Proper Orthogonal Procrustes Alignment

The Proper Orthogonal Procrustes problem finds the special orthogonal rotation matrix R (satisfying R^TR = I and det(R) = +1) and translation vector t that minimize the sum of squared Euclidean distances between labeled source points and target points.

Proper Rotation vs Reflection

Unconstrained orthogonal Procrustes allows both proper rotations (det(R) = +1) and improper reflections (det(R) = -1). When aligning physical rigid bodies in 2D or 3D, reflections reverse chirality (handedness) and are physically unrealizable. Restricting the optimization to SO(2) guarantees an orientation-preserving rigid motion.

Centroid Decoupling and Planar Rotation Formula

Subtracting the source centroid c_s and target centroid c_t decouples translation from rotation, yielding centered vectors u_i = x_i - c_s and v_i = y_i - c_t. In 2D, writing R(θ) with entries R_1,1 = cos(θ), R_1,2 = -sin(θ), R_2,1 = sin(θ), R_2,2 = cos(θ) reduces the trace maximization tr(R^TV U^T) to maximizing c · cos(θ) + s · sin(θ), where c = Σ(u_i,xv_i,x + u_i,yv_i,y) and s = Σ(u_i,xv_i,y - u_i,yv_i,x). When (c, s) ≠ (0, 0), the unique optimal angle is θ = atan2(s, c) and optimal translation is t = c_t - R c_s. When c = s = 0, every rotation yields the exact same residual cost.

Worked Example

Align the source triangle (-1, 0), (1, 0), (0, 2) to the Quarter turn target shifted by (1, -2), giving target vertices (1, -3), (1, -1), (-1, -2):

Centroids: Source centroid is c_s = (0, 2/3) and target centroid is c_t = (1/3, -2).
Centered inner products: Summing over the 3 pairs gives c = 0 and s = 14/3 > 0.
Rotation: θ = atan2(14/3, 0) = +90 degrees, giving R = [[0, -1], [1, 0]].
Translation and error: t = c_t - R c_s = (1/3, -2) - (-2/3, 0) = (1, -2), achieving an exact fit with squared error 0.

Experiment

Compare exact rotations, reflections, noisy perturbations, and degenerate collapses in https://riano.app/module/orthogonal-procrustes:

Select Reflection with shifts (-1, 1). Because vertices 1 and 2 are mirrored across the y-axis while det(R) = +1 forbids reflection, the best proper rotation stays at 0 degrees with squared fitting error 8.
Select Perturbed with shifts (0, 0) to observe a least-squares compromise rotation of -12.5288 degrees and squared error 1.0406.
Select Collapsed reference or Collapsed target to verify that c = s = 0 produces status Nonunique; identity representative.

Limitations

Closed-form Procrustes alignment requires known one-to-one point correspondences and uniform isotropic error weighting. Unknown correspondences require iterative assignment schemes such as Iterative Closest Point (ICP), and heavy outliers can skew least-squares centroids.

Common Pitfalls

Forgetting to center both point clouds before computing the cross-covariance sums c and s couples rotation about the origin with translation, producing suboptimal fits whenever centroids are non-zero.

References

The SVD and planar Kabsch–Umeyama derivation for proper orthogonal Procrustes alignment follow standard numerical linear algebra references:

NIST Technical Note: A Justification of the Kabsch–Umeyama Algorithm for the Proper Orthogonal Procrustes Problem

Proper Orthogonal Procrustes Alignment