Evolution equations on Gabor transforms and their applications

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 Hr. By using the left-invariant vector fields on Hr 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. Keywords: Evolution equations; Heisenberg group; Differential reassignment; Left-invariant vector fields; Diffusion on Lie groups; Gabor transforms; Medical imaging