Stratified Generalized Procrustes Analysis

Abstract

Generalized procrustes analysis computes the best set of transformations that relate matched shape data. In shape analysis the transformations are usually chosen as similarities, while in general statistical data analysis other types of transformation groups such as the affine group may be used. Generalized procrustes analysis has a nonlinear and nonconvex formulation. The classical approach alternates the computation of a so-called reference shape and the computation of transformations relating this reference shape to each shape datum in turn. We propose the stratified approach to generalized procrustes analysis. It first uses the affine transformation group to analyze the data and then upgrades the solution to the sought after group, whether Euclidean or similarity. We derive a convex formulation for each of these two steps, and efficient practical algorithms that gracefully handle missing data (incomplete shapes). Extensive experimental results show that our approaches perform well on simulated and real data. In particular our closed-form solution gives very accurate results for generalized procrustes analysis of Euclidean data.