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Left-invariant PDE-evolutions on the roto-translation group $SE(2)$ (andtheir resolvent equations) have been widely studied in the fields of corticalmodeling and image analysis. They include hypo-elliptic diffusion (for contourenhancement) proposed by Citti & Sarti, and Petitot, and they include thedirection process (for contour completion) proposed by Mumford. This paperpresents a thorough study and comparison of the many numerical approaches,which, remarkably, is missing in the literature. Existing numerical approachescan be classified into 3 categories: Finite difference methods, Fourier basedmethods (equivalent to $SE(2)$-Fourier methods), and stochastic methods (MonteCarlo simulations). There are also 3 types of exact solutions to thePDE-evolutions that were derived explicitly (in the spatial Fourier domain) inprevious works by Duits and van Almsick in 2005. Here we provide an overview ofthese 3 types of exact solutions and explain how they relate to each of the 3numerical approaches. We compute relative errors of all numerical approaches tothe exact solutions, and the Fourier based methods show us the best performancewith smallest relative errors. We also provide an improvement of Mathematicaalgorithms for evaluating Mathieu-functions, crucial in implementations of theexact solutions. Furthermore, we include an asymptotical analysis of thesingularities within the kernels and we propose a probabilistic extension ofunderlying stochastic processes that overcomes the singular behavior in theorigin of time-integrated kernels. Finally, we show retinal imagingapplications of combining left-invariant PDE-evolutions with invertibleorientation scores.

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