Procrustes Alignment

Definition

$$\arg \underset{s,\mathbf{R},\mathbf{t}}{\min }\sum _i\Vert s\mathbf{R}{\mathbf{x}}_i+\mathbf{t}-{\mathbf{y}}_i{\Vert }^2$$

for two point sets with known correspondences ${\mathbf{x}}_i\leftrightarrow {\mathbf{y}}_i$.

Solution

With centered point matrices $\bar{\mathbf{X}},\bar{\mathbf{Y}}$ and centroids $\bar{\mathbf{x}},\bar{\mathbf{y}}$:

$${\bar{\mathbf{Y}}}^T\bar{\mathbf{X}}=\mathbf{U}\Sigma {\mathbf{V}}^T,\mathbf{R}=\mathbf{U}{\mathbf{V}}^T,s=\frac{\text{tr}(\Sigma )}{\Vert \bar{\mathbf{X}}{\Vert }_F^2},\mathbf{t}=\bar{\mathbf{y}}-s\mathbf{R}\bar{\mathbf{x}}$$

Rotation from the SVD of the cross-covariance, scale as a quotient of sums, translation as the centroid difference.

Trap

Procrustes needs the correspondences as input. Finding them is the job of ICP. The optimal rotation does not depend on the scale.

Appears in

• Lecture 5, Procrustes Solution
• Lecture 5, Proof
• Lecture 5, ICP: step 3
• Lecture 1, course overview
• Lecture 9, Relative pose: aligning DUSt3R point maps

Related concepts

• Iterative Closest Point
• Rotation Matrix