Gabor Analysis and Algorithms

1. Front Matter

   Pages i-xvi

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2. Introduction

   • Hans G. Feichtinger, Thomas Strohmer

   Pages 1-31

3. The duality condition for Weyl-Heisenberg frames

   • A. J. E. M. Janssen

   Pages 33-84

4. Gabor systems and the Balian-Low Theorem

   • John J. Benedetto, Christopher Heil, David F. Walnut

   Pages 85-122

5. A Banach space of test functions for Gabor analysis

   • Hans G. Feichtinger, Georg Zimmermann

   Pages 123-170

6. Pseudodifferential operators, Gabor frames, and local trigonometric bases

   • Richard Rochberg, Kazuya Tachizawa

   Pages 171-192

7. Perturbation of frames and applications to Gabor frames

   • Ole Christensen

   Pages 193-209

8. Aspects of Gabor analysis on locally compact abelian groups

   • Karlheinz Gröchenig

   Pages 211-231

9. Quantization of TF lattice-invariant operators on elementary LCA groups

   • Hans G. Feichtinger, Werner Kozek

   Pages 233-266

10. Numerical algorithms for discrete Gabor expansions

    • Thomas Strohmer

    Pages 267-294

11. Oversampled modulated filter banks

    • Helmut Bölcskei, Franz Hlawatsch

    Pages 295-322

12. Adaptation of Weyl-Heisenberg frames to underspread environments

    • Werner Kozek

    Pages 323-352

13. Gabor representation and signal detection

    • Ariela Zeira, Benjamin Friedlander

    Pages 353-380

14. Multi-window Gabor schemes in signal and image representations

    • Yehoshua Y. Zeevi, Meir Zibulski, Moshe Porat

    Pages 381-407

15. Gabor kernels for affine—invariant object recognition

    • Jezekiel Ben-Arie, Zhiqian Wang

    Pages 409-426

16. Gabor’s signal expansion in optics

    • Martin J. Bastiaans

    Pages 427-451

17. Back Matter

    Pages 453-496

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In his paper Theory of Communication [Gab46], D. Gabor proposed the use of a family of functions obtained from one Gaussian by time-and frequency­ shifts. Each of these is well concentrated in time and frequency; together they are meant to constitute a complete collection of building blocks into which more complicated time-depending functions can be decomposed. The application to communication proposed by Gabor was to send the coeffi­ cients of the decomposition into this family of a signal, rather than the signal itself. This remained a proposal-as far as I know there were no seri­ ous attempts to implement it for communication purposes in practice, and in fact, at the critical time-frequency density proposed originally, there is a mathematical obstruction; as was understood later, the family of shifted and modulated Gaussians spans the space of square integrable functions [BBGK71, Per71] (it even has one function to spare [BGZ75] . . . ) but it does not constitute what we now call a frame, leading to numerical insta­ bilities. The Balian-Low theorem (about which the reader can find more in some of the contributions in this book) and its extensions showed that a similar mishap occurs if the Gaussian is replaced by any other function that is "reasonably" smooth and localized. One is thus led naturally to considering a higher time-frequency density.

• Analysis
• Signal
• algorithm
• algorithms
• calculus
• detection
• harmonic analysis
• image processing
• numerical analysis
• object recognition
• operator
• pattern recognition

"... a definitive survey of the subject shows connections and interactions between theory, numerical algorithms, and applications." —Mathematical Reviews

• Department of Mathematics, University of Vienna, Vienna, Austria

  Hans G. Feichtinger, Thomas Strohmer

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