Heat Diffusion Kernel and Distance on Surface Meshes and Point Sets

The heat diffusion distance and kernel have gained a central role in geometry processing and shape analysis. This paper addresses a novel discretization and spectrum-free computation of the diffusion kernel and distance on a 3D shape PP represented as a triangle mesh or a point set. After rewriting different discretizations of the Laplace-Beltrami operator in a unified way and using an intrinsic scalar product on the space of functions on PP, we derive a shape-intrinsic heat kernel matrix, together with the corresponding diffusion distances. Then, we propose an efficient computation of the heat distance and kernel through the solution of a set of sparse linear systems. In this way, we bypass the evaluation of the Laplacian spectrum, the selection of a specific subset of eigenpairs, and the use of multi-resolutive prolongation operators. The comparison with previous work highlights the main features of the proposed approach in terms of smoothness, stability to shape discretization, approximation accuracy, and computational cost