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

R. Duits, M.A. Almsick, van

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Abstract

We provide the solutions of linear, left-invariant, 2nd-order stochastic evolution equations on the 2D-Euclidean motion group. These solutions are given by group-convolution with the corresponding Green’s functions that we derive in explicit form in Fourier space. 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 base elements 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, 2nd-order stochastic evolution equations.
Original languageEnglish
Pages (from-to)27-67
JournalQuarterly of Applied Mathematics
Volume66
Issue number1
Publication statusPublished - 2008

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