Global Optimization for Optimal Generalized Procrustes Analysis

Abstract

This paper deals with generalized procrustes analysis. This is the problem of registering a set of shape data by estimating a reference shape and a set of rigid transformations given point correspondences. The transformed shape data must align with the reference shape as best possible. This is a difficult problem. The classical approach computes alternatively the reference shape, usually as the average of the transformed shapes, and each transformation in turn. We propose a global approach to generalized procrustes analysis for two- and three-dimensional shapes. It uses modern convex optimization based on the theory of Sum Of Squares functions. We show how to convert the whole procrustes problem, including missing data, into a semidefinite program. Our approach is statistically grounded: it finds the maximum likelihood estimate. We provide results on synthetic and real datasets. Compared to classical alternation our algorithm obtains lower errors. The discrepancy is very high when similarities are estimated or when the shape data have significant deformations.