Learning the Geodesic Embedding with Graph Neural Networks

TL;DR

GeGnn is a graph neural network-based method that efficiently computes approximate geodesic distances on mesh surfaces with high speed and accuracy, even on noisy or incomplete data.

Contribution

The paper introduces a novel GNN architecture with specialized modules for embedding meshes, enabling fast, accurate geodesic distance computation after minimal precomputation.

Findings

Achieves orders-of-magnitude faster computation than existing methods.

Maintains high accuracy comparable to traditional algorithms.

Demonstrates robustness on noisy and incomplete meshes.

Abstract

We present GeGnn, a learning-based method for computing the approximate geodesic distance between two arbitrary points on discrete polyhedra surfaces with constant time complexity after fast precomputation. Previous relevant methods either focus on computing the geodesic distance between a single source and all destinations, which has linear complexity at least or require a long precomputation time. Our key idea is to train a graph neural network to embed an input mesh into a high-dimensional embedding space and compute the geodesic distance between a pair of points using the corresponding embedding vectors and a lightweight decoding function. To facilitate the learning of the embedding, we propose novel graph convolution and graph pooling modules that incorporate local geodesic information and are verified to be much more effective than previous designs. After training, our method requires only one forward pass of the network per mesh as precomputation. Then, we can compute the geodesic distance between a pair of points using our decoding function, which requires only several matrix multiplications and can be massively parallelized on GPUs. We verify the efficiency and effectiveness of our method on ShapeNet and demonstrate that our method is faster than existing methods by orders of magnitude while achieving comparable or better accuracy. Additionally, our method exhibits robustness on noisy and incomplete meshes and strong generalization ability on out-of-distribution meshes. The code and pretrained model can be found on https://github.com/IntelligentGeometry/GeGnn.