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
this https URL

本研究提出了利用神经隐式函数表示 3D 网格模型上的测地线，并介绍了神经测地场（NeuroGFs），用于学习表示给定网格的全对测地线。使用 NeuroGFs，我们能高效准确地回答任意点对点的测地距离和路径查询。评估结果表明，NeuroGFs 在解决测地线问题上具有出色的性能，并具有将 3D 几何与测地线编码在统一表示中的独特优势。