Balazs, Peter;Feichtinger, Hans G.;Hampejs, Mario;Kracher, GŁnther

Double preconditioning for the Gabor frame operator

in 2006 IEEE International Conference on Acoustics, Speech and Signal Processing, 2006. ICASSP 2006 Proceedings. Volume 3, 14-19 May 2006 (2006) p.III-408 -- III-411

abstract

We present an application of the general idea of preconditioning in the
context of Gabor frames. While most (iterative) algorithms aim at a more or
less costly exact numerical calculation of the inverse Gabor frame matrix, we propose
here the use of ``cheap methods'' to find an approximation for it, based on (double) preconditioning. We thereby
obtain good approximations of the true dual Gabor atom at low
computational costs. Since the Gabor frame matrix commutes with certain
time-frequency shifts it is natural to make use of diagonal and circulant
preconditioners sharing this property. Part of the efficiency of the proposed
scheme results from the fact that all the matrices involved share a
well-known block matrix structure.
At least, for the smooth Gabor atoms
typically used, the combination of these two preconditioners leads
consistently to good results. These claims are supported by numerical
experiments in the second part of the paper.
For numerical evaluations we introduce two new matrix norms, which can be calculated efficiently by exploiting the structure of the frame matrix.
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