  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). These diffusion equations are forward
Kolmogorov equations for stochastic processes for contour enhancement and
completion. The solutions are group-convolutions with the corresponding Green's
function, which we derive in explicit form. We mainly focus on the 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), which we explicitly derive. Then we
compute completion distributions on SE(2) which are the product of a forward
and a backward resolvent evolved from resp. source and sink distribution on
SE(2). On the one hand, the modes of Mumford's direction process for contour
completion coincide with elastica curves minimizing $\int \kappa^{2} + \epsilon
ds$, related to zero-crossings of 2 left-invariant derivatives of the
completion distribution. On the other hand, the completion measure for the
contour enhancement concentrates on geodesics minimizing $\int \sqrt{\kappa^{2}
+ \epsilon} ds$. This motivates a comparison between geodesics and elastica,
which are quite similar. However, we derive more practical analytic solutions
for the geodesics. The theory is motivated by medical image analysis
applications where enhancement of elongated structures in noisy images is
required. We use left-invariant (non)-linear evolution processes for automated
contour enhancement on invertible orientation scores, obtained from an image by
means of a special type of unitary wavelet transform.
