Subspace Procrustes Analysis

Xavier Perez-Sala, Fernando De la Torre, Laura Igual, Sergio Escalera, Cecilio Angulo

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.