为 Voronoi 图着色

问题描述：

我正在尝试为使用创建的 Voronoi 图着色scipy.spatial.Voronoi。这是我的代码：

import numpy as np
import matplotlib.pyplot as plt
from scipy.spatial import Voronoi, voronoi_plot_2d

# make up data points
points = np.random.rand(15,2)

# compute Voronoi tesselation
vor = Voronoi(points)

# plot
voronoi_plot_2d(vor)

# colorize
for region in vor.regions:
    if not -1 in region:
        polygon = [vor.vertices[i] for i in region]
        plt.fill(*zip(*polygon))

plt.show()

生成的图像：

如您所见，图像边界上的一些 Voronoi 区域没有颜色。这是因为这些区域的 Voronoi 顶点的一些索引设置为-1，即 Voronoi 图之外的那些顶点。根据文档：

区域：（整数列表列表，形状（nregions，*））形成每个 Voronoi 区域的 Voronoi 顶点的索引。-1 表示 Voronoi 图之外的顶点。

为了给这些区域着色，我尝试过从多边形中删除这些“外部”顶点，但没用。我认为，我需要在图像区域的边界上填充一些点，但我似乎无法弄清楚如何合理地实现这一点。

有人可以帮忙吗？

解决方案 1：

Voronoi 数据结构包含构建“无穷远点”位置所需的所有信息。Qhull 还将它们简单地报告为-1索引，因此 Scipy 不会为您计算它们。

https://gist.github.com/pv/8036995

http://nbviewer.ipython.org/gist/pv/8037100

import numpy as np
import matplotlib.pyplot as plt
from scipy.spatial import Voronoi

def voronoi_finite_polygons_2d(vor, radius=None):
    """
    Reconstruct infinite voronoi regions in a 2D diagram to finite
    regions.

    Parameters
    ----------
    vor : Voronoi
        Input diagram
    radius : float, optional
        Distance to 'points at infinity'.

    Returns
    -------
    regions : list of tuples
        Indices of vertices in each revised Voronoi regions.
    vertices : list of tuples
        Coordinates for revised Voronoi vertices. Same as coordinates
        of input vertices, with 'points at infinity' appended to the
        end.

    """

    if vor.points.shape[1] != 2:
        raise ValueError("Requires 2D input")

    new_regions = []
    new_vertices = vor.vertices.tolist()

    center = vor.points.mean(axis=0)
    if radius is None:
        radius = vor.points.ptp().max()

    # Construct a map containing all ridges for a given point
    all_ridges = {}
    for (p1, p2), (v1, v2) in zip(vor.ridge_points, vor.ridge_vertices):
        all_ridges.setdefault(p1, []).append((p2, v1, v2))
        all_ridges.setdefault(p2, []).append((p1, v1, v2))

    # Reconstruct infinite regions
    for p1, region in enumerate(vor.point_region):
        vertices = vor.regions[region]

        if all(v >= 0 for v in vertices):
            # finite region
            new_regions.append(vertices)
            continue

        # reconstruct a non-finite region
        ridges = all_ridges[p1]
        new_region = [v for v in vertices if v >= 0]

        for p2, v1, v2 in ridges:
            if v2 < 0:
                v1, v2 = v2, v1
            if v1 >= 0:
                # finite ridge: already in the region
                continue

            # Compute the missing endpoint of an infinite ridge

            t = vor.points[p2] - vor.points[p1] # tangent
            t /= np.linalg.norm(t)
            n = np.array([-t[1], t[0]])  # normal

            midpoint = vor.points[[p1, p2]].mean(axis=0)
            direction = np.sign(np.dot(midpoint - center, n)) * n
            far_point = vor.vertices[v2] + direction * radius

            new_region.append(len(new_vertices))
            new_vertices.append(far_point.tolist())

        # sort region counterclockwise
        vs = np.asarray([new_vertices[v] for v in new_region])
        c = vs.mean(axis=0)
        angles = np.arctan2(vs[:,1] - c[1], vs[:,0] - c[0])
        new_region = np.array(new_region)[np.argsort(angles)]

        # finish
        new_regions.append(new_region.tolist())

    return new_regions, np.asarray(new_vertices)

# make up data points
np.random.seed(1234)
points = np.random.rand(15, 2)

# compute Voronoi tesselation
vor = Voronoi(points)

# plot
regions, vertices = voronoi_finite_polygons_2d(vor)
print "--"
print regions
print "--"
print vertices

# colorize
for region in regions:
    polygon = vertices[region]
    plt.fill(*zip(*polygon), alpha=0.4)

plt.plot(points[:,0], points[:,1], 'ko')
plt.xlim(vor.min_bound[0] - 0.1, vor.max_bound[0] + 0.1)
plt.ylim(vor.min_bound[1] - 0.1, vor.max_bound[1] + 0.1)

plt.show()

解决方案 2：

我对这个问题有一个更简单的解决方案，那就是在调用 Voronoi 算法之前将 4 个远距离虚拟点添加到你的点列表中。

根据您的代码，我添加了两行。

import numpy as np
import matplotlib.pyplot as plt
from scipy.spatial import Voronoi, voronoi_plot_2d

# make up data points
points = np.random.rand(15,2)

# add 4 distant dummy points
points = np.append(points, [[999,999], [-999,999], [999,-999], [-999,-999]], axis = 0)

# compute Voronoi tesselation
vor = Voronoi(points)

# plot
voronoi_plot_2d(vor)

# colorize
for region in vor.regions:
    if not -1 in region:
        polygon = [vor.vertices[i] for i in region]
        plt.fill(*zip(*polygon))

# fix the range of axes
plt.xlim([0,1]), plt.ylim([0,1])

plt.show()

那么得到的图形就如下所示。

解决方案 3：

我认为，如果不重新进行至少部分 voronoi 计算，vor 结构中可用的数据信息不足以解决这个问题。既然如此，以下是原始 voronoi_plot_2d 函数的相关部分，您应该能够使用这些部分提取与 vor.max_bound 或 vor.min_bound 相交的点（这些点是图表的左下角和右上角），以便找出多边形的其他坐标。

for simplex in vor.ridge_vertices:
    simplex = np.asarray(simplex)
    if np.all(simplex >= 0):
        ax.plot(vor.vertices[simplex,0], vor.vertices[simplex,1], 'k-')

ptp_bound = vor.points.ptp(axis=0)
center = vor.points.mean(axis=0)
for pointidx, simplex in zip(vor.ridge_points, vor.ridge_vertices):
    simplex = np.asarray(simplex)
    if np.any(simplex < 0):
        i = simplex[simplex >= 0][0]  # finite end Voronoi vertex

        t = vor.points[pointidx[1]] - vor.points[pointidx[0]]  # tangent
        t /= np.linalg.norm(t)
        n = np.array([-t[1], t[0]])  # normal

        midpoint = vor.points[pointidx].mean(axis=0)
        direction = np.sign(np.dot(midpoint - center, n)) * n
        far_point = vor.vertices[i] + direction * ptp_bound.max()

        ax.plot([vor.vertices[i,0], far_point[0]],
                [vor.vertices[i,1], far_point[1]], 'k--')