NumPy / SciPy或相关库中是否有类似Matlab的procrustes函数？
以供参考。 Procrustes分析的目的是对齐2组点（换句话说，是2个形状），以通过消除比例，平移和旋转扭曲分量来最小化它们之间的平方距离。
Matlab中的示例：
X = [0 1; 2 3; 4 5; 6 7; 8 9]; % first shape
R = [1 2; 2 1]; % rotation matrix
t = [3 5]; % translation vector
Y = X * R + repmat(t, 5, 1); % warped shape, no scale and no distortion
[d Z] = procrustes(X, Y); % Z is Y aligned back to X
Z
Z =
0.0000 1.0000
2.0000 3.0000
4.0000 5.0000
6.0000 7.0000
8.0000 9.0000
NumPy中的相同任务：
X = arange(10).reshape((5, 2))
R = array([[1, 2], [2, 1]])
t = array([3, 5])
Y = dot(X, R) + t
Z = ???
注意：我只对对齐的形状感兴趣，因为平方误差（在Matlab代码中为变量d）很容易从2个形状中计算出来。
参考方案
我不知道Python中是否已有任何实现，但是使用edit procrustes.m轻松查看MATLAB代码并将其移植到Numpy很容易：
def procrustes(X, Y, scaling=True, reflection='best'):
"""
A port of MATLAB's `procrustes` function to Numpy.
Procrustes analysis determines a linear transformation (translation,
reflection, orthogonal rotation and scaling) of the points in Y to best
conform them to the points in matrix X, using the sum of squared errors
as the goodness of fit criterion.
d, Z, [tform] = procrustes(X, Y)
Inputs:
------------
X, Y
matrices of target and input coordinates. they must have equal
numbers of points (rows), but Y may have fewer dimensions
(columns) than X.
scaling
if False, the scaling component of the transformation is forced
to 1
reflection
if 'best' (default), the transformation solution may or may not
include a reflection component, depending on which fits the data
best. setting reflection to True or False forces a solution with
reflection or no reflection respectively.
Outputs
------------
d
the residual sum of squared errors, normalized according to a
measure of the scale of X, ((X - X.mean(0))**2).sum()
Z
the matrix of transformed Y-values
tform
a dict specifying the rotation, translation and scaling that
maps X --> Y
"""
n,m = X.shape
ny,my = Y.shape
muX = X.mean(0)
muY = Y.mean(0)
X0 = X - muX
Y0 = Y - muY
ssX = (X0**2.).sum()
ssY = (Y0**2.).sum()
# centred Frobenius norm
normX = np.sqrt(ssX)
normY = np.sqrt(ssY)
# scale to equal (unit) norm
X0 /= normX
Y0 /= normY
if my < m:
Y0 = np.concatenate((Y0, np.zeros(n, m-my)),0)
# optimum rotation matrix of Y
A = np.dot(X0.T, Y0)
U,s,Vt = np.linalg.svd(A,full_matrices=False)
V = Vt.T
T = np.dot(V, U.T)
if reflection is not 'best':
# does the current solution use a reflection?
have_reflection = np.linalg.det(T) < 0
# if that's not what was specified, force another reflection
if reflection != have_reflection:
V[:,-1] *= -1
s[-1] *= -1
T = np.dot(V, U.T)
traceTA = s.sum()
if scaling:
# optimum scaling of Y
b = traceTA * normX / normY
# standarised distance between X and b*Y*T + c
d = 1 - traceTA**2
# transformed coords
Z = normX*traceTA*np.dot(Y0, T) + muX
else:
b = 1
d = 1 + ssY/ssX - 2 * traceTA * normY / normX
Z = normY*np.dot(Y0, T) + muX
# transformation matrix
if my < m:
T = T[:my,:]
c = muX - b*np.dot(muY, T)
#transformation values
tform = {'rotation':T, 'scale':b, 'translation':c}
return d, Z, tform
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