Gabor Analysis and Algorithms

Theory and Applications

H.G. Feichtinger, University of Vienna (Ed.)

T. Strohmer, University of Vienna (Ed.)

Birkhäuser 1998 * Hardcover * 500 pages * 29 Illustrations

Series: Applied and Numerical Harmonic Analysis

The field's leading international experts have come together to give a detailed survey of the theory of Gabor analysis, a method of time-frequency analysis and its applications in signal and image processing. This book is a collection of surveys thematically organized, showing the connections and interactions between theory, numerical algorithms, and applications. It gives an overview of the different branches of Gabor analysis, and contains many original results which are published for the first time.

The first part of the book is devoted to the mathematical foundations of Gabor theory. The duality condition of Weyl-Heisenberg frames, connections to filter banks, the Balian-Low Theorem and Wilson bases are presented. Irregular Gabor systems and Gabor analysis for distributions are also covered. In the second part important concepts like the uncertainty principle, the Zak transform and density conditions for Weyl-Heisenberg frames are generalized for locally compact abelian groups. Further, the group theoretical approach allows a unified view on discrete and continuous, finite and infinite, one-dimensional, and higher-dimensional Gabor analysis. The third part of the book covers applications in signal and image processing. Each chapter provides an up-to-date introduction for the important role of Gabor analysis in such applications as filter bank design, signal detection, mobile communication, image representation, pattern recognition, and optics.

Most important features:

- Self-contained introduction to basic Gabor analysis
- Survey of fundamental results in Gabor theory
- Efficient numerical algorithms
- Gabor expansions in signal and image processing
- Applications in pattern recognition, filter bank design and optics
- Exhaustive bibliography
The book provides an introduction to mathematicians and engineers who want to learn about the different approaches and aspects of Gabor analysis or want to apply Gabor-based techniques to tasks in signal and image processing. It is an especially useful reference for research specialists in harmonic analysis, applied mathematics, numerical analysis, engineering, signal and image processing, optics, and pattern recognition.

Contents:

Foreword /

Ingrid Daubechies

Preface

Contributors

Introduction with readers guide/H.G. Feichtinger&T. Strohmer

- 1.
The duality condition for Weyl-Heisenberg frames/A.J.E.M. Janssen- 1.1 Introduction
- 1.2 Time-continuous shift-invariant systems
- 1.3 Weyl-Heisenberg systems as shift-invariant systems
- 1.4 Weyl-Heisenberg systems in the time-frequency domain
- 1.5 Rational Weyl-Heisenberg systems in the Zak transform domain
- 1.6 Time-discrete Weyl-Heisenberg systems

- 2.
Gabor systems and the Balian-Low Theorem/J.J. Benedetto, C. Heil & D.F. Walnut- 2.1 Introduction
- 2.2 Background
- 2.3 The Zak Transform and the Amalgam BLT
- 2.4 Wilson bases
- 2.5 Distributional calculations and the continuity of the Zak transform
- 2.6 The Uncertainty Principle approach to the BLT
- 2.7 Appendix: Notation

- 3.
A Banach space of test functions for Gabor analysis/H.G. Feichtinger & G. Zimmermann- 3.1 Introduction
- 3.2 Characterizations of the Segal algebra
S_{o}(R^{d})- 3.3 Continuity of Gabor operators
- 3.4 Riesz bases and frames for Banach spaces
- 3.5 Dual pairs and biorthogonal systems
- 3.6 Dual pairs in
S_{o}

- 4.
Pseudodifferential operators, Gabor frames, and local trigonometric bases//R. Rochberg & K. Tachizawa- 4.1 Introduction
- 4.2 Main results
- 4.3 Analysis of elliptic pseudodifferential operators
- 4.4 Approximate diagonalization of
o(x, D)- 4.5 The boundedness of
o(x, D)on the Sobolev spaces- 4.6 Estimates for singular values
- 4.7 Size estimates for eigenfunctions

- 5.
Perturbation of frames and applications to Gabor frames/O. Christensen- 5.1 Introduction
- 5.2 Frames and Riesz bases
- 5.3 Perturbation of frames
- 5.4 Applications to Gabor frames
- 5.5 Banach frames

- 6.
Aspects of Gabor analysis on locally compact abelian groups/K. Gröchenig- 6.1 Introduction
- 6.2 Basics on locally compact abelian groups
- 6.3 Uncertainty Principles and Lieb's inequalities
- 6.4 Zak transform, Gabor frames, and the Balian-Low phenomenon
- 6.5 Density conditions

- 7.
Quantization of TF lattice-invariant operators on elementary LCA groups/H.G. Feichtinger & W. Kozek- 7.1 Introduction
- 7.2 Elementary LCA groups and their TF-shift
- 7.3 The Gelfand triple
(S_{O}, L^{2}, S'_{0})(G)- 7.4 The operator Gelfand triple
(B,H,B')- 7.5 The Generalized KN correspondence
- 7.6 Spreading function
- 7.7 TF-Lattice invariant operators
- 7.8 KN versus Weyl quantization

- 8.
Numerical algorithms for discrete Gabor expansions/T.Strohmer- 8.1 Introduction
- 8.2 An Algebraic setting for discrete Gabor theory
- 8.3 Unitary factorizations of the Gabor frame operator
- 8.4 Finite Gabor expansions and number theory
- 8.5 Design of adaptive dual windows
- 8.6 Conjugate Gradient methods for Gabor expansions
- 8.7 Preconditioners and Approximate Inverses

- 9.
Oversampled modulated filter banks/H. Bölcskei & F. Hlawatsch- 9.1 Introduction and outline
- 9.2 Oversampled filter banks and frames
- 9.3 Oversampled DFT filter banks
- 9.4 Oversampled cosine modulated filter banks
- 9.5 Conclusion

- 10.
Adaptation of Weyl-Heisenberg frames to underspread environments/W.Kozek- 10.1 Introduction
- 10.2 Time-frequency operator representation
- 10.3 Operator analysis and synthesis via STFT
- 10.4 Adaptation of continuous WH frames
- 10.5 Underspread operators
- 10.6 Applying adapted continuous frames
- 10.7 Adaptation of discrete WH frames/bases
- 10.8 Numerical simulation

- 11.
Gabor representation and signal detection/A. Zeira & B. Friedlander- 11.1 Introduction
- 11.2 Background
- 11.3 Detection in the transform domain
- 11.4 Detection in the data domain
- 11.5 Sensitivity to mismatch
- 11.6 Robust matched subspace detectors
- 11.7 Summary and conclusions

- 12.
Multi-window Gabor schemes in signal and image representations/Y.Y. Zeevi, M. Zibulski & M. Porat- 12.1 Motivation for using Gabor-type schemes
- 12.2 Generalized Gabor-type schemes
- 12.3 Applications in image processing and computer vision
- 12.4 Summary and discussion

- 13.
Gabor kernels for affine-invariant object recognition/J. Ben-Arie & Z. Wang- 13.1 Introduction
- 13.2 Affine-invariant spectral signatures (AISSs)
- 13.3 Affine-invariant recognition by multi-dimensional indexing (MDI)
- 13.4 Experimental results

- 14.
Gabor's signal expansion in optics/M.J. Bastiaans- 14.1 Introduction
- 14.2 Some optics fundamentals
- 14.3 Gabor's signal expansion in optics
- 14.4 Degrees of freedom of an optical signal
- 14.5 Coherent-optical generation of the Gabor transform via the Zak transform
Bibliography

Index