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.

翻译：我们提出GeGnn，一种基于学习的方法，用于计算离散多面体表面上任意两点间的近似测地线距离，在快速预计算后具有常数时间复杂度。以往的相关方法要么专注于计算单个源点到所有目标点的测地线距离（至少具有线性复杂度），要么需要较长的预计算时间。我们的核心思想是训练一个图神经网络将输入网格嵌入到高维嵌入空间，并通过相应的嵌入向量与轻量级解码函数计算点对间的测地线距离。为促进嵌入学习，我们提出融合局部测地线信息的新型图卷积与图池化模块，经验证其有效性远超以往设计。训练完成后，我们的方法仅需对每个网格进行一次网络前向传播作为预计算，随后即可通过仅需若干矩阵乘法且可在GPU上大规模并行化的解码函数计算点对间的测地线距离。我们在ShapeNet上验证了该方法的效率与有效性，证明其速度比现有方法快数个数量级，同时达到可比或更高的精度。此外，该方法对噪声和不完整网格具有鲁棒性，并对分布外网格表现出强泛化能力。代码与预训练模型可在https://github.com/IntelligentGeometry/GeGnn获取。