Optimal iterative algorithms in Gabor analysis

Gabor expansions of discrete signals and images have a wide range of applications in signal analysis and pattern recognition. It is well known that the difficulty in expanding a 1-D or a 2-D signal into a Gabor series is the non-orthogonality of the building blocks, which are time-frequency shifted versions (along some lattice in the TF-plane) of a, given Gabor atom. The theory of frames (Gabor or Weyl-Heisenberg frames) has reached fame as the appropriate tool for resolving the problem. In particular, the dual frame turns out to be again a Gabor frame with respect to the same lattice, and its generating function is called the dual Gabor atom g˜. It is obtained by applying the inverse of the frame operator S to the original frame (or just the Gabor atom), i.e. g˜= S^-1g. There is also another important use for the dual Gabor window. Given the sampled STFT (short time or sliding window Fourier transform) of some signal x with respect to the window g over the same lattice it is possible to recover the signal x using a simple Shannon-type reconstruction formula. We present some basic ideas behind a new family of iterative algorithms for determining the dual Gabor atom in the finite (discrete and periodic) case. They are mainly based on the conjugate gradient methods in combination with structural properties of the Gabor frame operator