获取 Voronoi 图中特定点对应区域的面积

Question

使用this answer，我可以创建一个有界Voronoi 图（此代码归功于@Flabetvibes）：

import matplotlib.pyplot as plt
import numpy as np
import scipy as sp
import scipy.spatial
import sys

eps = sys.float_info.epsilon

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]])

points = np.array([[0.17488374, 0.36498964],
   [0.94904866, 0.80085891],
   [0.89265224, 0.4160692 ],
   [0.17035869, 0.82769497],
   [0.30274931, 0.04572908],
   [0.40515272, 0.1445514 ],
   [0.23191921, 0.08250689],
   [0.48713553, 0.94806717],
   [0.77714412, 0.46517511],
   [0.25945989, 0.76444964]])

vor = voronoi(points,(0,1,0,1))

fig = plt.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-')

现在，我想获取包含蓝色原始点之一的区域区域，例如点 [0]。在本例中，points[0] 是点 (0.17488374, 0.36498964)。我想我可以使用以下代码找到这一点的区域：

area = ConvexHull(vor.vertices[vor.filtered_regions[0], :]).volume

因为我认为 points[0] 中的 0 索引将与 vor.filtered_regions[0] 中的 0 索引相对应。但它没有—— vor.filtered_regions[9] 实际上是我正在寻找的（我手动计算出来的，但我希望它是自动化的）。在另一个示例中，索引为 2 的区域是我要查找的区域，因此它看起来也不一致。

有没有办法找到 vor.filtered_regions 的索引，这会给我我想要的区域？或者还有其他方法可以解决这个问题吗？即使我正在创建包含所有 10 个点的整个 Voronoi 图，但带有点 [0] 的区域区域是我真正要寻找的（虽然仍然是有界的），所以我假设可能会有更快这样做的方法，但我不知道那可能是什么。

Answer 1

scipy Voronoi 图的point_region 属性告诉您哪个区域与哪个点相关联。因此，您可以使用该数据来查找相关区域。

这是您的 voronoi 函数的一个非常简化的版本，它使用该属性来确保 filted_points 和 filtered_regions 的构造一致，即第一个区域是与第一个点相关联的区域。

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 = []
    [vor.point_region[i] for i in range(10)]

    vor.filtered_points = points_center
    vor.filtered_regions = [vor.regions[vor.point_region[i]] for i in range(len(points_center))]
    return vor