Speaker: Dr. Hans Feichtinger, Professor of Mathematics, University of Vienna
Abstract: Gabor analysis is concerned with the atomic decomposition of a given (e.g. audio-) signal into components comparable with the score of a piece of music. It was a standard formulation found in many older papers that a Gabor decomposition of a signal, e.g. using a double series of time-frequency shifted Gaussians along some lattice of the form aZ x bZ with ab < 1 is interesting, due to the information carried by the individual coefficient, but unfortunately the non-orthogonality of this system makes it difficult to compute the coefficients. Of course the problem becomes worse in the 2D-setting, where one has local plane waves with smooth blob-like contours as building blocks.
However the research carried out in Gabor analysis in the last 25 years have provided a lot of insight, showing that there are various invariance properties involved in the description of Gabor systems or the Gabor frame operator. Let us only mention the Wexler-Raz and the Ron-Shen principle and the Janssen representation of the Gabor frame operator. We will indicate how e.g. commutation laws help to significantly reduce the computational costs for regular Gabor families, even if the corresponding lattice is not just the separable one mentioned above.