Centroidal Voronoi Tessellation to generate sample: Lloyd's algorithm

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
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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()

Here is a 2D example starting with Sobol' points.