TetCNN: Convolutional Neural Networks on Tetrahedral Meshes

TL;DR

This paper introduces TetCNN, a novel graph convolutional neural network framework designed specifically for tetrahedral meshes, leveraging volumetric Laplace-Beltrami operators for improved analysis of 3D structures like brain images.

Contribution

The work presents a new interpretable CNN model for tetrahedral meshes, including novel pooling methods and visualization techniques, tailored for volumetric data analysis.

Findings

Outperforms traditional methods on cortical tetrahedral meshes

Effectively visualizes regions of interest as biomarkers

Demonstrates superiority over unitary cortical thickness and graph Laplacian representations

Abstract

Convolutional neural networks (CNN) have been broadly studied on images, videos, graphs, and triangular meshes. However, it has seldom been studied on tetrahedral meshes. Given the merits of using volumetric meshes in applications like brain image analysis, we introduce a novel interpretable graph CNN framework for the tetrahedral mesh structure. Inspired by ChebyNet, our model exploits the volumetric Laplace-Beltrami Operator (LBO) to define filters over commonly used graph Laplacian which lacks the Riemannian metric information of 3D manifolds. For pooling adaptation, we introduce new objective functions for localized minimum cuts in the Graclus algorithm based on the LBO. We employ a piece-wise constant approximation scheme that uses the clustering assignment matrix to estimate the LBO on sampled meshes after each pooling. Finally, adapting the Gradient-weighted Class Activation Mapping algorithm for tetrahedral meshes, we use the obtained heatmaps to visualize discovered regions-of-interest as biomarkers. We demonstrate the effectiveness of our model on cortical tetrahedral meshes from patients with Alzheimer's disease, as there is scientific evidence showing the correlation of cortical thickness to neurodegenerative disease progression. Our results show the superiority of our LBO-based convolution layer and adapted pooling over the conventionally used unitary cortical thickness, graph Laplacian, and point cloud representation.