Robot vision
Precise Numerical Methods Using C++ with Cdrom
Precise Numerical Methods Using C++ with Cdrom
Introduction to Algorithms
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
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Handbook of Mathematical Functions, With Formulas, Graphs, and Mathematical Tables,
Short note: O(N) implementation of the fast marching algorithm
Journal of Computational Physics
Coined quantum walks lift the cospectrality of graphs and trees
EMMCVPR'05 Proceedings of the 5th international conference on Energy Minimization Methods in Computer Vision and Pattern Recognition
Image enhancement and denoising by complex diffusion processes
IEEE Transactions on Pattern Analysis and Machine Intelligence
The complex wave representation of distance transforms
EMMCVPR'11 Proceedings of the 8th international conference on Energy minimization methods in computer vision and pattern recognition
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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.