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

Duits, Remco; van Almsick, Markus

doi:10.1090/S0033-569X-07-01066-0

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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