具有已知对应关系的两个点云的刚性配准

Question

想象一下我有两个（python）列表（具有有限的）3D 点数量。如何找到一个刚性变换来尽可能紧密地匹配这些点。对于每个列表的每个点，已知该点对应于哪个其他点。有这方面的算法/库吗？

我找到了迭代最近点算法，但这假设没有已知的对应关系，并且它似乎是针对大型点云而设计的。我说的是 3 到 8 点的有限组。

可能的点集（根据列表中的索引对应）

a = [[0,0,0], [1,0,0],[0,1,0]]
b = [[0,0,0.5], [1,0,1],[0,0,2]]

Answer 1

事实证明确实存在一个解析解。Arun 等人 1987 年的论文《两个 3D 点集的最小二乘拟合》对此进行了描述

我编写了一个测试脚本来测试他们的算法，它似乎工作得很好（如果你想要一个最小化平方误差之和的解决方案，如果你有一个异常值，这可能不理想）：

import numpy as np

##Based on Arun et al., 1987

#Writing points with rows as the coordinates
p1_t = np.array([[0,0,0], [1,0,0],[0,1,0]])
p2_t = np.array([[0,0,1], [1,0,1],[0,0,2]]) #Approx transformation is 90 degree rot over x-axis and +1 in Z axis

#Take transpose as columns should be the points
p1 = p1_t.transpose()
p2 = p2_t.transpose()

#Calculate centroids
p1_c = np.mean(p1, axis = 1).reshape((-1,1)) #If you don't put reshape then the outcome is 1D with no rows/colums and is interpeted as rowvector in next minus operation, while it should be a column vector
p2_c = np.mean(p2, axis = 1).reshape((-1,1))

#Subtract centroids
q1 = p1-p1_c
q2 = p2-p2_c

#Calculate covariance matrix
H=np.matmul(q1,q2.transpose())

#Calculate singular value decomposition (SVD)
U, X, V_t = np.linalg.svd(H) #the SVD of linalg gives you Vt

#Calculate rotation matrix
R = np.matmul(V_t.transpose(),U.transpose())

assert np.allclose(np.linalg.det(R), 1.0), "Rotation matrix of N-point registration not 1, see paper Arun et al."

#Calculate translation matrix
T = p2_c - np.matmul(R,p1_c)

#Check result
result = T + np.matmul(R,p1)
if np.allclose(result,p2):
    print("transformation is correct!")
else:
    print("transformation is wrong...")