A Schrodinger wave mechanics formalism for the eikonal problem and its associated gradient density computation.

摘要

Many computational techniques based on classical mechanics exist but surprisingly there isn't a concomitant borrowing from quantum mechanics. Our work shows an application of the Schrodinger formalism to solve the classical eikonal problem---a nonlinear, first order, partial differential equation of the form || ▿ S || = f, where the forcing function f(X) is a positive valued bounded function and ▿ denotes the gradient operator. Hamiltonian Jacobi based solvers like the fast marching and fast sweeping methods solve for S by the Godunov upwind discretization scheme. In sharp contrast to that, we present a Schrodinger wave mechanics formalism to solve the eikonal equation by recasting it as a limiting case of a quantum wave equation. We show that a solution to the non-linear eikonal equation is obtained in the limit as Planck's constant h (treated as a free parameter) tends to zero of the solution to the corresponding linear Schrodinger equation.;We begin with, by considering the Euclidean distance function problem, a special case of the eikonal equation where the forcing function is everywhere identically equal to one. We show that the solution to the Schrodinger wave function can be expressed as a discrete convolution between two functions efficiently computable by the Fast Fourier Transforms (FFT). The Euclidean distance function can then be recovered from the exponent of the wave function. Since the wave function is computed for a small but non-zero h, the obtained solution is an approximation. We show convergence of our approximate closed form solution for the Euclidean distance function problem to the true solution as h→0 and also bound the error for a given value of h. Moreover the differentiability of our solution allows us to compute its first and second derivatives in closed form, also computable by a series of convolutions. In order to determine the sign of the distance function (positive inside a close region and negative outside), we compute the winding number in 2D and topological degree in 3D, by explicitly showing that their computations can also be done via convolutions. We show an application our of method by computing the medial axes for a set of 2D silhouettes. A major advantage of our approach over the other classical methods is that, we do not require a spatial discretization of gradient operators as we obtain a closed-form solution for the wave function.;For the general eikonal problem where the forcing can be an arbitrary but positive and bounded function, the Schrodinger equation turns out to be a generalized, screened Poisson equation. Despite being linear, it does not have a closed-form solution. We use a standard perturbation analysis approach to compute the solution which is guaranteed to converge for all positive and bounded forcing functions. The perturbation technique requires a sequence of discrete convolutions which can be performed using the FFT.;Finally using stationary phase approximations we establish a mathematical result relating the density of the gradient(s) of distance function S and the scaled power spectrum of the wave function for small values of h, when the scalar field S appears as the phase of the wave function. By providing rigorous mathematical proofs, we justify our result for an arbitrary thrice differentiable function in one dimension and for distance transforms in two dimensions. We also furnish anecdotal visual evidences to corroborate our claim. Our result gives a new signature for the distance transforms and potentially serve as its gradient density estimator.