A Schrödinger Equation for the Fast Computation of Approximate Euclidean Distance Functions

Computational techniques adapted from classical mechanics and used in image analysis run the gamut from Lagrangian action principles to HamiltonJacobi field equations: witness the popularity of the fast marching and fast sweeping methods which are essentially fast Hamilton-Jacobi solvers. In sharp contrast, there are very few applications of quantum mechanics inspired computational methods. Given the fact that most of classical mechanics can be obtained as a limiting case of quantum mechanics (as Planck’s constant h tends to zero), this paucity of quantum mechanics inspired methods in image analysis is surprising. In this work, we derive relationships between nonlinear Hamilton-Jacobi and linear Schrödinger equations for the Euclidean distance function problem (in 1D, 2D and 3D). We then solve the Schrödinger wave equation instead of the corresponding Hamilton-Jacobi equation. We show that the Schrödinger equation has a closed form solution and that this solution can be efficiently computed in O(N logN), N being the number of grid points. The Euclidean distance can then be recovered from the wave function. Since the wave function is computed for a small but non-zero h, the obtained Euclidean distance function is an approximation. We derive analytic bounds for the error of the approximation and experimentally compare the results of our approach with the exact Euclidean distance function on real and synthetic data.