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A colloquium on discrete curvature

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Spectral GMDS: A Quasi-Linear Approach for Surface Matching

Y. Aflalo, A. Dubrovina, & R. Kimmel

The Laplace-Beltrami operator has become a ubiquitous tool in surface processing. In particular, when applied to surface coordinates, it provides its mean curvature vector. Here we show how a natural surface spectral decomposition provided by the Laplace-Beltrami operator can be exploited to efficiently obtain dense mapping between two surfaces.

Toward that end, we discuss the Multidimensional scaling (MDS), which is a family of methods that embed a given set of points into a simple, usually flat, domain. The points are assumed to be sampled from some metric space, and the mapping attempts to preserve the distances between each pair of points in the set. Distances in the target space can be computed analytically in this setting. Generalized MDS is an extension that allows mapping one metric space into another, that is, multidimensional scaling into target spaces in which distances are evaluated numerically rather than analytically. Here, we propose an efficient approach for computing such mappings between surfaces based on their natural spectral decomposition given by the Laplace-Beltrami operator, where the surfaces are treated as sampled metric-spaces. The resulting spectral-GMDS procedure enables efficient embedding by implicitly incorporating smoothness of the mapping into the problem, thereby substantially reducing the complexity involved in its solution while practically overcoming its non-convex nature. The method is compared to existing techniques that compute dense correspondence between shapes. Numerical experiments of the proposed method demonstrate its efficiency and accuracy compared to state-of-the-art approaches.