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 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.

Curvature Estimation for Enhancement of Crossing Curves

“…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…”

Left Invariant Evolution Equations on Gabor Transforms

“…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.…”

Nonlinear Diffusion on the 2D Euclidean Motion Group

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

“…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.…”

“…If there is no strong orientation, the diffusion should be isotropic in the spatial plane, i.e. D 22 = D 33 should be large as well as D 11 . Curvature is not defined at such positions so κ = 0.…”