update

umeyama.py

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+import numpy as np
+
+def umeyama(src, dst, estimate_scale=True):
+    """Umeyama algorithm to estimate similarity transformation."""
+    assert src.shape == dst.shape
+
+    # Compute the mean of the source and destination points
+    src_mean = np.mean(src, axis=0)
+    dst_mean = np.mean(dst, axis=0)
+
+    # Subtract the means from the points
+    src_centered = src - src_mean
+    dst_centered = dst - dst_mean
+
+    # Compute the covariance matrix
+    cov_matrix = np.dot(dst_centered.T, src_centered) / src.shape[0]
+
+    # Singular Value Decomposition
+    U, D, Vt = np.linalg.svd(cov_matrix)
+
+    # Compute the rotation matrix
+    R = np.dot(U, Vt)
+    if np.linalg.det(R) < 0:
+        Vt[-1, :] *= -1
+        R = np.dot(U, Vt)
+
+    # Compute the scale factor
+    if estimate_scale:
+        var_src = np.var(src_centered, axis=0).sum()
+        scale = 1.0 / var_src * np.sum(D)
+    else:
+        scale = 1.0
+
+    # Compute the translation vector
+    t = dst_mean - scale * np.dot(R, src_mean)
+
+    # Create the transformation matrix
+    T = np.identity(3)
+    T[:2, :2] = scale * R
+    T[:2, 2] = t
+
+    return T
+
+# Generate 20 random 2D points
+np.random.seed(42)  # For reproducibility
+src_points = np.random.rand(20, 2)
+
+# Define a known rotation matrix R and translation vector t
+theta = np.pi / 4  # 45 degrees rotation
+R = np.array([
+    [np.cos(theta), -np.sin(theta)],
+    [np.sin(theta), np.cos(theta)]
+])
+t = np.array([1.0, 2.0])
+
+# Apply the transformation to generate the destination points
+dst_points = np.dot(src_points, R.T) + t
+
+# Perform Umeyama to estimate the transformation
+T = umeyama(src_points, dst_points)
+
+# Apply the resulting transformation to the source points
+src_points_hom = np.hstack((src_points, np.ones((src_points.shape[0], 1))))
+aligned_points = np.dot(T, src_points_hom.T).T[:, :2]
+
+# Calculate the difference between the destination points and the aligned points
+difference = np.linalg.norm(dst_points - aligned_points)
+
+print("Original Source Points:\n", src_points)
+print("Transformed Destination Points:\n", dst_points)
+print("Recovered Aligned Points:\n", aligned_points)
+print("\nDifference between destination and aligned points:", difference)
+