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Heat diffusion has been widely used in brain imaging for surface fairing,mesh regularization and cortical data smoothing. Motivated by diffusionwavelets and convolutional neural networks on graphs, we present a new fast andaccurate numerical scheme to solve heat diffusion on surface meshes. This isachieved by approximating the heat kernel convolution using high degreeorthogonal polynomials in the spectral domain. We also derive the closed-formexpression of the spectral decomposition of the Laplace-Beltrami operator anduse it to solve heat diffusion on a manifold for the first time. The proposedfast polynomial approximation scheme avoids solving for the eigenfunctions ofthe Laplace-Beltrami operator, which is computationally costly for large meshsize, and the numerical instability associated with the finite element methodbased diffusion solvers. The proposed method is applied in localizing the maleand female differences in cortical sulcal and gyral graph patterns obtainedfrom MRI in an innovative way. The MATLAB code is available athttp://www.stat.wisc.edu/~mchung/chebyshev.

Fast Polynomial Approximation of Heat Kernel Convolution on Manifolds and Its Application to Brain Sulcal and Gyral Graph Pattern Analysis