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.

## Handbook of Mathematical Methods in Imaging - Semantic Scholar

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.

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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.

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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.

The chapter is organized as follows. Section 2 presents central tools from functional analysis in Hilbert spaces, e.

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In Sect. Gabor frames on L2 Rd are introduced in Sect. Fingerprint Photoacoustic Tomography.

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