Gabor systems with good TF-localization and applications to image processing

The basic design freedom of a (generalized) Gabor transform is the choice of (i) the time-frequency lattice constants and (ii) the analysis prototype (atom, window). The design of the synthesis prototype is subject to linear constraints (Wexler-Raz (1990) condition) depending on the redundancy of the presentation. For multidimensional signals the design freedom can be greatly increased by the consideration of nonseparable situations: (i) nonseparable prototypes and/or (ii) nonseparable position-frequency sampling lattices. We present such a general theory for the Gabor expansion of 2D signals. Our main result is a generalized biorthogonality condition connecting the analysis and synthesis prototype. The theory is illustrated by a simple numerical experiment