Schr\"{o}dinger Diffusion for Shape Analysis with Texture

Abstract

In recent years, quantities derived from the heat equation have become popular in shape processing and analysis of triangulated surfaces. Such measures are often robust with respect to different kinds of perturbations, including near-isometries, topological noise and partialities. Here, we propose to exploit the semigroup of a Schr\"{o}dinger operator in order to deal with texture data, while maintaining the desirable properties of the heat kernel. We define a family of Schr\"{o}dinger diffusion distances analogous to the ones associated to the heat kernels, and show that they are continuous under perturbations of the data. As an application, we introduce a method for retrieval of textured shapes through comparison of Schr\"{o}dinger diffusion distance histograms with the earth's mover distance, and present some numerical experiments showing superior performance compared to an analogous method that ignores the texture.