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 an eikonal equation, resulting in a first-order-accurate scheme in terms of a typical triangle's edge length denoted as h. Figure 1 : Geodesic distance from a single source point on a surface. Our method produces highly accurate distance maps while operating in quasi-linear time. It can be easily shown that the exact geodesic distances computed on a polygonal mesh approximating a continuous surface would be at most a second order approximation of the corresponding distances on the surface (see Appendix A.1). To overcome the second order limitation, we extend the numerical support about each vertex beyond the classical one ring approximation, and utilize the universal approximation properties of neural networks. We develop a neural network based local solver that overcomes the second order approximation limitation induced by geodesics restricted to polyhedral meshes. We exploit the low complexity of the well-known dynamic programming update scheme (Dijkstra, 1959) , and combine it with a novel neural network-based solver, resulting in an efficient and accurate method. In a related effort (Lichtenstein et al., 2019), a neural network based local solver for the computation of geodesic distances was proposed. We improve upon Lichtenstein's O(h 2 ) approach by extending the local neighborhood numerical support, and refining the network's architecture to obtain O(h 3 ) accuracy at similar linear complexity. The proposed local solver is trained in a supervised manner using ground truth examples. Since geodesics can not be derived analytically except for a limited set of surfaces like spheres and planes, we propose a multi-hierarchy ground truth generation technique. The suggested framework uses distance approximations on high resolution sampled meshes to better approximate distances on lower resolution meshes. We thereby utilize our ability to compute high order solvers to generate accurate training examples at low resolution.

1.1. CONTRIBUTIONS

We develop a fast and accurate geodesic distance approximation method on surfaces. • For fast computation, we use a distance update scheme (Algorithm 1) that guarantees quasilinear computational complexity. • For accurate approximation, we develop a neural network based local solver with a wide local neighborhood support that operates directly on the sampled mesh vertices. • To provide accurate ground truth distances required for training our solver, we propose a novel data generation bootstrapping procedure.

2. RELATED EFFORTS

Given a domain Ω ⊂ R n and a curve Γ ∈ Ω, the predominant approach for generating distance functions from the curve Γ to all other points in Ω, is to find a function ϕ : Ω → R which satisfies the eikonal equation, |∇ϕ(x)| = 1, x ∈ Ω \ Γ ϕ(x) = 0, x ∈ Γ . Due to the non-linearity and hyperbolicity of this partial differential equation (PDE), solving it directly is a challenge. Common solvers sample the continuous domain and approximate the solution on the corresponding discretized domain while being consistent with viscosity solutions. Fast Eikonal Solvers. In Sethian (1996) ; Tsitsiklis (1995), quasi-linear algorithms for approximating distances on regularly sampled grids were introduced. These algorithms involve O(N log(N )) complexity, where N is the number of points on the grid. For example, the fast marching algorithm

