Left-invariant parabolic evolutions on $SE(2)$ and contour enhancement via invertible orientation scores Part I: Linear left-invariant diffusion equations on $SE(2)$

We provide the explicit solutions of linear, left-invariant, diffusion equations and the corresponding resolvent equations on the 2D-Euclidean motion group $SE(2)=\mathbb {R}^2 \rtimes \mathbb {T}$. These parabolic equations are forward Kolmogorov equations for well-known stochastic processes for contour enhancement and contour completion. The solutions are given by group convolution with the corresponding Green’s functions. In earlier work we have solved the forward Kolmogorov equations (or Fokker-Planck equations) for stochastic processes on contour completion. Here we mainly focus on the forward Kolmogorov equations for contour enhancement processes which do not include convection. We derive explicit formulas for the Green’s functions (i.e., the heat kernels on $SE(2)$) of the left-invariant partial differential equations related to the contour enhancement process. By applying a contraction we approximate the left-invariant vector fields on $SE(2)$ by left-invariant generators of a Heisenberg group, and we derive suitable approximations of the Green’s functions. The exact Green’s functions are used in so-called collision distributions on $SE(2)$, which are the product of two left-invariant resolvent diffusions given an initial distribution on $SE(2)$. We use the left-invariant evolution processes for automated contour enhancement in noisy medical image data using a so-called orientation score, which is obtained from a grey-value image by means of a special type of unitary wavelet transformation. Here the real part of the (invertible) orientation score serves as an initial condition in the collision distribution.