Question

我有点（例如，lat，lon对的细胞塔位置），我需要得到它们形成的Voronoi细胞的多边形。

from scipy.spatial import Voronoi

tower = [[ 24.686 ,  46.7081],
       [ 24.686 ,  46.7081],
       [ 24.686 ,  46.7081]]

c = Voronoi(towers)

现在，我需要为每个单元格的lat，lon坐标获取多边形边界（以及该多边形所包围的质心）。我需要这个Voronoi也是有限的。意味着边界不会进入无限远，而是在边界框内。

Answer 1

给定一个矩形边界框，我的第一个想法是在这个边界框和由scipy.spatial.Voronoi产生的Voronoï图之间定义一种交叉运算。一个想法不一定很好，因为这需要编码计算几何的大量基本函数。

然而，这是我想到的第二个想法（黑客？）：计算平面中一组n点的Voronoï图的算法的时间复杂度为O(n ln(n)) 。如何添加点来限制初始点的Voronoï单元格位于边界框中？

有界Voronoï图的解决方案

一张图片值得一个伟大的演讲：

我在这做了什么？这很简单！初始点（蓝色）位于[0.0, 1.0] x [0.0, 1.0]。然后我根据[-1.0, 0.0] x [0.0, 1.0]（边界框的左边缘）通过反射对称得到左边的点（蓝色）（即x = 0.0）。根据{{1}}，x = 1.0和y = 0.0（边界框的其他边缘）的反射对称性，我得到了我需要完成工作的所有点（蓝色）。

然后我跑y = 1.0。上图描绘了生成的Voronoï图（我使用scipy.spatial.voronoi_plot_2d）。

下一步该怎么做？只需根据边界框过滤点，边或面。并根据众所周知的公式计算每个面部的质心来计算centroid of polygon。这是结果的图像（质心是红色的）：

在显示代码之前有些乐趣

大！它似乎工作。如果在一次迭代后我尝试在质心（红色）而不是初始点（蓝色）上重新运行算法怎么办？如果我一次又一次地尝试怎么办？

第2步

第10步

第25步

酷！ Voronoï细胞倾向于最小化它们的能量 ......

这是代码

scipy.spatial.Voronoi

Answer 2

在使用scipy的voronoi函数和创建CVD时，我遇到了很多麻烦，因此这些精彩的帖子和评论极大地帮助了我。作为一名编程新手，我试图理解Flabetvvibes答案中的代码，并且我将分享我对它如何与Energya以及我自己的修改一起工作的解释。我还在此答案的底部完整地发布了我的代码版本

import matplotlib.pyplot as pl
import numpy as np
import scipy as sp
import scipy.spatial
import sys
import copy

eps = sys.float_info.epsilon

# Returns a new np.array of towers that within the bounding_box
def in_box(towers, bounding_box):
    return np.logical_and(np.logical_and(bounding_box[0] <= towers[:, 0],
                                         towers[:, 0] <= bounding_box[1]),
                          np.logical_and(bounding_box[2] <= towers[:, 1],
                                         towers[:, 1] <= bounding_box[3]))

in_box函数使用numpy的logical_and方法返回一个布尔数组，该布尔数组表示塔中边界框内的坐标。

# Generates a bounded vornoi diagram with finite regions in the bounding box
def bounded_voronoi(towers, bounding_box):
    # Select towers inside the bounding box
    i = in_box(towers, bounding_box)

    # Mirror points left, right, above, and under to provide finite regions for the
    # edge regions of the bounding box
    points_center = towers[i, :]

    points_left = np.copy(points_center)
    points_left[:, 0] = bounding_box[0] - (points_left[:, 0] - bounding_box[0])

    points_right = np.copy(points_center)
    points_right[:, 0] = bounding_box[1] + (bounding_box[1] - points_right[:, 0])

    points_down = np.copy(points_center)
    points_down[:, 1] = bounding_box[2] - (points_down[:, 1] - bounding_box[2])

    points_up = np.copy(points_center)
    points_up[:, 1] = bounding_box[3] + (bounding_box[3] - points_up[:, 1])

    points = np.append(points_center,
                       np.append(np.append(points_left,
                                           points_right,
                                           axis=0),
                                 np.append(points_down,
                                           points_up,
                                           axis=0),
                                 axis=0),
                       axis=0)

Flabetvvibes镜像这些点，以使边界框内边缘的区域有限。 Scipy的voronoi方法针对未定义的顶点返回-1，因此对这些点进行镜像可以使边界框内的所有区域都是有限的，并且所有无限区域都位于边界框外的镜像区域中，稍后将丢弃。

# Compute Voronoi
vor = sp.spatial.Voronoi(points)

# creates a new attibute for points that form the diagram within the region
vor.filtered_points = points_center 
# grabs the first fifth of the regions, which are the original regions
vor.filtered_regions = np.array(vor.regions)[vor.point_region[:vor.npoints//5]]

return vor

bounded_voronoi方法的最后这一点调用scipy的voronoi函数，并为边界框中的已过滤点和区域添加新属性。 Energya建议删除Flabetvvibe的代码，该代码使用一个衬板手动找到边界框内的所有有限区域，该衬板获得区域的前五分之一，这是原始输入以及组成边界框的点。

def generate_CVD(points, iterations, bounding_box):
    p = copy.copy(points)

    for i in range(iterations):
        vor = bounded_voronoi(p, bounding_box)
        centroids = []

        for region in vor.filtered_regions:
            # grabs vertices for the region and adds a duplicate
            # of the first one to the end
            vertices = vor.vertices[region + [region[0]], :] 
            centroid = centroid_region(vertices)
            centroids.append(list(centroid[0, :]))

        p = np.array(centroids)

    return bounded_voronoi(p, bounding_box)

