tupui · August 9, 2021 05:40 · tupui · Aug 2, 2021
diff --git a/lloyd_sampling.py b/lloyd_sampling.py
 """Centroidal Voronoi Tessellation to generate sample: Lloyd's algorithm.

 Based on the implementation of Stéfan van der Walt

 	https://github.com/stefanv/lloyd

 which is:

  Copyright (c) 2021-04-21 Stéfan van der Walt https://github.com/stefanv/lloyd
  MIT License


 ---------------------------

 MIT License

 Copyright (c) 2021 Pamphile Tupui ROY

 Permission is hereby granted, free of charge, to any person obtaining a copy
 of this software and associated documentation files (the "Software"), to deal
 in the Software without restriction, including without limitation the rights
 to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
 copies of the Software, and to permit persons to whom the Software is
 furnished to do so, subject to the following conditions:

 The above copyright notice and this permission notice shall be included in all
 copies or substantial portions of the Software.

 THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
 IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
 FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
 AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
 LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
 OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
 SOFTWARE.

 """
 import functools

 import numpy as np

 from scipy import spatial
 from scipy.stats import qmc
 from scipy import optimize

 import pandas as pd
 import matplotlib.pyplot as plt
 import seaborn as sns


 rng = np.random.default_rng()


 def _decay(n_iters):
    """Exponential decay.

    Fit an exponential to be 2 at 0 and 1 at `n_iters`.
    The decay is used for relaxation.
    """
    res = optimize.root_scalar(lambda x: np.exp(-n_iters/x) + 1 - 1.1,
                               bracket=[1, n_iters])

    return (np.exp(-x / res.root) + 0.9 for x in range(n_iters))


 def _points_contain_duplicates(points):
    """Check whether `points` contains duplicates."""
    vals, count = np.unique(points, return_counts=True)
    return np.any(vals[count > 1])


 def _jitter_points(points, scalar=.000000001):
    """Randomly jitter points until they are unique.

    If the points are not unique, the number of regions in our
    tessellation will be less than the number of input points.
    """
    while _points_contain_duplicates(points):
        offset = rng.uniform(-scalar, scalar, size=(len(points), points.shape[1]))
        points = points + offset
    return points


 def lloyd_centroidal_voronoi_tessellation(points):
    """Centroidal Voronoi Tessellation.

    Perturb a sample of points using Lloyd's algorithm.
    """
    points = _jitter_points(points)

    centroids = np.empty_like(points)

    # Add exterior corners before tesselation
    d = centroids.shape[1]
    arrays = np.tile([0, 1], (d, 1))
    hypercube_corners = np.stack(np.meshgrid(*arrays), axis=-1).reshape(-1, d)

    n_hypercube_corners = len(hypercube_corners)
    points = np.concatenate((points, hypercube_corners))

    voronoi = spatial.Voronoi(points)

    decay_ = next(decay)

    for ii, idx in enumerate(voronoi.point_region[:-n_hypercube_corners]):
        # the region is a series of indices into self.voronoi.vertices
        # remove point at infinity, designated by index -1
        region = [i for i in voronoi.regions[idx] if i != -1]

        # enclose the polygon
        region = region + [region[0]]

        # get the vertices for this region
        verts = voronoi.vertices[region]

        is_valid = _within_hypercube(verts)
        verts = verts[is_valid]
        #verts = np.clip(verts, 0, 1)

        centroid = _centroid(verts)
        centroids[ii] = points[ii] + (centroid - points[ii]) * decay_
        centroids[ii] = centroid

    is_valid = _within_hypercube(centroids)

    points = points[:-n_hypercube_corners]
    points[is_valid] = centroids[is_valid]


    # points = np.clip(centroids, 0, 1)

    return points


 def _centroid(polygon):
    """Centroid in n-D is the mean for uniformly distributed nodes of a geometry."""
    return np.mean(polygon, axis=0)


 def _within_hypercube(points):
    return np.all(np.logical_and(points >= 0, points <= 1), axis=1)


 n_iters = 1000
 decay = _decay(n_iters)
 points = qmc.Sobol(d=2, scramble=True).random(128)

 points_lloyd = points.copy()
 for _ in range(n_iters):
    points_lloyd = lloyd_centroidal_voronoi_tessellation(points_lloyd)

 sns.pairplot(pd.DataFrame(points_lloyd), corner=True, diag_kws={"bins": 8})
 plt.show()
	"""Centroidal Voronoi Tessellation to generate sample: Lloyd's algorithm.

