A statistical model of Riemannian metric variation for deformable shape analysis

The analysis of deformable 3D shape is often cast in terms of the shape´s intrinsic geometry due to its invariance to a wide range of non-rigid deformations. However, object´s plasticity in non-rigid transformation often result in transformations that are not completely isometric in the surface´s geometry and whose mode of deviation from isometry is an identifiable characteristic of the shape and its deformation modes. In this paper, we propose a novel generative model of the variations of the intrinsic metric of deformable shapes, based on the spectral decomposition of the Laplace-Beltrami operator. To this end, we assume two independent models for the eigenvectors and the eigenvalues of the graph-Laplacian of a 3D mesh which are learned in a supervised way from a set of shapes belonging to the same class. We show how this model can be efficiently learned given a set of 3D meshes, and evaluate the performance of the resulting generative model in shape classification and retrieval tasks. Comparison with state-of-the-art solutions for these problems confirm the validity of the approach.