Nonstationary Gabor Frames - Approximately Dual Frames and Reconstruction Errors

TL;DR

This paper explores approximate reconstruction methods for nonstationary Gabor frames, which generalize classical Gabor frames for adaptive signal analysis, by using approximately dual frames to compensate for the lack of perfect reconstruction.

Contribution

It introduces a framework for approximate reconstruction in nonstationary Gabor frames using approximately dual frames, especially for almost painless frames with compactly supported windows.

Findings

Good approximate reconstruction is achievable with approximately dual frames.

Constructive examples provided for almost painless nonstationary frames.

Theoretical results supported by computational and numerical examples.

Abstract

Nonstationary Gabor frames, recently introduced in adaptive signal analysis, represent a natural generalization of classical Gabor frames by allowing for adaptivity of windows and lattice in either time or frequency. Due to the lack of a complete lattice structure, perfect reconstruction is in general not feasible from coefficients obtained from nonstationary Gabor frames. In this paper it is shown that for nonstationary Gabor frames that are related to some known frames for which dual frames can be computed, good approximate reconstruction can be achieved by resorting to approximately dual frames. In particular, we give constructive examples for so-called almost painless nonstationary frames, that is, frames that are closely related to nonstationary frames with compactly supported windows. The theoretical results are illustrated by concrete computational and numerical examples.