Non-Invertible Gabor Transforms

TL;DR

This paper explores methods for signal recovery from non-invertible Gabor transforms, proposing three techniques based on consistency, error minimization, and subspace assumptions, with practical implementations via filter banks.

Contribution

It introduces novel recovery procedures for non-invertible Gabor representations, including algorithms based on different criteria and their implementation using filter banks.

Findings

Recovery methods effectively shape reconstructed signals.

Twisted convolution operations are key to processing Gabor coefficients.

Simulation results compare the advantages and weaknesses of each method.

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 $L_2$. 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.