Fast polynomial approximation to heat diffusion in manifolds

Abstract

Heat diffusion has been widely used in image processing for surface fairing, mesh regularization and surface data smoothing. We present a new fast and accurate numerical method to solve heat diffusion on curved surfaces. This is achieved by approximating the heat kernel using high degree orthogonal polynomials in the spectral domain. The proposed polynomial expansion method avoids solving for the eigenfunctions of the Laplace-Beltrami operator, which is computationally costly for large-scale surface meshes, and the numerical instability associated with the finite element method based diffusion solvers. We apply the proposed method to localize the sex differences in cortical brain sulcal and gyral curve patterns.