根据概率密度函数p(x，y，z)随机填充3D网格 [英] Randomly fill a 3D grid according to a probability density function p(x,y,z)

问题描述

如何按给定的概率密度函数指定的顺序填充3D网格?

我想使用python，以随机顺序放置点，但要根据该区域上的某些指定概率分布，而无需重复点.

Using python, I'd like to lay down points in a random order, but according to some specified probability distribution over that region, with no repeated points.

依次:

• 创建离散的3D网格
• 指定每个网格点pdf(x，y，z)的概率密度函数
• 放置一个点(x0，y0，z0)，其随机位置与pdf(x，y，z)成正比
• 继续添加点(不重复)，直到所有位置都填满为止

期望的结果是网格中所有点的所有点的列表(无重复)，以便填充它们.

The desired result is a list of all points (no repeats) of all the points in the grid, in order that they were filled.

推荐答案

下面是使用高斯pdf的示例(请参见图).此代码很容易适应任何指定的pdf:

Here's an example, using a gaussian pdf (see plot). This code is easily adapted to any specified pdf:

%matplotlib qt 
import numpy as np
import matplotlib.pyplot as plt
from mpl_toolkits.mplot3d import Axes3D

#number of points to lay down:
n = 4000;

#create meshgrid:
min, max, L = -5, 5, 91;
[x_grid,y_grid,z_grid] = np.meshgrid(np.linspace(min,max,L),np.linspace(min,max,L),np.linspace(min,max,L))
xi,yi,zi = x_grid.ravel(),y_grid.ravel(),z_grid.ravel()

#create normalized pdf (gaussian here):
pdf = np.exp(-(x_grid**2 + y_grid**2 + z_grid**2));
pdf = pdf/np.sum(pdf);

#obtain indices of randomly selected points, as specified by pdf:
randices = np.random.choice(np.arange(x_grid.ravel().shape[0]), n, replace = False,p = pdf.ravel())

#random positions:
x_rand = xi[randices]
y_rand = yi[randices]
z_rand = zi[randices]

fig = plt.figure();
ax = fig.add_subplot(111, projection='3d',aspect='equal')
svals = 16;
ax.scatter(x_rand, y_rand, z_rand, s=svals, alpha=.1)

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