{'fraction_non_alphanumeric': 0.0163098878695209, 'fraction_numerical': 0.0030581039755351682, 'mean_word_length': 5.251592356687898, 'pattern_counts': {'":': 0, '<': 0, '<?xml version=': 0, '>': 0, 'https://': 0, 'lorem ipsum': 0, 'www.': 0, 'xml': 0}, 'pii_count': 0, 'substrings_counts': 0, 'word_list_counts': {'cursed_substrings.json': 0, 'profanity_word_list.json': 0, 'sexual_word_list.json': 0, 'zh_pornsignals.json': 0}}
A Schrodinger wave mechanics formalism for the eikonal problem and its associated gradient density computation

of Dissertation Presented to the Graduate School of the University of Florida in Partial Fulfillment of the Requirements for the Degree of Doctor of Philosophy A SCHRÖDINGER WAVE MECHANICS FORMALISM FOR THE EIKONAL PROBLEM AND ITS ASSOCIATED GRADIENT DENSITY COMPUTATION By Karthik S. Gurumoorthy August 2011 Chair: Arunava Banerjee Cochair: Anand Rangarajan Major: Computer Engineering 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 Schrödinger formalism to solve the classical eikonal problem—a nonlinear, first order, partial differential equation of the form ‖ 5 S‖ = f , where the forcing function f (X ) is a positive valued bounded function and 5 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 Schrödinger 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 ~ (treated as a free parameter) tends to zero of the solution to the corresponding linear Schrödinger 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 Schrödinger 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