Robot vision
Fronts propagating with curvature-dependent speed: algorithms based on Hamilton-Jacobi formulations
Journal of Computational Physics
A Hamiltonian Approach to the Eikonal Equation
EMMCVPR '99 Proceedings of the Second International Workshop on Energy Minimization Methods in Computer Vision and Pattern Recognition
Numerical Geometry of Images: Theory, Algorithms, and Applications
Numerical Geometry of Images: Theory, Algorithms, and Applications
Fast Sweeping Methods for Static Hamilton--Jacobi Equations
SIAM Journal on Numerical Analysis
Computational Geometry: Algorithms and Applications
Computational Geometry: Algorithms and Applications
Construction of neuroanatomical shape complex atlas from 3D brain MRI
MICCAI'10 Proceedings of the 13th international conference on Medical image computing and computer-assisted intervention: Part III
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Computational techniques adapted from classical mechanics and used in image analysis run the gamut from Lagrangian action principles to Hamilton-Jacobi 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 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.