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.

翻译：卷积神经网络（CNN）已在图像、视频、图结构及三角网格上得到广泛研究，然而针对四面体网格的研究却极为罕见。鉴于体网格在脑图像分析等应用中的优势，我们提出了一种面向四面体网格结构的可解释新型图CNN框架。受ChebyNet启发，我们的模型利用体积拉普拉斯-贝尔特拉米算子（LBO）在常用图拉普拉斯（缺乏三维流形黎曼度量信息）上定义滤波器。为适配池化操作，我们基于LBO提出了Graclus算法中局部最小割的新目标函数。采用分段常数逼近方案，利用聚类分配矩阵估计每次池化后采样网格上的LBO。最后，通过将梯度加权类激活映射算法适配于四面体网格，我们利用所得热力图将检测到的感兴趣区域可视化作为生物标志物。我们在阿尔茨海默病患者皮层四面体网格上验证了模型有效性——已有科学证据表明皮层厚度与神经退行性疾病进展存在相关性。结果表明，基于LBO的卷积层与适配池化在性能上优于传统使用的单一皮层厚度、图拉普拉斯及点云表示方法。