Constant-time all-pairs geodesic distance query on triangle meshes

Abstract

Computing discrete geodesics on polyhedral surfaces plays an important role in computer graphics. In contrast to the well-studied "single-source, all-destination" discrete geodesic problem, little progress has been reported to the all-pairs geodesic, i.e., computing the geodesic distance between arbitrary two points on the surface. To our knowledge, the existing all-pairs geodesic algorithms have very high computational cost, thus, can not be applied to real-world models, which usually contain thousands of vertices. In this paper, we propose an efficient algorithm to approximate the all-pairs geodesic on triangular meshes. The pre-processing step takes O(mn2 log n) time for the input mesh with n vertices and m samples, where m (≪ n) is specified by the user, usually between a few hundred and several thousand. In the query step, our algorithm can compute the approximate geodesic distance between arbitrary pair of points (not necessarily mesh vertices) in O(1) time. Furthermore, the geodesic path and the geodesic distance field can be approximated in linear time. Both theoretical analysis and experimental results on real-world models demonstrate that our algorithm is efficient and accurate. We demonstrate the efficacy of our algorithm on the interactive texture mapping by using discrete exponential map.