A Schrödinger Wave Equation Approach to the Eikonal Equation: Application to Image Analysis

Abstract

As Planck’s constant $\hbar$ (treated as a free parameter) tends to zero, the solution to the eikonal equation $|\nabla S(X)|=f(X)$ can be increasingly closely approximated by the solution to the corresponding Schrödinger equation. When the forcing function f(X) is set to one, we get the Euclidean distance function problem. We show that the corresponding Schrödinger equation has a closed form solution which can be expressed as a discrete convolution and efficiently computed using a Fast Fourier Transform (FFT). The eikonal equation has several applications in image analysis, viz. signed distance functions for shape silhouettes, surface reconstruction from point clouds and image segmentation being a few. We show that the sign of the distance function, its gradients and curvature can all be written in closed form, expressed as discrete convolutions and efficiently computed using FFTs. Of note here is that the sign of the distance function in 2D is expressed as a winding number computation. For the general eikonal problem, we present a perturbation series approach which results in a sequence of discrete convolutions once again efficiently computed using FFTs. We compare the results of our approach with those obtained using the fast sweeping method, closed-form solutions (when available) and Dijkstra’s shortest path algorithm.