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. We show that exact geodesic distances restricted to the polygon are at most second-order accurate with respect to the distances on the corresponding continuous surface. By order of accuracy we refer to the convergence rate as a function of the average distance between sampled points. Next, a higher-order accurate deep learning method for computing geodesic distances on surfaces is introduced. Traditionally, one considers two main components when 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. Classical minimal path methods often exploit a dynamic programming principle with quasi-linear computational complexity in the number of sampled points. The quality of the distance approximation is determined by the local solver that is revisited in this paper. To improve state of the art accuracy, we consider a neural network-based local solver which implicitly approximates the structure of the continuous surface. We supply numerical evidence that the proposed learned update scheme provides better accuracy compared to the best possible polyhedral approximations and previous learning-based methods. The result is a third-order accurate solver with a bootstrapping-recipe for further improvement.

翻译：计算连续曲面距离的常见方法是考虑一个近似曲面的离散多边形网格，并在多边形上估计距离。我们证明，限制在多边形上的精确测地距离相对于相应连续曲面上的距离至多具有二阶精度。此处精度阶数指的是收敛速率作为采样点间平均距离的函数。随后，我们提出了一种用于计算曲面上测地距离的高阶精度深度学习方法。传统上，在计算曲面距离时主要考虑两个组成部分：局部逼近距离函数的数值求解器，以及更新曲面点的高效因果排序方案。经典的最短路径方法通常利用动态规划原理，其计算复杂度在采样点数量上呈拟线性。距离逼近的质量由本文重新审视的局部求解器决定。为了提升现有技术水平下的精度，我们采用基于神经网络的局部求解器，该求解器隐式地逼近连续曲面的结构。我们提供的数值证据表明，与最佳可能的多面体逼近方法及先前基于学习的方法相比，所提出的学习型更新方案具有更高的精度。最终我们获得了一个三阶精度的求解器，并提供了可进一步改进的自举优化方案。