已知对应的两点云的刚性配准

Question

假设我有两个（python）列表（数量有限）的3D点。如何找到刚性变换以尽可能接近地匹配点。对于每个列表的每个点，都知道该点对应的其他点。是否有用于此的算法/库

我找到了迭代的最近点算法，但这假设没有已知的对应关系，并且它似乎适用于大型点云。我说的是3到8点的有限集合

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

事实上，有一个解析解。这在Arun et al., 1987, Least square fitting of two 3D point sets的论文中有描述

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（如果您想要一个最小化平方误差总和的解决方案，如果您有一个异常值，这可能并不理想）：

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...")