2007 # The explicit solutions of linear left-invariant second order stochastic evolution equations on the 2D Euclidean motion group

**Abstract:** 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 approxim…

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“…However, one can derive left-invariant differential operators from the Lie algebra of SE(2) at the unity elements. This results in (see [4] for a derivation) the differential operators {∂ θ , ∂ ξ , ∂ η }, where…”

confidence: 99%

“…However, one can derive left-invariant differential operators from the Lie algebra of SE(2) at the unity elements. This results in (see [4] for a derivation) the differential operators {∂ θ , ∂ ξ , ∂ η }, where…”

confidence: 99%

“…We adapt the group theoretical approach developed for the Euclidean motion groups in the recent works [9,18,12,13,15,11], thus illustrating the scope of the methods devised for general Lie groups in [10] in signal and image processing. Reassignment will be seen to be a special case of left-invariant convection.…”

confidence: 99%

“…with g = (x, θ), are all left-invariant, see [11] for a derivation. Consequently, all combinations of the operators…”

confidence: 99%

“…D 22 should be large, D 11 and D 33 should be small, and the curvature measurement of the previous section should be taken into account. If there is no strong orientation, the diffusion should be isotropic in the spatial plane, i.e.…”

confidence: 99%