# TetCNN: Convolutional Neural Networks on Tetrahedral Meshes

arXiv: https://arxiv.org/abs/2302.03830 · HTML: https://mlsanity.com/paper/2302.03830/

## Key facts

- Cited by 5 papers in the map (data through 14 Oct 2026).
- Builds on 12 of its 32 references that are in the map.
- First posted on arXiv 8 Feb 2023.

## Builds on (most cited first, 12 of 12)

- [Adam: A Method for Stochastic Optimization](https://mlsanity.com/paper/1412.6980/) (2014)
- [Batch Normalization: Accelerating Deep Network Training by Reducing Internal Covariate Shift](https://mlsanity.com/paper/1502.03167/) (2015)
- [PointNet: Deep Learning on Point Sets for 3D Classification and Segmentation](https://mlsanity.com/paper/1612.00593/) (2016)
- [Fast Graph Representation Learning with PyTorch Geometric](https://mlsanity.com/paper/1903.02428/) (2019)
- [Dynamic Graph CNN for Learning on Point Clouds](https://mlsanity.com/paper/1801.07829/) (2018)
- [Convolutional Neural Networks on Graphs with Fast Localized Spectral Filtering](https://mlsanity.com/paper/1606.09375/) (2016)
- [Geometric Deep Learning: Grids, Groups, Graphs, Geodesics, and Gauges](https://mlsanity.com/paper/2104.13478/) (2021)
- [Fully Convolutional Mesh Autoencoder using Efficient Spatially Varying Kernels](https://mlsanity.com/paper/2006.04325/) (2020)
- [Interpretation of Brain Morphology in Association to Alzheimer's Disease Dementia Classification Using Graph Convolutional Networks on Triangulated Meshes](https://mlsanity.com/paper/2008.06151/) (2020)
- [Brain Multigraph Prediction using Topology-Aware Adversarial Graph Neural Network](https://mlsanity.com/paper/2105.02565/) (2021)
- [Revisiting convolutional neural network on graphs with polynomial approximations of Laplace-Beltrami spectral filtering](https://mlsanity.com/paper/2010.13269/) (2020)
- [Spectral Coarsening of Geometric Operators](https://mlsanity.com/paper/1905.05161/) (2019)

## Built on by (most cited first, 5 of 5)

- [Advancing Graph Neural Networks with HL-HGAT: A Hodge-Laplacian and Attention Mechanism Approach for Heterogeneous Graph-Structured Data](https://mlsanity.com/paper/2403.06687/) (2024)
- [A Recipe for Geometry-Aware 3D Mesh Transformers](https://mlsanity.com/paper/2411.00164/) (2024)
- [Enhancing Alzheimer's Diagnosis: Leveraging Anatomical Landmarks in Graph Convolutional Neural Networks on Tetrahedral Meshes](https://mlsanity.com/paper/2503.05031/) (2025)
- [Hierarchical Mesh Transformers with Topology-Guided Pretraining for Morphometric Analysis of Brain Structures](https://mlsanity.com/paper/2604.05215/) (2026)
- [From Surfaces to Volumes: Registered Geometry for Protein Representation Learning](https://mlsanity.com/paper/2609.36277/) (2026)

## Abstract (arXiv)

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.