Geodesics are essential in many geometry processing applications. However, traditional algorithms for computing geodesic distances and paths on 3D mesh models are often inefficient and slow. This makes them impractical for scenarios that require extensive querying of arbitrary point-to-point geodesics. Although neural implicit representations have emerged as a popular way of representing 3D shape geometries, there is still no research on representing geodesics with deep implicit functions. To bridge this gap, this paper presents the first attempt to represent geodesics on 3D mesh models using neural implicit functions. Specifically, we introduce neural geodesic fields (NeuroGFs), which are learned to represent the all-pairs geodesics of a given mesh. By using NeuroGFs, we can efficiently and accurately answer queries of arbitrary point-to-point geodesic distances and paths, overcoming the limitations of traditional algorithms. Evaluations on common 3D models show that NeuroGFs exhibit exceptional performance in solving the single-source all-destination (SSAD) and point-to-point geodesics, and achieve high accuracy consistently. Moreover, NeuroGFs offer the unique advantage of encoding both 3D geometry and geodesics in a unified representation. Code is made available at https://github.com/keeganhk/NeuroGF/tree/master.

翻译：测地线在许多几何处理应用中至关重要。然而，在三维网格模型上计算测地距离与路径的传统算法通常效率低下且速度缓慢，这使得它们在需要大量查询任意点对测地线的场景中缺乏实用性。尽管神经隐式表示已成为三维形状几何表示的流行方法，但目前仍未有利用深度隐式函数表示测地线的研究。为填补这一空白，本文首次尝试利用神经隐式函数表示三维网格模型上的测地线。具体而言，我们引入了神经测地场（NeuroGFs），通过学习表示给定网格的全源-全目标测地线。借助NeuroGFs，我们能够高效且精确地回答任意点对测地距离与路径的查询，从而克服传统算法的局限性。在常见三维模型上的评估表明，NeuroGFs在解决单源全目标（SSAD）及点对点测地线问题上表现出卓越性能，并始终达到高精度。此外，NeuroGFs还具有将三维几何与测地线统一编码为同一表示的独特优势。代码已发布于https://github.com/keeganhk/NeuroGF/tree/master。