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Biharmonic Distance
ACM Transactions on Graphics, June 2010

Yaron Lipman, Raif Rustamov, Thomas Funkhouser


Biharmonic distance from a source point (darkest blue). Red points are furthest from the source. White lines are equally spaced in distance

Abstract

Measuring distances between pairs of points on a 3D surface is a fundamental problem in computer graphics and geometric processing. For most applications, the important properties of a distance are that it is a metric, smooth, locally isotropic, globally “shape-aware,” isometry invariant, insensitive to noise and small topology changes, parameter-free, and practical to compute on a discrete mesh. However, the basic methods currently popular in computer graphics (e.g., geodesic and diffusion distances) do not have these basic properties. In this paper, we propose a new distance measure based on the biharmonic differential operator that has all the desired properties. This new surface distance is related to the diffusion and commute-time distances, but applies different (inverse squared) weighting to the eigenvalues of the Laplace-Beltrami operator, which provides a nice trade-off between nearly geodesic distances for small distances and global shape-awareness for large distances. The paper provides theoretical and empirical analysis for a large number of meshes.

Citation (BibTeX)

Yaron Lipman, Raif Rustamov, and Thomas Funkhouser. Biharmonic Distance. ACM Transactions on Graphics 29(3), June 2010.

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