Discrete Curvature

Theory and Applications


A colloquium on discrete curvature

Laplace-Beltrami Operator for Nonrigid Shape Comparison and Correspondence

Alon Shtern

An important operation in geometry processing is nding the correspondences between pairs of shapes. The Gromov-Hausdorff distance, a measure of dissimilarity between metric spaces, has been found to be highly useful for nonrigid shape comparison. Here, we explore the applicability of related shape similarity measures to the problem of shape correspondence, adopting spectral type distances. We propose to evaluate the spectral kernel distance, the spectral embedding distance and the novel spectral quasi-conformal distance, comparing the manifolds from di fferent viewpoints. By matching the shapes in the spectral domain, utilizing the eigen-functions and eigenvalues of the Laplace-Beltrami operator, important attributes of surface structure are being aligned. For the purpose of testing our ideas, we introduce a fully automatic framework for fi nding intrinsic correspondence between two shapes. The proposed method achieves state-of-the-art results on the Princeton isometric shape matching protocol applied, as usual, to the TOSCA and SCAPE benchmarks.

This is a joint work with Ron Kimmel

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