Abstract

Three-dimensional ob ject shape is commonly represented in terms of deformations of a triangular mesh from an exemplar shape. Ex- isting models, however, are based on a Euclidean representation of shape deformations. In contrast, we argue that shape has a manifold structure: For example, summing the shape deformations for two people does not necessarily yield a deformation corresponding to a valid human shape, nor does the Euclidean difference of these two deformations provide a meaningful measure of shape dissimilarity. Consequently, we define a novel manifold for shape representation, with emphasis on body shapes, using a new Lie group of deformations. This has several advantages. First we define triangle deformations exactly, removing non-physical deforma- tions and redundant degrees of freedom common to previous methods. Second, the Riemannian structure of Lie Bodies enables a more mean- ingful definition of body shape similarity by measuring distance between bodies on the manifold of body shape deformations. Third, the group structure allows the valid composition of deformations. This is important for models that factor body shape deformations into multiple causes or represent shape as a linear combination of basis shapes. Finally, body shape variation is modeled using statistics on manifolds. Instead of mod- eling Euclidean shape variation with Principal Component Analysis we capture shape variation on the manifold using Principal Geodesic Analy- sis. Our experiments show consistent visual and quantitative advantages of Lie Bodies over traditional Euclidean models of shape deformation and our representation can be easily incorporated into existing methods.