我有点（例如，lat、lon 对蜂窝塔位置），我需要获取它们形成的 Voronoi 单元的多边形。
from scipy.spatial import Voronoi
tower = [[ 24.686 , 46.7081],
[ 24.686 , 46.7081],
[ 24.686 , 46.7081]]
c = Voronoi(towers)
现在，我需要获取每个单元格的纬度、经度坐标中的多边形边界（以及该多边形包围的质心是什么）。我也需要这个 Voronoi 有界。这意味着边界不会无限远，而是在边界框内。
【问题讨论】：
标签： python computational-geometry polygons voronoi
给定一个矩形边界框，我的第一个想法是在这个边界框和scipy.spatial.Voronoi 生成的 Voronoï 图之间定义一种相交操作。这个想法不一定很好，因为这需要编写大量计算几何的基本函数。
但是，这是我想到的第二个想法（hack？）：计算平面中一组n 点的 Voronoï 图的算法具有O(n ln(n)) 的时间复杂度。添加点以约束初始点的 Voronoï 单元位于边界框内怎么样？
一张照片胜过一场精彩的演讲：
我在这里做了什么？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x = 0.0（边界框的左边缘）通过反射对称得到左侧的点（蓝色）（即[-1.0, 0.0] x [0.0, 1.0]）。根据x = 1.0、y = 0.0 和y = 1.0（边界框的其他边缘）的反射对称性，我得到了完成这项工作所需的所有点（蓝色）。
然后我运行scipy.spatial.Voronoi。上图描绘了生成的 Voronoï 图（我使用 scipy.spatial.voronoi_plot_2d）。
接下来要做什么？只需根据边界框过滤点、边或面。并根据众所周知的公式得到每个人脸的质心计算centroid of polygon。这是结果的图像（质心为红色）：
太棒了！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呢？
第 2 步
第 10 步
第 25 步
酷！ Voronoï 细胞倾向于最小化它们的能量...
import matplotlib.pyplot as pl
import numpy as np
import scipy as sp
import scipy.spatial
import sys
eps = sys.float_info.epsilon
n_towers = 100
towers = np.random.rand(n_towers, 2)
bounding_box = np.array([0., 1., 0., 1.]) # [x_min, x_max, y_min, y_max]
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]))
def voronoi(towers, bounding_box):
# Select towers inside the bounding box
i = in_box(towers, bounding_box)
# Mirror points
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)
# Filter regions
regions = []
for region in vor.regions:
flag = True
for index in region:
if index == -1:
flag = False
break
else:
x = vor.vertices[index, 0]
y = vor.vertices[index, 1]
if not(bounding_box[0] - eps <= x and x <= bounding_box[1] + eps and
bounding_box[2] - eps <= y and y <= bounding_box[3] + eps):
flag = False
break
if region != [] and flag:
regions.append(region)
vor.filtered_points = points_center
vor.filtered_regions = regions
return vor
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]])
vor = voronoi(towers, bounding_box)
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-')
# Compute and plot centroids
centroids = []
for region in vor.filtered_regions:
vertices = vor.vertices[region + [region[0]], :]
centroid = centroid_region(vertices)
centroids.append(list(centroid[0, :]))
ax.plot(centroid[:, 0], centroid[:, 1], 'r.')
ax.set_xlim([-0.1, 1.1])
ax.set_ylim([-0.1, 1.1])
pl.savefig("bounded_voronoi.png")
sp.spatial.voronoi_plot_2d(vor)
pl.savefig("voronoi.png")
【讨论】：
scipy.spatial.Voronoi() 的点的前 1/5。这意味着您可以使用为每个点提供的Voronoi.point_region 属性，它所属的区域的索引：vor.filtered_regions = np.array(vor.regions)[vor.point_region[:vor.npoints//5]]。这 a) 节省了一堆手动检查，并且 b) 保证区域以与其匹配点相同的顺序返回（这对我的用例至关重要 :)）
我在使用 scipy 的 voronoi 函数和创建 CVD 时遇到了很多麻烦，所以这些精彩的帖子和 cmets 帮助很大。作为一个编程新手，我试图理解来自 Flabetvvibes 答案的代码，我将分享我对它如何与 Energya 和我自己的修改一起工作的解释。我还在此答案的底部完整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，该数组表示来自塔的哪些坐标在边界框中。
# 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 镜像点以允许沿边界框内边缘的区域是有限的。 Scipy 的 voronoi 方法对于未定义的顶点返回 -1，因此镜像点允许边界框内的所有区域都是有限的，并且所有无限区域都在边界框外的镜像区域中，稍后将被丢弃。
# 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 方法的最后一位调用 scipy 的 voronoi 函数并为边界框内的过滤点和区域添加新属性。 Energya 建议删除 Flabetvvibe 的代码，该代码手动找到边界框内的所有有限区域，并使用一条线获得前五分之一的区域，这些区域是原始输入以及构成边界框的点。
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，并将其形成为一种易于使用的方法。对于每次迭代，调用先前的 bounded_voronoi 函数，然后为每个单元找到质心，它们成为下一次迭代的新点集。 vertices = vor.vertices[region + [region[0]], :] 简单地抓取当前区域的所有顶点并将第一个顶点复制到末尾，这样第一个和最后一个顶点相同用于计算质心。
感谢 Flabetvvibes 和 Energya。您的帖子/答案教会了我如何比其文档更好地使用 scipy 的 voronoi 方法。我还将代码作为一个单独的主体发布给任何其他寻找复制/粘贴的人。
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
【讨论】：