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

Abstract

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.

[Retracted] Swimming Training Evaluation Method Based on Convolutional Neural Network

“…In the AlexNet network structure, the size of the first layer of convolution kernel is 11 × 11, the size of the second layer of convolution kernel is 5 × 5, and the size of the third, fourth, and fifth layers of convolution kernel is 3 × 3. Related scholars pointed out that a large convolution kernel can be stacked by using multiple small convolution kernels [26,27]. erefore, in order to simplify the design process, the convolution kernel size in the convolution layer used in this article is all 3 × 3. e size of the feature map is adjusted by the nonoverlapping maximum pooling layer with a size of 2 × 2 and a step size of 2.…”

Labeling lateral prefrontal sulci using spherical data augmentation and context-aware training

“…Indentations in the outer surface of the cerebrum, known as sulci, are key phenotypical biomarkers for linking brain structure and function ( Armstrong et al, 1995 ; Cachia et al, 2008 ; De Winter et al, 2015 ; Huang et al, 2020 ; Le Goualher et al, 1999 ; Lyu et al, 2018a; 2018b; Mangin et al, 2004 ; Miller et al, 2020b ; Weiner et al, 2014 ; Welker, 1990 ; Zilles et al, 1988 ). It is well known that deep sulci, which emerge early in gestation, are key landmarks linking structure, function, and behavior in primary sensory cortices ( Armstrong et al, 1995 ; Ono et al, 1990 ; Sanides, 1964 ; Schwarzkopf and Rees, 2013 ; Welker, 1990 ).…”

Discussion of ‘Event history and topological data analysis’

“…Compared to topological data analysis methods, geometric methods are more adept at detecting localized signals in trees. Figure 2 displays the sucal and gyral trees obtained from brain surface meshes ( Huang et al, 2020 ). Trees are treated as a heat source with value +1 on gyral trees and a heat sink with value −1 on sulcal trees.…”

“…The major advantage of this approach is that such maps can be easily compared across different subjects. In Huang et al (2020) , a two-sample t -statistic is calculated at each mesh vertex and is used in localizing the sex difference, 268 females, 176 males, near the temporal lobes of the brain. Such localized signal detection is not possible with many existing topological data analysis methods.…”