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Numerical Approaches for Linear Left-invariant Diffusions on SE(2), their Comparison to Exact Solutions, and their Applications in Retinal Imaging

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Numerical Mathematics(Theory,Methods and Applications) 2016年第1期9卷 1-50页

作者： Jiong Zhang Remco Duits Gonzalo Sanguinetti bart mter haar romeny Department of Biomedical Engineering Biomedical Image Analysis(BMIA) Department of Mathematics and Computer Science Image Science and Technology(IST/e) Eindhoven University of Technology P.O.Box 5135600 MBEindhovenThe Netherlands Northeastern University ShenyangChina

Left-invariant PDE-evolutions on the roto-translation group SE(2)(and their resolvent equations)have been widely studied in the fields of cortical model-ing and image *** include hypo-elliptic diffusion(for contour enhance-ment)proposed by Citti&Sarti,and Petitot,and they include the direction process(for contour completion)proposed by *** paper presents a thorough study and comparison of the many numerical approaches,which,remarkably,are missing in the *** numerical approaches can be classified into 3 categories:Finite difference methods,Fourier based methods(equivalent to SE(2)-Fourier methods),and stochastic methods(Monte Carlo simulations).There are also 3 types of exact solutions to the PDE-evolutions that were derived explicitly(in the spatial Fourier domain)in previous works by Duits and van Almsick in *** we provide an overview of these 3 types of exact solutions and explain how they relate to each of the 3 numerical *** compute relative errors of all nu-merical approaches to the exact solutions,and the Fourier based methods show us the best performance with smallest relative *** also provide an improvement of Mathematica algorithms for evaluating Mathieu-functions,crucial in implemen-tations of the exact ***,we include an asymptotical analysis of the singularities within the kernels and we propose a probabilistic extension of un-derlying stochastic processes that overcomes the singular behavior in the origin of time-integrated ***,we show retinal imaging applications of combining left-invariant PDE-evolutions with invertible orientation scores.

关键词： Brownian motion Euclidean motion group PDE’s on SE(2) Mathieu operators contour completion contour enhancement retinal imaging

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