Subspace Procrustes Analysis

Abstract

Procrustes Analysis (PA) has been a popular technique to align and build $2$ -D statistical models of shapes. Given a set of $2$ -D shapes PA is applied to remove rigid transformations. Then, a non-rigid $2$ -D model is computed by modeling (e.g., PCA) the residual. Although PA has been widely used, it has several limitations for modeling $2$ -D shapes: occluded landmarks and missing data can result in local minima solutions, and there is no guarantee that the $2$ -D shapes provide a uniform sampling of the $3$ -D space of rotations for the object. To address previous issues, this paper proposes Subspace PA (SPA). Given several instances of a $3$ -D object, SPA computes the mean and a $2$ -D subspace that can simultaneously model all rigid and non-rigid deformations of the $3$ -D object. We propose a discrete (DSPA) and continuous (CSPA) formulation for SPA, assuming that $3$ -D samples of an object are provided. DSPA extends the traditional PA, and produces unbiased $2$ -D models by uniformly sampling different views of the $3$ -D object. CSPA provides a continuous approach to uniformly sample the space of $3$ -D rotations, being more efficient in space and time. Experiments using SPA to learn $2$ -D models of bodies from motion capture data illustrate the benefits of our approach.