initial

mathlib/umeyama.py

@@ -0,0 +1,71 @@
+import numpy as np
+
+def umeyama(src, dst, estimate_scale):
+    """Estimate N-D similarity transformation with or without scaling.
+    Parameters
+    ----------
+    src : (M, N) array
+        Source coordinates.
+    dst : (M, N) array
+        Destination coordinates.
+    estimate_scale : bool
+        Whether to estimate scaling factor.
+    Returns
+    -------
+    T : (N + 1, N + 1)
+        The homogeneous similarity transformation matrix. The matrix contains
+        NaN values only if the problem is not well-conditioned.
+    References
+    ----------
+    .. [1] "Least-squares estimation of transformation parameters between two
+            point patterns", Shinji Umeyama, PAMI 1991, DOI: 10.1109/34.88573
+    """
+
+    num = src.shape[0]
+    dim = src.shape[1]
+
+    # Compute mean of src and dst.
+    src_mean = src.mean(axis=0)
+    dst_mean = dst.mean(axis=0)
+
+    # Subtract mean from src and dst.
+    src_demean = src - src_mean
+    dst_demean = dst - dst_mean
+
+    # Eq. (38).
+    A = np.dot(dst_demean.T, src_demean) / num
+
+    # Eq. (39).
+    d = np.ones((dim,), dtype=np.double)
+    if np.linalg.det(A) < 0:
+        d[dim - 1] = -1
+
+    T = np.eye(dim + 1, dtype=np.double)
+
+    U, S, V = np.linalg.svd(A)
+
+    # Eq. (40) and (43).
+    rank = np.linalg.matrix_rank(A)
+    if rank == 0:
+        return np.nan * T
+    elif rank == dim - 1:
+        if np.linalg.det(U) * np.linalg.det(V) > 0:
+            T[:dim, :dim] = np.dot(U, V)
+        else:
+            s = d[dim - 1]
+            d[dim - 1] = -1
+            T[:dim, :dim] = np.dot(U, np.dot(np.diag(d), V))
+            d[dim - 1] = s
+    else:
+        T[:dim, :dim] = np.dot(U, np.dot(np.diag(d), V.T))
+
+    if estimate_scale:
+        # Eq. (41) and (42).
+        scale = 1.0 / src_demean.var(axis=0).sum() * np.dot(S, d)
+    else:
+        scale = 1.0
+
+    T[:dim, dim] = dst_mean - scale * np.dot(T[:dim, :dim], src_mean.T)
+    T[:dim, :dim] *= scale
+
+    return T