Module procrustes 

Procrustes analysis for aligning geometric configurations.

Overview

Procrustes analysis finds the optimal orthogonal transformation (rotation and optionally reflection and scaling) that maps one matrix onto another in the Frobenius-norm sense.

Orthogonal Procrustes Problem

Given matrices A (n × d) and B (n × d), find:

min_{R: Rᵀ R = I}  ||s · A R + 1 tᵀ − B||_F

Solution via SVD of Bᵀ A = U Σ Vᵀ:

• R = V Uᵀ (or V diag(1,…,det(VUᵀ)) Uᵀ to prevent reflections)
• Optimal scale s = trace(Σ) / ||A||_F² (when centering and scaling enabled)

Generalized Procrustes Analysis

Aligns multiple matrices to a common mean (consensus) shape via iterative pairwise Procrustes alignment, similar to the GPA algorithm of Gower (1975).

References

• Schönemann (1966): A generalized solution of the orthogonal Procrustes problem
• Gower (1975): Generalized Procrustes analysis
• Golub & Van Loan (1996): Matrix Computations, §12.4

Structs

ProcrustesConfig

Configuration for Procrustes alignment.

ProcrustesResult

Result of a Procrustes alignment.

Functions

generalized_procrustes

Generalized Procrustes Analysis (GPA): align multiple matrices to a common mean.

orthogonal_procrustes

Solve the orthogonal Procrustes problem.