Subspace Procrustes Analysis
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.
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