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# ------------------------------------------------------------------------------
# Adapted from https://github.com/akanazawa/hmr
# Original licence: Copyright (c) 2018 akanazawa, under the MIT License.
# ------------------------------------------------------------------------------
import numpy as np
def compute_similarity_transform(source_points, target_points):
"""Computes a similarity transform (sR, t) that takes a set of 3D points
source_points (N x 3) closest to a set of 3D points target_points, where R
is an 3x3 rotation matrix, t 3x1 translation, s scale. And return the
transformed 3D points source_points_hat (N x 3). i.e. solves the orthogonal
Procrutes problem.
Note:
Points number: N
Args:
source_points (np.ndarray): Source point set with shape [N, 3].
target_points (np.ndarray): Target point set with shape [N, 3].
Returns:
np.ndarray: Transformed source point set with shape [N, 3].
"""
assert target_points.shape[0] == source_points.shape[0]
assert target_points.shape[1] == 3 and source_points.shape[1] == 3
source_points = source_points.T
target_points = target_points.T
# 1. Remove mean.
mu1 = source_points.mean(axis=1, keepdims=True)
mu2 = target_points.mean(axis=1, keepdims=True)
X1 = source_points - mu1
X2 = target_points - mu2
# 2. Compute variance of X1 used for scale.
var1 = np.sum(X1**2)
# 3. The outer product of X1 and X2.
K = X1.dot(X2.T)
# 4. Solution that Maximizes trace(R'K) is R=U*V', where U, V are
# singular vectors of K.
U, _, Vh = np.linalg.svd(K)
V = Vh.T
# Construct Z that fixes the orientation of R to get det(R)=1.
Z = np.eye(U.shape[0])
Z[-1, -1] *= np.sign(np.linalg.det(U.dot(V.T)))
# Construct R.
R = V.dot(Z.dot(U.T))
# 5. Recover scale.
scale = np.trace(R.dot(K)) / var1
# 6. Recover translation.
t = mu2 - scale * (R.dot(mu1))
# 7. Transform the source points:
source_points_hat = scale * R.dot(source_points) + t
source_points_hat = source_points_hat.T
return source_points_hat
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