python - 质心 Voronoi 镶嵌

Question

我正在尝试使用 scipy 包构建一个有界 Voronoi 图，并且在每次迭代中我计算 Voronoi 单元的质心，并向质心移动一些三角形，并通过更新生成器点重新计算 Voronoi 图。当我尝试绘制更新的点时，我得到一个奇怪的错误，因为我绘制的点不在预期的位置。这是代码

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

np.random.seed(1)
eps = sys.float_info.epsilon

n_robots = 10
robots = np.random.rand(n_robots, 2)
#print(robots)
bounding_box = np.array([0., 1., 0., 1.]) 

def in_box(robots, bounding_box):
    return np.logical_and(np.logical_and(bounding_box[0] <= robots[:, 0],
                                         robots[:, 0] <= bounding_box[1]),
                          np.logical_and(bounding_box[2] <= robots[:, 1],
                                         robots[:, 1] <= bounding_box[3]))


def voronoi(robots, bounding_box):
    i = in_box(robots, bounding_box)
    points_center = robots[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 = []
    ind = np.arange(points.shape[0])
    ind = np.expand_dims(ind,axis= 1)


    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):

    A = 0

    C_x = 0

    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]])

def plot(r,index):
    vor = voronoi(r, bounding_box)

    fig = pl.figure()
    ax = fig.gca()
# Plot initial points
    ax.plot(vor.filtered_points[:, 0], vor.filtered_points[:, 1], 'b.')
    print("initial",vor.filtered_points)
# 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.')
    centroids = np.asarray(centroids)
    rob = np.copy(vor.filtered_points)
    # the below code is for the plotting purpose the update happens in the update function
    interim_x = np.asarray(centroids[:,0] - rob[:,0])
    interim_y = np.asarray(centroids[:,1] - rob[:,1])
    magn = [np.linalg.norm(centroids[i,:] - rob[i,:]) for i in range(rob.shape[0])]
    x = np.copy(interim_x)
    x = np.asarray([interim_x[i]/magn[i] for i in range(interim_x.shape[0])])
    y = np.copy(interim_y)
    y = np.asarray([interim_y[i]/magn[i] for i in range(interim_y.shape[0])])
    nor = np.copy(rob)
    for i in range(x.shape[0]):
        nor[i,0] = x[i]
        nor[i,1] = y[i]
    temp = np.copy(rob)
    temp[:,0] = [rob[i,0] + 0.5*interim_x[i] for i in range(rob.shape[0])]
    temp[:,1] = [rob[i,1] + 0.5*interim_y[i] for i in range(rob.shape[0])]
    ax.plot(temp[:,0] ,temp[:,1], 'y.' )
    ax.set_xlim([-0.1, 1.1])
    ax.set_ylim([-0.1, 1.1])
    pl.savefig("voronoi" + str(index) + ".png")
    return centroids

def update(rob,centroids):

  interim_x = np.asarray(centroids[:,0] - rob[:,0])
  interim_y = np.asarray(centroids[:,1] - rob[:,1])
  magn = [np.linalg.norm(centroids[i,:] - rob[i,:]) for i in range(rob.shape[0])]
  x = np.copy(interim_x)
  x = np.asarray([interim_x[i]/magn[i] for i in range(interim_x.shape[0])])
  y = np.copy(interim_y)
  y = np.asarray([interim_y[i]/magn[i] for i in range(interim_y.shape[0])])
  nor = [np.linalg.norm([x[i],y[i]]) for i in range(x.shape[0])]
  temp = np.copy(rob)
  temp[:,0] = [rob[i,0] + 0.5*interim_x[i] for i in range(rob.shape[0])]
  temp[:,1] = [rob[i,1] + 0.5*interim_y[i] for i in range(rob.shape[0])]
  return np.asarray(temp)

if __name__ == '__main__':
    for i in range(1):
        centroids = plot(robots,i)
        robots = update(robots,centroids)

这里还有一张图片，展示了代码的作用。蓝色点是生成器点，红色是质心，黄色应该是蓝色点和红色点之间的中点。但正如您所见，黄色点不在蓝色点和红色点之间。

Answer 1

问题是当你的 points 被馈送到 Voronoi 时，在镶嵌构造的过程中膨胀了，当你稍后过滤掉它们时，点的顺序是错误的.因此，当您在 voronoi() 中设置 vor.filtered_points = points_center 时，与区域的顺序相比，点会被打乱。因此，当您正确计算中点时，您使用的是错误的点对。

我在这里用绿色圈出了两个正确的配对，用红色圈出了一个不正确的配对: 从红色圆圈可以看出，边缘单元格中的基点与相邻单元格的质心成对。

解决方案很简单:当您过滤区域并找到要保留的区域时，您需要收集落在相应区域内的点。您可以通过将 vor.points 与 vor.point_region 匹配并找到相应的区域来完成此操作，为此您需要枚举您的 区域:

# Compute Voronoi
vor = sp.spatial.Voronoi(points)
# Filter regions and select corresponding points
regions = []
points_to_filter = [] # we'll need to gather points too
ind = np.arange(points.shape[0])
ind = np.expand_dims(ind,axis= 1)

for i,region in enumerate(vor.regions): # enumerate the regions
    if not region: # nicer to skip the empty region altogether
        continue

    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 flag:
        regions.append(region)

        # find the point which lies inside
        points_to_filter.append(vor.points[vor.point_region == i][0,:])

vor.filtered_points = np.array(points_to_filter)
vor.filtered_regions = regions

通过这些修改，平均效果很好: