wFEM Heat Kernel: Discretization and Applications to Shape Analysis and Retrieval

Recent results in geometry processing have shown that shape segmentation, comparison, and analysis can be successfully addressed through the heat diffusion kernel. In this paper, we focus our attention on the properties (e.g., scale-invariance, semi-group property, robustness to noise) of the wFEM heat kernel, recently proposed in Patanè and Falcidieno (2010), and its application to shape comparison and feature-driven approximation. After proving that the wFEM heat kernel is intrinsically scale-covariant (i.e., without shape or kernel normalization) and scale-invariant through a normalization of the Laplacian eigenvalues, we experimentally verify that the wFEM heat kernel descriptors are more robust against shape/scale changes and provide better matching performances with respect to previous work. In the space F(M) of piecewise linear scalar functions defined on a triangle mesh M, we introduce the wFEM heat kernel Kt , which is used to increase the degree of flexibility in the design of geometry-aware basis functions. Furthermore, we efficiently compute scale-based representations of maps on M by specializing the Chebyshev method through the solution of a set of sparse linear systems, thus avoiding the spectral decomposition of the Laplacian matrix. Finally, the scalar product induced by Kt makes F(M) a Reproducing Kernel Hilbert Space, whose (reproducing) kernel is the linear FEM heat kernel, and induces the FEM diffusion distances on M.