DEEP ACCURATE SOLVER FOR THE GEODESIC PROBLEM

Abstract

A high order accurate deep learning method for computing geodesic distances on surfaces is introduced. We consider two main components for computing distances on surfaces; A numerical solver that locally approximates the distance function and an efficient causal ordering scheme by which surface points are updated. The proposed method exploits a dynamic programming principle which lends itself to a scheme with quasi-linear computational complexity. The quality of the distance approximation is determined by the local solver and is the main focus of this paper. A common approach to compute distances on continuous surfaces is by considering a discretized polygonal mesh approximating the surface, and estimating distances on the polygon. With such an approximation, the exact geodesic distances restricted to the polygon are at most second order accurate with respect to the distances on the corresponding continuous surface. Here, by order of accuracy we refer to the rate of convergence as a function of the average distance between sampled points. To improve the accuracy, we consider a neural network based local solver which implicitly approximates the structure of the continuous surface. The proposed solver circumvents the polyhedral representation, by directly consuming sampled mesh vertices for approximation of distances on the sampled continuous surfaces. We supply numerical evidence that the proposed learned update scheme, with appropriate local numerical support, provides better accuracy compared to the best possible polyhedral approximations and previous learning based methods. We introduce a trained solver which is third order accurate, with quasi-linear complexity in the number of sampled points.

1. INTRODUCTION

Geodesic distance is defined as the length of the shortest path connecting two points on a surface. It can be considered as a generalization of the Euclidean distance to curved manifolds. The approximation of geodesic distances is used as a building block in many applications. It can be found in robot navigation (Kimmel et al., 1998; Kimmel & Sethian, 2001) , and shape matching (Ion et al., 2008; Elad & Kimmel, 2001; Shamai & Kimmel, 2017; Panozzo et al., 2013) , to name just a few examples. Thus, for effective and reliable use, computation of geodesics is expected to be both fast and accurate. Over the years, many methods have been proposed for computing distances on polygonal meshes that compromise between the accuracy of the distance approximation and the complexity of the algorithm. One family of algorithms for computing distances in this domain is based on solutions to the exact discrete geodesic problem introduced by Mitchell et al. (1987) . This problem is defined as that of finding the exact distances on a polyhedral mesh. The algorithms introduced so far for solving the discrete geodesic problem involve substantially higher than linear complexity which makes them impractical for operating on surfaces sampled by a large number of vertices. At the other end, a popular family of methods for efficient approximation of distances known as fast marching, involves quasi-linear computational complexity. These methods are based on the solution of the eikonal equation and consists of two main components, a heap sorting strategy and a local causal numerical solver, often referred to as a numerical update procedure. Fast marching, originally introduced for regularly sampled grids (Sethian, 1996; Tsitsiklis, 1995) , was extended to triangulated surfaces in Kimmel & Sethian (1998) . While operating on curved surfaces approximated by triangulated mesh, the first proximity neighbors of a vertex in the mesh are used to locally approximate the solution of 1

