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Non-Invertible Gabor Transforms (2009)
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BibTeX
@MISC{Matusiak09non-invertiblegabor,
author = {Ewa Matusiak and Tomer Michaeli and Yonina C. Eldar},
title = { Non-Invertible Gabor Transforms},
year = {2009}
}
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Abstract
Time-frequency analysis, such as the Gabor transform, plays an important role in many signal processing applications. The redundancy of such representations is often directly related to the computational load of any algorithm operating in the transform domain. To reduce complexity, it may be desirable to increase the time and frequency sampling intervals beyond the point where the transform is invertible, at the cost of an inevitable recovery error. In this paper we initiate the study of recovery procedures for non-invertible Gabor representations. We propose using fixed analysis and synthesis windows, chosen e.g., according to implementation constraints, and to process the Gabor coefficients prior to synthesis in order to shape the reconstructed signal. We develop three methods to tackle this problem. The first follows from the consistency requirement, namely that the recovered signal has the same Gabor representation as the input signal. The second, is based on the minimization of a worst-case error criterion. Last, we develop a recovery technique based on the assumption that the input signal lies in some subspace of L2. We show that for each of the criteria, the manipulation of the transform coefficients amounts to a 2D twisted convolution operation, which we show how to perform using a filter-bank. When the under-sampling factor is an integer, the processing reduces to standard 2D convolution. We provide simulation results to demonstrate the advantages and weaknesses of each of the algorithms.
Keyphrases
1non-invertible gabor transforms important role under-sampling factor worst-case error criterion processing reduces twisted convolution operation gabor representation gabor coefficient gabor transform inevitable recovery error consistency requirement recovered signal algorithm operating time-frequency analysis input signal lie transform domain recovery technique implementation constraint input signal simulation result recovery procedure reconstructed signal non-invertible gabor representation synthesis window many signal processing application computational load