我采用了Flabetvvibe的代码，该代码执行了loyd算法的迭代，并将其形成一种易于使用的方法。对于每次迭代，将调用先前的bounded_voronoi函数，然后为每个单元找到质心，它们将成为下一次迭代的新点集。 vertices = vor.vertices[region + [region[0]], :]只需获取当前区域的所有顶点，并将第一个顶点复制到末尾，以使第一个和最后一个顶点用于计算质心相同。

感谢Flabetvvibes和Energya。您的帖子/答案教会了我如何更好地使用scipy的voronoi方法而不是其文档。我还将代码作为一个单独的代码发布在其他任何要复制/粘贴的代码下面。

import matplotlib.pyplot as pl
import numpy as np
import scipy as sp
import scipy.spatial
import sys
import copy

eps = sys.float_info.epsilon

# Returns a new np.array of towers that within the bounding_box
def in_box(towers, bounding_box):
    return np.logical_and(np.logical_and(bounding_box[0] <= towers[:, 0],
                                         towers[:, 0] <= bounding_box[1]),
                          np.logical_and(bounding_box[2] <= towers[:, 1],
                                         towers[:, 1] <= bounding_box[3]))


# Generates a bounded vornoi diagram with finite regions
def bounded_voronoi(towers, bounding_box):
    # Select towers inside the bounding box
    i = in_box(towers, bounding_box)

    # Mirror points left, right, above, and under to provide finite regions for the edge regions of the bounding box
    points_center = towers[i, :]

    points_left = np.copy(points_center)
    points_left[:, 0] = bounding_box[0] - (points_left[:, 0] - bounding_box[0])

    points_right = np.copy(points_center)
    points_right[:, 0] = bounding_box[1] + (bounding_box[1] - points_right[:, 0])

    points_down = np.copy(points_center)
    points_down[:, 1] = bounding_box[2] - (points_down[:, 1] - bounding_box[2])

    points_up = np.copy(points_center)
    points_up[:, 1] = bounding_box[3] + (bounding_box[3] - points_up[:, 1])

    points = np.append(points_center,
                       np.append(np.append(points_left,
                                           points_right,
                                           axis=0),
                                 np.append(points_down,
                                           points_up,
                                           axis=0),
                                 axis=0),
                       axis=0)

    # Compute Voronoi
    vor = sp.spatial.Voronoi(points)

    vor.filtered_points = points_center # creates a new attibute for points that form the diagram within the region
    vor.filtered_regions = np.array(vor.regions)[vor.point_region[:vor.npoints//5]] # grabs the first fifth of the regions, which are the original regions

    return vor


# Finds the centroid of a region. First and last point should be the same.
def centroid_region(vertices):
    # Polygon's signed area
    A = 0
    # Centroid's x
    C_x = 0
    # Centroid's y
    C_y = 0
    for i in range(0, len(vertices) - 1):
        s = (vertices[i, 0] * vertices[i + 1, 1] - vertices[i + 1, 0] * vertices[i, 1])
        A = A + s
        C_x = C_x + (vertices[i, 0] + vertices[i + 1, 0]) * s
        C_y = C_y + (vertices[i, 1] + vertices[i + 1, 1]) * s
    A = 0.5 * A
    C_x = (1.0 / (6.0 * A)) * C_x
    C_y = (1.0 / (6.0 * A)) * C_y
    return np.array([[C_x, C_y]])


# Performs x iterations of loyd's algorithm to calculate a centroidal vornoi diagram
def generate_CVD(points, iterations, bounding_box):
    p = copy.copy(points)

    for i in range(iterations):
        vor = bounded_voronoi(p, bounding_box)
        centroids = []

        for region in vor.filtered_regions:
            vertices = vor.vertices[region + [region[0]], :] # grabs vertices for the region and adds a duplicate of the first one to the end
            centroid = centroid_region(vertices)
            centroids.append(list(centroid[0, :]))

        p = np.array(centroids)

    return bounded_voronoi(p, bounding_box)


# returns a pyplot of given voronoi data
def plot_vornoi_diagram(vor, bounding_box, show_figure):
    # Initializes pyplot stuff
    fig = pl.figure()
    ax = fig.gca()

    # Plot initial points
    ax.plot(vor.filtered_points[:, 0], vor.filtered_points[:, 1], 'b.')

    # Plot ridges points
    for region in vor.filtered_regions:
        vertices = vor.vertices[region, :]
        ax.plot(vertices[:, 0], vertices[:, 1], 'go')

    # Plot ridges
    for region in vor.filtered_regions:
        vertices = vor.vertices[region + [region[0]], :]
        ax.plot(vertices[:, 0], vertices[:, 1], 'k-')

    # stores references to numbers for setting axes limits
    margin_percent = .1
    width = bounding_box[1]-bounding_box[0]
    height = bounding_box[3]-bounding_box[2]

    ax.set_xlim([bounding_box[0]-width*margin_percent, bounding_box[1]+width*margin_percent])
    ax.set_ylim([bounding_box[2]-height*margin_percent, bounding_box[3]+height*margin_percent])

    if show_figure:
        pl.show()
    return fig

从Voronoi单元获取有界多边形坐标

2 个答案:

有界Voronoï图的解决方案

在显示代码之前有些乐趣

这是代码