将 Voronoi 图单元区域转换为像素坐标列表

Question

我正在使用 Voronoi 图进行图像处理 (procedurally generated stippling)。 为此，我需要创建一个包含元组（x,y 像素位置）的列表（coords_within_cell）的列表（单元格）。

我已经开发了几个蛮力算法来实现这一点（见下文），但它们处理超过 ~10 个点的速度太慢。 scipy 空间实用程序的效率似乎提高了 1000 倍以上。因此，我想使用 scipy 来生成 Voronoi 图： https://docs.scipy.org/doc/scipy/reference/generated/scipy.spatial.Voronoi.html

使用 scipy 生成 Voronoi 图相当简单，但不幸的是我无法弄清楚如何将单元格区域转换为像素坐标。最好的方法是什么？

我找到了一个相关的问题，但没有答案，已被删除：https://web.archive.org/web/20200120151304/https://stackoverflow.com/questions/57703129/converting-a-voronoi-diagram-into-bitmap

蛮力算法 1（太慢）

import math
import random
from PIL import Image 

def distance(x1, y1, x2, y2):
    return math.hypot(x2 - x1, y2 - y1)

# define the size of the x and y bounds
screen_width = 1260
screen_height = 1260

# define the number of points that should be used
number_of_points = 16

# randomly generate a list of n points within the given x and y bounds
point_x_coordinates = random.sample(range(0, screen_width), number_of_points)
point_y_coordinates = random.sample(range(0, screen_height), number_of_points)
points = list(zip(point_x_coordinates, point_y_coordinates))

# each point needs to have a corresponding list of pixels
point_pixels = []
for i in range(len(points)):
    point_pixels.append([]) 

# for each pixel within bounds, determine which point it is closest to and add it to the corresponding list in point_pixels
for pixel_y_coordinate in range(screen_height):
    for pixel_x_coordinate in  range(screen_width):
        distance_to_closest_point = float('inf')
        closest_point_index = 1

        for point_index, point in enumerate(points):
            distance_to_point = distance(pixel_x_coordinate, pixel_y_coordinate, point[0], point[1])
            if(distance_to_point < distance_to_closest_point):
                closest_point_index = point_index
                distance_to_closest_point = distance_to_point
        
        point_pixels[closest_point_index].append((pixel_x_coordinate, pixel_y_coordinate))

# each point needs to have a corresponding centroid
point_pixels_centroid = []

for pixel_group in point_pixels:
    x_sum = 0
    y_sum = 0
    for pixel in pixel_group:
        x_sum += pixel[0]
        y_sum += pixel[1]
    
    x_average = x_sum / len(pixel_group)
    y_average = y_sum / len(pixel_group)

    point_pixels_centroid.append((round(x_average), round(y_average)))



# display the resulting voronoi diagram
display_voronoi = Image.new("RGB", (screen_width, screen_height), "white")
for pixel_group in point_pixels:
    rgb = random.sample(range(0, 255), 3)
    for pixel in pixel_group:
        display_voronoi.putpixel( pixel, (rgb[0], rgb[1], rgb[2], 255) )

for centroid in point_pixels_centroid:
    print(centroid)
    display_voronoi.putpixel( centroid, (1, 1, 1, 255) )

display_voronoi.show()

蛮力算法 2（也太慢了）： Based on this concept.

import math
import random
from PIL import Image 

def distance(x1, y1, x2, y2):
    return math.hypot(x2 - x1, y2 - y1)

# define the size of the x and y bounds
screen_width = 500
screen_height = 500

# define the number of points that should be used
number_of_points = 4

# randomly generate a list of n points within the given x and y bounds
point_x_coordinates = random.sample(range(0, screen_width), number_of_points)
point_y_coordinates = random.sample(range(0, screen_height), number_of_points)
points = list(zip(point_x_coordinates, point_y_coordinates))

# each point needs to have a corresponding list of pixels
point_pixels = []
for i in range(len(points)):
    point_pixels.append([]) 

# for each pixel within bounds, determine which point it is closest to and add it to the corresponding list in point_pixels
# do this by continuously growing circles outwards from the points
# if circles overlap then whoever was their first claims the location

# keep track of whether pixels have been used or not
# this is done via a 2D list of booleans
is_drawn_on = []
for i in range(screen_width):
    is_drawn_on.append([]) 
    for j in range(screen_height):
        is_drawn_on[i].append(False)

circles_are_growing = True
radius = 1
while(circles_are_growing):
    circles_are_growing = False
    for point_index, point in enumerate(points):
        for i in range(point[0] - radius, point[0] + radius):
            for j in range(point[1] - radius, point[1] + radius):
                # print(str(i)+" vs "+str(len(is_drawn_on)))
                if(i >= 0 and i < len(is_drawn_on)):
                    if(j >= 0 and j < len(is_drawn_on[i])):
                        if(not is_drawn_on[i][j] and distance(i, j, point[0], point[1]) <= radius):
                            point_pixels[point_index].append((i, j))
                            circles_are_growing = True
                            is_drawn_on[i][j] = True
    radius += 1

