Left-invariant Parabolic Evolutions on Se(2) and Contour Enhancement via Invertible Orientation Scores Part I: Linear Left-invariant Diffusion Equations

We provide the explicit solutions of linear, left-invariant, diffusion equations and the corresponding resolvent equations on the 2D-Euclidean motion group SE(2) = R 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 Received March 14, 2008. 2000 Mathematics Subject Classification. Primary 58J65, 49Q20, 22E30; Secondary 34B30, 37L05, 47D06.