Heat diffusion kernel and distance on surface meshes and point sets

• Spectrum-free computation of the heat diffusion kernel and distances.
• The computation is independent of the selected eigenpairs and prolongation operators.
• The approximation accuracy is lower than 10−r (e.g., r≔5,7)r≔5,7), r degree of the rational Chebyshev polynomial.
• The approach is robust to surface discretization and free of user-defined parameters.

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.

Stability of the computation of the Chebyshev approximation of the heat diffusion distances on partially sampled surfaces.Figure optionsDownload high-quality image (332 K)Download as PowerPoint slide