  Heat diffusion has been widely used in brain imaging for surface fairing,
mesh regularization and cortical data smoothing. Motivated by diffusion
wavelets and convolutional neural networks on graphs, we present a new fast and
accurate numerical scheme to solve heat diffusion on surface meshes. This is
achieved by approximating the heat kernel convolution using high degree
orthogonal polynomials in the spectral domain. We also derive the closed-form
expression of the spectral decomposition of the Laplace-Beltrami operator and
use it to solve heat diffusion on a manifold for the first time. The proposed
fast polynomial approximation scheme avoids solving for the eigenfunctions of
the Laplace-Beltrami operator, which is computationally costly for large mesh
size, and the numerical instability associated with the finite element method
based diffusion solvers. The proposed method is applied in localizing the male
and female differences in cortical sulcal and gyral graph patterns obtained
from MRI in an innovative way. The MATLAB code is available at
http://www.stat.wisc.edu/~mchung/chebyshev.
