  We introduce a systematic approach to the design, implementation and analysis
of left-invariant evolution schemes acting on Gabor transform, primarily for
applications in signal and image analysis. Within this approach we relate
operators on signals to operators on Gabor transforms. In order to obtain a
translation and modulation invariant operator on the space of signals, the
corresponding operator on the reproducing kernel space of Gabor transforms must
be left invariant, i.e. it should commute with the left regular action of the
reduced Heisenberg group H_r. By using the left-invariant vector fields on H_r
in the generators of our evolution equations on Gabor transforms, we naturally
employ the essential group structure on the domain of a Gabor transform. Here
we distinguish between two tasks. Firstly, we consider non-linear adaptive
left-invariant convection (reassignment) to sharpen Gabor transforms, while
maintaining the original signal. Secondly, we consider signal enhancement via
left-invariant diffusion on the corresponding Gabor transform. We provide
numerical experiments and analytical evidence for our methods and we consider
an explicit medical imaging application.
