Paper Info
Title
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.
Year
DOI
Venue
2009
10.1007/978-3-642-03641-5_11
EMMCVPR
Keywords
Field
DocType
dinger wave equation approach,closed-form solution,corresponding schr,eikonal equation,distance function,closed form,general eikonal problem,closed form solution,dinger equation,discrete convolution,euclidean distance function problem,image analysis,shortest path algorithm,surface reconstruction,wave equation,fast fourier transform,point cloud,euclidean distance,winding number,image segmentation
Mathematical optimization,Mathematical analysis,Convolution,Signed distance function,Eikonal equation,Schrödinger equation,Closed-form expression,Winding number,Fast Fourier transform,Mathematics,Free parameter
Conference
Volume
ISSN
Citations 
5681
0302-9743
3
PageRank 
References 
Authors
0.50
7
2
Name
Order
Citations
PageRank
A Rangarajan13698367.52
karthik s gurumoorthy25210.09