  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.
