The Explicit Solutions of Linear Left-invariant 2nd-order Evolution Equations on the 2D-Euclidean motion group

We provide the solutions of Linear Left-invariant 2nd-order 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 Forward Kolmogorov equation of the direction process, the exact solution of which was strongly required in the field of image analysis. We also provide this tangible solution which indeed coincides with earlier conjectures on the structure of such a solution. Moreover, by approximating the left invariant base elements of the generators by left invariant generators of Heisenberg-type we derive nice analytic approximations of the Green’s functions. We put the explicit connections between these approximations and the exact solutions and give numerical error estimations for various parameter ranges. Finally, we explain the connection between the exact solutions and previous numerical implementations, yielding more efficient numerical computation schemes.