Abstract
In contrast to classical Fourier analysis, time–frequency analysis is concerned with localized Fourier transforms. Gabor analysis is an important branch of time–frequency analysis. Although significantly different, it shares with the wavelet transform methods the ability to describe the smoothness of a given function in a location-dependent way.
The main tool is the sliding window Fourier transform or short-time Fourier transform (STFT) in the context of audio signals. It describes the correlation of a signal with the time–frequency shifted copies of a fixed function (or window or atom). Thus, it characterizes a function by its transform over phase space, which is the time–frequency plane (TF-plane) in a musical context or the location–wave-number domain in the context of image processing.
Since the transition from the signal domain to the phase space domain introduces an enormous amount of data redundancy, suitable subsampling of the continuous transform allows for complete recovery of the signal from the sampled STFT. The knowledge about appropriate choices of windows and sampling lattices has increased significantly during the last three decades. Since the suggestion goes back to the idea of D. Gabor [45], this branch of TF analysis is called Gabor analysis . Gabor expansions are not only of interest due to their very natural interpretation but also algorithmically convenient due to a good understanding of algebraic and analytic properties of Gabor families.
In this chapter, we describe some of the generalities relevant for an understanding of Gabor analysis of functions on Rd. We pay special attention to the case d = 2, which is the most important case for image processing and image analysis applications.
The chapter is organized as follows. Section 2 presents central tools from functional analysis in Hilbert spaces, e.g., the pseudo-inverse of a bounded operator and the central facts from frame theory. In Sect. 3, we introduce several operators that play important roles in Gabor analysis. Gabor frames on L2(Rd) are introduced in Sect. 4, and their discrete counterpart are treated in Sect. 5. Finally, the application of Gabor expansions to image representation is considered in Sect. 6.
The main tool is the sliding window Fourier transform or short-time Fourier transform (STFT) in the context of audio signals. It describes the correlation of a signal with the time–frequency shifted copies of a fixed function (or window or atom). Thus, it characterizes a function by its transform over phase space, which is the time–frequency plane (TF-plane) in a musical context or the location–wave-number domain in the context of image processing.
Since the transition from the signal domain to the phase space domain introduces an enormous amount of data redundancy, suitable subsampling of the continuous transform allows for complete recovery of the signal from the sampled STFT. The knowledge about appropriate choices of windows and sampling lattices has increased significantly during the last three decades. Since the suggestion goes back to the idea of D. Gabor [45], this branch of TF analysis is called Gabor analysis . Gabor expansions are not only of interest due to their very natural interpretation but also algorithmically convenient due to a good understanding of algebraic and analytic properties of Gabor families.
In this chapter, we describe some of the generalities relevant for an understanding of Gabor analysis of functions on Rd. We pay special attention to the case d = 2, which is the most important case for image processing and image analysis applications.
The chapter is organized as follows. Section 2 presents central tools from functional analysis in Hilbert spaces, e.g., the pseudo-inverse of a bounded operator and the central facts from frame theory. In Sect. 3, we introduce several operators that play important roles in Gabor analysis. Gabor frames on L2(Rd) are introduced in Sect. 4, and their discrete counterpart are treated in Sect. 5. Finally, the application of Gabor expansions to image representation is considered in Sect. 6.
| Original language | English |
|---|---|
| Title of host publication | Handbook of Mathematical Methods in Imaging |
| Editors | Otmar Scherzer |
| Volume | 1 |
| Publisher | Springer |
| Publication date | 2015 |
| Edition | 2 |
| Pages | 1717-1757 |
| ISBN (Print) | 978-1-4939-0789-2 |
| ISBN (Electronic) | 978-1-4939-0790-8 |
| DOIs | |
| Publication status | Published - 2015 |