A fast eikonal equation solver using the Schrödinger wave equation

Abstract. We use a Schrödinger wave equation formalism to solve the eikonal equation. We show that a solution to the eikonal equation is obtained in the limit as ~ → 0 of the solution to the corresponding linear Schrödinger equation. The Schrödinger equation corresponding to the eikonal turns out to be a generalized, screened Poisson equation. Despite being linear, it does not have a closed-form solution for arbitrary forcing functions. We use a standard perturbation analysis approach to derive a new algorithm which is guaranteed to converge provided the forcing function is bounded and positive. The perturbation technique requires a sequence of discrete convolutions which can be performed in O(N logN) using the Fast Fourier Transform (FFT) where N is the number of grid points. A major advantage of our approach over most other methods is that we do not require a spatial discretization of gradient operators and this contributes to the increased accuracy of our technique. The solution to the eikonal solution is recovered from the exponent of the wave function. Since the wave function is computed for a small but non-zero ~, the obtained solution is an approximation. We provide evidence for the usefulness of our technique by comparing the results of our approach with those obtained from Hamilton-Jacobi solvers such as the fast sweeping algorithm and the Dijkstra single source shortest path algorithm as well as with exact solutions. The latter are available for the Euclidean distance function problem—a special case of the eikonal when the forcing function is set to one.