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 analysis.They include hypo-elliptic diffusion(for ...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 analysis.They 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 Mumford.This paper presents a thorough study and comparison of the many numerical approaches,which,remarkably,are missing in the literature.Existing 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 2005.Here we provide an overview of these 3 types of exact solutions and explain how they relate to each of the 3 numerical approaches.We 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 errors.We also provide an improvement of Mathematica algorithms for evaluating Mathieu-functions,crucial in implemen-tations of the exact solutions.Furthermore,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 kernels.Finally,we show retinal imaging applications of combining left-invariant PDE-evolutions with invertible orientation scores.展开更多
文摘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 analysis.They 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 Mumford.This paper presents a thorough study and comparison of the many numerical approaches,which,remarkably,are missing in the literature.Existing 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 2005.Here we provide an overview of these 3 types of exact solutions and explain how they relate to each of the 3 numerical approaches.We 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 errors.We also provide an improvement of Mathematica algorithms for evaluating Mathieu-functions,crucial in implemen-tations of the exact solutions.Furthermore,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 kernels.Finally,we show retinal imaging applications of combining left-invariant PDE-evolutions with invertible orientation scores.
