Evolution equations on Gabor transforms and their applications

Remco Duits, Hartmut Führ, Bart Janssen, Mark Bruurmijn, Luc Florack, Hans van Assen
2013 Applied and Computational Harmonic Analysis  
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
more » ... e 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.
doi:10.1016/j.acha.2012.11.007 fatcat:zk53ohxcd5hn7pwf6f5a64xhge