	Based on the implementation of Stéfan van der Walt

	https://github.com/stefanv/lloyd

	which is:

	Copyright (c) 2021-04-21 Stéfan van der Walt https://github.com/stefanv/lloyd
	MIT License


	---------------------------

	MIT License

	Copyright (c) 2021 Pamphile Tupui ROY

	Permission is hereby granted, free of charge, to any person obtaining a copy
	of this software and associated documentation files (the "Software"), to deal
	in the Software without restriction, including without limitation the rights
	to use, copy, modify, merge, publish, distribute, sublicense, and/or sell
	copies of the Software, and to permit persons to whom the Software is
	furnished to do so, subject to the following conditions:

	The above copyright notice and this permission notice shall be included in all
	copies or substantial portions of the Software.

	THE SOFTWARE IS PROVIDED "AS IS", WITHOUT WARRANTY OF ANY KIND, EXPRESS OR
	IMPLIED, INCLUDING BUT NOT LIMITED TO THE WARRANTIES OF MERCHANTABILITY,
	FITNESS FOR A PARTICULAR PURPOSE AND NONINFRINGEMENT. IN NO EVENT SHALL THE
	AUTHORS OR COPYRIGHT HOLDERS BE LIABLE FOR ANY CLAIM, DAMAGES OR OTHER
	LIABILITY, WHETHER IN AN ACTION OF CONTRACT, TORT OR OTHERWISE, ARISING FROM,
	OUT OF OR IN CONNECTION WITH THE SOFTWARE OR THE USE OR OTHER DEALINGS IN THE
	SOFTWARE.

	"""
	import functools

	import numpy as np

	from scipy import spatial
	from scipy.stats import qmc
	from scipy import optimize

	import pandas as pd
	import matplotlib.pyplot as plt
	import seaborn as sns


	rng = np.random.default_rng()


	def _decay(n_iters):
	"""Exponential decay.

	Fit an exponential to be 2 at 0 and 1 at `n_iters`.
	The decay is used for relaxation.
	"""
	res = optimize.root_scalar(lambda x: np.exp(-n_iters/x) + 1 - 1.1,
	bracket=[1, n_iters])

	return (np.exp(-x / res.root) + 0.9 for x in range(n_iters))


	def _points_contain_duplicates(points):
	"""Check whether `points` contains duplicates."""
	vals, count = np.unique(points, return_counts=True)
	return np.any(vals[count > 1])


	def _jitter_points(points, scalar=.000000001):
	"""Randomly jitter points until they are unique.

	If the points are not unique, the number of regions in our
	tessellation will be less than the number of input points.
	"""
	while _points_contain_duplicates(points):
	offset = rng.uniform(-scalar, scalar, size=(len(points), points.shape[1]))
	points = points + offset
	return points


	def lloyd_centroidal_voronoi_tessellation(points):
	"""Centroidal Voronoi Tessellation.

	Perturb a sample of points using Lloyd's algorithm.
	"""
	points = _jitter_points(points)

	centroids = np.empty_like(points)

	# Add exterior corners before tesselation
	d = centroids.shape[1]
	arrays = np.tile([0, 1], (d, 1))
	hypercube_corners = np.stack(np.meshgrid(*arrays), axis=-1).reshape(-1, d)

	n_hypercube_corners = len(hypercube_corners)
	points = np.concatenate((points, hypercube_corners))

	voronoi = spatial.Voronoi(points)

	decay_ = next(decay)

	for ii, idx in enumerate(voronoi.point_region[:-n_hypercube_corners]):
	# the region is a series of indices into self.voronoi.vertices
	# remove point at infinity, designated by index -1
	region = [i for i in voronoi.regions[idx] if i != -1]

	# enclose the polygon
	region = region + [region[0]]

	# get the vertices for this region
	verts = voronoi.vertices[region]

	is_valid = _within_hypercube(verts)
	verts = verts[is_valid]
	#verts = np.clip(verts, 0, 1)

	centroid = _centroid(verts)
	centroids[ii] = points[ii] + (centroid - points[ii]) * decay_
	centroids[ii] = centroid

	is_valid = _within_hypercube(centroids)

	points = points[:-n_hypercube_corners]
	points[is_valid] = centroids[is_valid]


	# points = np.clip(centroids, 0, 1)

	return points


	def _centroid(polygon):
	"""Centroid in n-D is the mean for uniformly distributed nodes of a geometry."""
	return np.mean(polygon, axis=0)


	def _within_hypercube(points):
	return np.all(np.logical_and(points >= 0, points <= 1), axis=1)


	n_iters = 1000
	decay = _decay(n_iters)
	points = qmc.Sobol(d=2, scramble=True).random(128)

	points_lloyd = points.copy()
	for _ in range(n_iters):
	points_lloyd = lloyd_centroidal_voronoi_tessellation(points_lloyd)

	sns.pairplot(pd.DataFrame(points_lloyd), corner=True, diag_kws={"bins": 8})
	plt.show()