
|
|
|
|
This is done as follows: Recall that the n-th region in regions
|
|
surrounds the n-th generator in points and that the k-th
|
|
Voronoi vertex in vertices is the circumcenter of the k-th triangle
|
|
in self._simplices. For each region n, we choose the first triangle
|
|
(=Voronoi vertex) in self._simplices and a vertex of that triangle
|
|
not equal to the center n. These determine a unique neighbor of that
|
|
triangle, which is then chosen as the second triangle. The second
|
|
triangle will have a unique vertex not equal to the current vertex or
|
|
the center. This determines a unique neighbor of the second triangle,
|
|
which is then chosen as the third triangle and so forth. We proceed
|
|
through all the triangles (=Voronoi vertices) belonging to the
|
|
generator in points and obtain a sorted version of the vertices
|
|
of its surrounding region.
|
|
"""
|
|
if self._dim != 3:
|
|
raise TypeError("Only supported for three-dimensional point sets")
|
|
_voronoi.sort_vertices_of_regions(self._simplices, self.regions)
|
|
|
|
def _calculate_areas_3d(self):
|
|
self.sort_vertices_of_regions()
|
|
sizes = [len(region) for region in self.regions]
|
|
csizes = np.cumsum(sizes)
|
|
num_regions = csizes[-1]
|
|
|
|
# We create a set of triangles consisting of one point and two Voronoi
|
|
# vertices. The vertices of each triangle are adjacent in the sorted
|
|
# regions list.
|
|
point_indices = [i for i, size in enumerate(sizes)
|
|
for j in range(size)]
|
|
|
|
nbrs1 = np.array([r for region in self.regions for r in region])
|
|
|
|
# The calculation of nbrs2 is a vectorized version of:
|
|
# np.array([r for region in self.regions for r in np.roll(region, 1)])
|
|
nbrs2 = np.roll(nbrs1, 1)
|
|
indices = np.roll(csizes, 1)
|
|
indices[0] = 0
|
|
nbrs2[indices] = nbrs1[csizes - 1]
|
|
|
|
# Normalize points and vertices.
|
|
pnormalized = (self.points - self.center) / self.radius
|
|
vnormalized = (self.vertices - self.center) / self.radius
|
|
|
|
# Create the complete set of triangles and calculate their solid angles
|
|
triangles = np.hstack([pnormalized[point_indices],
|
|
vnormalized[nbrs1],
|
|
vnormalized[nbrs2]
|
|
]).reshape((num_regions, 3, 3))
|
|
triangle_solid_angles = calculate_solid_angles(triangles)
|
|
|
|
# Sum the solid angles of the triangles in each region
|
|
solid_angles = np.cumsum(triangle_solid_angles)[csizes - 1]
|
|
solid_angles[1:] -= solid_angles[:-1]
|
|
|
|
# Get polygon areas using A = omega * r**2
|
|
return solid_angles * self.radius**2
|
|
|
|
def _calculate_areas_2d(self):
|
|
# Find start and end points of arcs
|
|
arcs = self.points[self._simplices] - self.center
|
|
|
|
# Calculate the angle subtended by arcs
|
|
cosine = np.einsum('ij,ij->i', arcs[:, 0], arcs[:, 1])
|
|
sine = np.abs(np.linalg.det(arcs))
|
|
theta = np.arctan2(sine, cosine)
|
|
|
|
# Get areas using A = r * theta
|
|
areas = self.radius * theta
|
|
|
|
# Correct arcs which go the wrong way (single-hemisphere inputs)
|
|
signs = np.sign(np.einsum('ij,ij->i', arcs[:, 0],
|
|
self.vertices - self.center))
|
|
indices = np.where(signs < 0)
|
|
areas[indices] = 2 * np.pi * self.radius - areas[indices]
|
|
return areas
|
|
|
|
def calculate_areas(self):
|
|
"""Calculates the areas of the Voronoi regions.
|
|
|
|
For 2D point sets, the regions are circular arcs. The sum of the areas
|
|
is `2 * pi * radius`.
|
|
|
|
For 3D point sets, the regions are spherical polygons. The sum of the
|
|
areas is `4 * pi * radius**2`.
|
|
|
|
.. versionadded:: 1.5.0
|
|
|
|
Returns
|
|
-------
|
|
areas : double array of shape (npoints,)
|
|
The areas of the Voronoi regions.
|
|
"""
|
|
if self._dim == 2:
|
|
return self._calculate_areas_2d()
|
|
elif self._dim == 3:
|
|
return self._calculate_areas_3d()
|
|
else:
|
|
raise TypeError("Only supported for 2D and 3D point sets")
|