# each point needs to have a corresponding centroid
point_pixels_centroid = []

for pixel_group in point_pixels:
    x_sum = 0
    y_sum = 0
    for pixel in pixel_group:
        x_sum += pixel[0]
        y_sum += pixel[1]
    
    x_average = x_sum / len(pixel_group)
    y_average = y_sum / len(pixel_group)

    point_pixels_centroid.append((round(x_average), round(y_average)))



# display the resulting voronoi diagram
display_voronoi = Image.new("RGB", (screen_width, screen_height), "white")
for pixel_group in point_pixels:
    rgb = random.sample(range(0, 255), 3)
    for pixel in pixel_group:
        display_voronoi.putpixel( pixel, (rgb[0], rgb[1], rgb[2], 255) )

for centroid in point_pixels_centroid:
    print(centroid)
    display_voronoi.putpixel( centroid, (1, 1, 1, 255) )

display_voronoi.show()

Answer 1

与直接构建和查询 Voronoi 图相比，构建和查询标准搜索树更容易。下面是我使用 scipy.spatial.KDTree 对您的代码进行的修改，以确定每个像素位置的最近点，然后是结果图像（具有 500 个 Voronoi 点的 500x500 图像）。

代码仍然有点慢，但现在可以很好地扩展 Voronoi 点的数量。如果您避免为每个 Voronoi 单元构建像素位置列表，而是直接在图像中设置数据，这可能会更快。

最快的解决方案可能涉及构建 Voronoi 图表并一次遍历一个像素并关联最近的 Voronoi 单元，在需要时查看相邻的 Voronoi 单元（因为前一个像素为找到 Voronoi 单元提供了很好的猜测用于下一个像素）。但这将涉及编写更多像这样天真地使用 KDTree 的代码，并且可能不会产生巨大的收益：此时代码的缓慢部分是构建所有可以独立清理的每像素数组/数据。

import math
import random
from PIL import Image 
from scipy import spatial
import numpy as np

# define the size of the x and y bounds
screen_width = 500
screen_height = 500

# define the number of points that should be used
number_of_points = 500

# randomly generate a list of n points within the given x and y bounds
point_x_coordinates = random.sample(range(0, screen_width), number_of_points)
point_y_coordinates = random.sample(range(0, screen_height), number_of_points)
points = list(zip(point_x_coordinates, point_y_coordinates))

# each point needs to have a corresponding list of pixels
point_pixels = []
for i in range(len(points)):
    point_pixels.append([]) 

# build a search tree
tree = spatial.KDTree(points)

# build a list of pixed coordinates to query
pixel_coordinates = np.zeros((screen_height*screen_width, 2));
i = 0
for pixel_y_coordinate in range(screen_height):
    for pixel_x_coordinate in  range(screen_width):
        pixel_coordinates[i] = np.array([pixel_x_coordinate, pixel_y_coordinate])
        i = i+1

# for each pixel within bounds, determine which point it is closest to and add it to the corresponding list in point_pixels
[distances, indices] = tree.query(pixel_coordinates)

i = 0
for pixel_y_coordinate in range(screen_height):
    for pixel_x_coordinate in  range(screen_width):
        point_pixels[indices[i]].append((pixel_x_coordinate, pixel_y_coordinate))
        i = i+1

# each point needs to have a corresponding centroid
point_pixels_centroid = []

for pixel_group in point_pixels:
    x_sum = 0
    y_sum = 0
    for pixel in pixel_group:
        x_sum += pixel[0]
        y_sum += pixel[1]

    x_average = x_sum / max(len(pixel_group),1)
    y_average = y_sum / max(len(pixel_group),1)

    point_pixels_centroid.append((round(x_average), round(y_average)))


# display the resulting voronoi diagram
display_voronoi = Image.new("RGB", (screen_width, screen_height), "white")
for pixel_group in point_pixels:
    rgb = random.sample(range(0, 255), 3)
    for pixel in pixel_group:
        display_voronoi.putpixel( pixel, (rgb[0], rgb[1], rgb[2], 255) )

for centroid in point_pixels_centroid:
    #print(centroid)
    display_voronoi.putpixel( centroid, (1, 1, 1, 255) )

#display_voronoi.show()
display_voronoi.save("test.png")