Interpolated eigenfunctions for volumetric shape processing

This paper introduces a set of volumetric functions suitable for geometric processing of volumes. We start with Laplace-Beltrami eigenfunctions on the bounding surface and interpolate them into the interior using barycentric coordinates. The interpolated eigenfunctions: (1) can be computed efficiently by using the boundary mesh only; (2) can be seen as a shape-aware generalization of barycentric coordinates; (3) can be used for efficiently representing volumetric functions; (4) can be naturally plugged into existing spectral embedding constructions such as the diffusion embedding to provide their volumetric counterparts. Using the interior diffusion embedding, we define the interior Heat Kernel Signature (iHKS) and examine its performance for the task of volumetric point correspondence. We show that the three main qualities of the surface Heat Kernel Signature-being informative, multiscale, and insensitive to pose-are inherited by this volumetric construction. Next, we construct a bag of features based shape descriptor that aggregates the iHKS signatures over the volume of a shape, and evaluate its performance on a public shape retrieval benchmark. We find that while, theoretically, strict isometry invariance requires concentrating on the intrinsic surface properties alone, yet, practically, pose insensitive shape retrieval can be achieved using volumetric information.