Quarterly of Applied Mathematics

Quarterly of Applied Mathematics

Online ISSN 1552-4485; Print ISSN 0033-569X



The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D Euclidean motion group

Authors: Remco Duits and Markus van Almsick
Journal: Quart. Appl. Math. 66 (2008), 27-67
MSC (2000): Primary 22E25, 37L05, 68U10; Secondary 34B30, 47D06
DOI: https://doi.org/10.1090/S0033-569X-07-01066-0
Published electronically: December 12, 2007
MathSciNet review: 2396651
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Abstract: We provide the solutions of linear, left-invariant, second order stochastic evolution equations on the 2D Euclidean motion group. These solutions are given by group-convolution with the corresponding Green's functions which we derive in explicit form. A particular case coincides with the hitherto unsolved forward Kolmogorov equation of the so-called direction process, the exact solution of which is required in the field of image analysis for modeling the propagation of lines and contours. By approximating the left-invariant basis of the generators by left-invariant generators of a Heisenberg-type group, we derive simple, analytic approximations of the Green's functions. We provide the explicit connection and a comparison between these approximations and the exact solutions. Finally, we explain the connection between the exact solutions and previous numerical implementations, which we generalize to cope with all linear, left-invariant, second order stochastic evolution equations.

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Additional Information

Remco Duits
Affiliation: Department of Mathematics/Computer Science and Department of Biomedical Engineering, Eindhoven University of Technology, Den Dolech 2, P.O. Box 513, 5600MB Eindhoven, The Netherlands
Email: R.Duits@tue.nl

Markus van Almsick
Affiliation: Department of Biomedical Engineering, Eindhoven University of Technology, Den Dolech 2, P.O. Box 513, 5600MB Eindhoven, The Netherlands
Email: M.v.Almsick@tue.nl

DOI: https://doi.org/10.1090/S0033-569X-07-01066-0
Keywords: Lie groups, stochastic evolution equations, image analysis, direction process, completion field
Received by editor(s): May 2, 2006
Published electronically: December 12, 2007
Additional Notes: The Netherlands Organization for Scientific Research is gratefully acknowledged for financial support.
Article copyright: © Copyright 2007 Brown University
The copyright for this article reverts to public domain 28 years after publication.

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