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.

@article{Lipman:2010:BD,
   author = "Yaron Lipman and Raif Rustamov and Thomas Funkhouser",
   title = "Biharmonic Distance",
   journal = "ACM Transactions on Graphics",
   year = "2010",
   month = jun,
   volume = "29",
   number = "3"
}