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. ► New systematic approach to the design of evolutions acting on Gabor transforms. ► We introduce/compare left-invariant convection schemes on Gabor transforms. ► We introduce non-linear left-invariant diffusion on Gabor transforms for denoising/frequency propagation. ► Novel Gabor technique for deriving cardiac wall deformations from cardiac MRI. ► We provide a complete analytic description of reassignment of chirp signals.