Computing Distance Functions and Geodesics

In this paper we review computational techniques for efficiently computing distance functions and geodesics, thereby addressing optimal trajectory problems. These techniques are based on solving the Eikonal equation. Following [30] we first describe how we can numerically solve this equation in a Cartesian grid in O(N) operations, N being the number of grid points. This optimal run-time cost is obtained while keeping an error bound of the same order of magnitude as the original O(N logN) algorithms. We also describe, following [19, 20], how we can use this to robustly solve optimal trajectories on surfaces, defined either in implicit form or via point cloud samples.