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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

Part of: Multivariate analysis Stochastic analysis Partial differential equations, initial value and time-dependent initial-boundary value problems Miscellaneous topics - Partial differential equations

Published online by Cambridge University Press:  15 February 2016

Jiong Zhang ,
Remco Duits ,
Gonzalo Sanguinetti  and
Bart M. ter Haar Romeny
Jiong Zhang*
Affiliation:
Department of Biomedical Engineering, Biomedical Image Analysis (BMIA)
Remco Duits*
Affiliation:
Eindhoven University of Technology, P.O. Box 513, 5600 MB, Eindhoven, The Netherlands
Gonzalo Sanguinetti
Affiliation:
Department of Mathematics and Computer Science, Image Science and Technology (IST/e)
Bart M. ter Haar Romeny
Affiliation:
Department of Biomedical Engineering, Biomedical Image Analysis (BMIA) Northeastern University, Shenyang, China
*
*Corresponding and joint main authors.Email addresses:J.Zhang1@tue.nl (J. Zhang), R.Duits@tue.nl (R. Duits), g.r.sanguinetti@tue.nl (G. Sanguinetti), B.M.terhaarRomeny@tue.nl (B.-t.-H. Romeny)
*Corresponding and joint main authors.Email addresses:J.Zhang1@tue.nl (J. Zhang), R.Duits@tue.nl (R. Duits), g.r.sanguinetti@tue.nl (G. Sanguinetti), B.M.terhaarRomeny@tue.nl (B.-t.-H. Romeny)
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Abstract

Left-invariant PDE-evolutions on the roto-translation group SE(2)(and their resolvent equations) have been widely studied in the fields of cortical modeling and image analysis. They include hypo-elliptic diffusion (for contour enhancement) 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 numerical 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 implementations of the exact solutions. Furthermore, we include an asymptotical analysis of the singularities within the kernels and we propose a probabilistic extension of underlying 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.

Keywords

Brownian motionEuclidean motion groupPDE's on SE(2)Mathieu operatorscontour completioncontour enhancementretinal imaging

MSC classification

Secondary: 35R03: Partial differential equations on Heisenberg groups, Lie groups, Carnot groups, etc. 65M80: Fundamental solutions, Green's function methods, etc. 65M06: Finite difference methods 65M12: Stability and convergence of numerical methods 62H35: Image analysis 60H30: Applications of stochastic analysis (to PDE, etc.)
Type
Research Article
Information
Numerical Mathematics: Theory, Methods and Applications , Volume 9 , Issue 1 , February 2016 , pp. 1 - 50
Copyright
Copyright © Global-Science Press 2016 

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