Multi-Scale Feature Spaces for Shape Processing and Analysis

In digital geometry processing and shape modeling, the Laplace-Beltrami and the heat diffusion operator, together with the corresponding Laplacian eigenmaps, harmonic and geometry-aware functions, have been used in several applications, which range from surface parameterization, deformation, and compression to segmentation, clustering, and comparison. Using the linear FEM approximation of the Laplace-Beltrami operator, we derive a discrete heat kernel that is linear, stable to an irregular sampling density of the input surface, and scale covariant. With respect to previous work, this last property makes the kernel particularly suitable for shape analysis and comparison; in fact, local and global changes of the surface correspond to a re-scaling of the time parameter without affecting its spectral component. Finally, we study the scale spaces that are induced by the proposed heat kernel and exploited to provide a multi-scale approximation of scalar functions defined on 3D shapes.