Left-invariant Stochastic Evolution Equations on SE(2) and its Applications to Contour Enhancement and Contour Completion via Invertible Orientation Scores

We provide the explicit solutions of linear, left-invariant, (convection)-diffusion equations and the corresponding resolvent equations on the 2D-Euclidean motion group SE(2) = RoT. These diffusion equations are forward Kolmogorov equations for well-known stochastic processes for contour enhancement and contour completion. The solutions are given by groupconvolution with the corresponding Green’s functions which we derive in explicit form. We have solved the Kolmogorov equations for stochastic processes on contour completion, in earlier work [19]. Here we mainly focus on the Forward Kolmogorov equations for contour enhancement processes which, in contrast to the Kolmogorov equations for contour completion, do not include convection. The Green’s functions of these left-invariant partial differential equations coincide with the heat-kernels on SE(2). Nevertheless, our exact formulae do not seem to appear in literature. Furthermore, by approximating the left-invariant basis of the generators on SE(2) by left-invariant generators of a Heisenberg-type group, we derive approximations of the Green’s functions. The Green’s functions are used in so-called completion distributions on SE(2) which are the product of a forward resolvent evolved from a source distribution on SE(2) and a backward resolvent evolution evolved from a sink distribution on SE(2). Such completion distributions on SE(2) represent the probability density that a random walker from a forward proces collides with a random walker from a backward process. On the one hand, the modes of Mumford’s direction process (for contour completion) coincides with elastica curves minimizing R κ+ ds, and they are closely related to zero-crossings of two left-invariant derivatives of the completion distribution. On the other hand, the completion measure for the contour enhancement proposed by Citti and Sarti, [11] concentrates on the geodesics minimizing R √ κ2 + ds if the expected life time 1/α of a random walker in SE(2) tends to zero. This motivates a comparison between the geodesics and elastica. For reasonable parameter settings they turn out to be quite similar. However, we apply the results by Bryant and Griffiths[9] on Marsden-Weinstein reduction on Euler-Lagrange equations associated to the elastica functional, to the case of the geodesic functional. This yields rather simple practical analytic solutions for the geodesics, which in contrast to the formula for the elastica, do not involve special functions. The theory is directly motivated by several medical image analysis applications where enhancement of elongated structures, such as catheters and bloodvessels, in noisy medical image data is required. Within this article we show how the left-invariant evolution processes