Gabor And Wavelet Frames (Lecture Notes, Institute for Mathematical Sciences, National University of Singapore) (Lecture Notes, Institute for Mathematical Sciences, National University of Singapore)

Gabor and wavelet analyses have found widespread applications in signal analysis, image processing and many other information-related areas. Both deliver representations that are simultaneously local in time and in frequency. Due to their significance and success in practical applications, they formed some of the core topics of the program Mathematics and Computation in Imaging Science and Information Processing , which was held at the Institute for Mathematical Sciences, National University of Singapore, from July to December 2003 and in August 2004. As part of the program, tutorial lectures were conducted by international experts, and they covered a wide spectrum of topics in mathematical image, signal and information processing. This volume includes exposition articles by the tutorial speakers on the foundations of Gabor analysis, subband filters and wavelet algorithms, and operator-theoretic interpolation of wavelets and frames. It also presents research papers on Gabor analysis, written by specialists in their respective areas. The volume takes graduate students and researchers new to the field on a valuable learning journey from introductory Gabor and wavelet analyses to advanced topics of current research CONTENTS 6 Foreword 8 Preface 10 A Guided Tour from Linear Algebra to the Foundations of Gabor Analysis Hans G. Feichtinger, Franz Luef and Tobias Werther 14 1. Introduction 15 2. Basics in Linear Algebra 17 3. Finite Dimensional Gabor Analysis 21 4. Frames and Riesz Bases 27 5. Gabor Analysis on L2 31 6. Time-Frequency Representations 35 7. The Gelfand Triple 38 8. The Spreading Function 47 9. Conclusion and Outlook 58 References 60 Some Iterative Algorithms to Compute Canonical Win- dows for Gabor Frames A. J. E. M. Janssen 63 1. Introduction 63 2. Overview 64 3. Basic Tools 67 4. Analysis of Recursion I to Approximate gt 71 5. Proposing Iterations Without Inversions 73 5.1. Iterations for gt 73 5.2. Iterations for gd 75 6. Analysis of Recursion II to Approximate gt 78 7. Analysis of Recursion IV to Approximate gd 82 8. Summary of Results for Iterations III and V 83 9. Concluding Remarks 86 Acknowledgments 86 References 87 Gabor Analysis, Noncommutative Tori and Feichtinger's Algebra Franz Luef 89 1. Introduction 89 2. Operator Algebras of Time-Frequency Shifts 93 3. Noncommutative Tori and Feichtinger's Algebra 96 4. Feichtinger's Algebra as Bimodule for C ( ) and C ( 0) 106 5. Application to Gabor Analysis: Biorthogonality Relation of Wexler-Raz 111 6. Conclusions 115 Acknowledgment 116 References 116 Unitary Matrix Functions,Wavelet Algorithms, and Struc- tural Properties of Wavelets Palle E. T. Jorgensen 119 1. Introduction 119 1.1. Index of terminology in mathematics and in engineering 120 1.2. Motivation 135 1.2.1. Some points of history 141 1.2.2. Some early applications 144 2. Signal Processing 145 2.1. Filters in communications engineering 148 2.2. Algorithms for signals and for wavelets 149 2.2.2. Subdivision algorithms 154 2.2.3. Wavelet packet algorithms 156 2.2.4. Lifting algorithms: Sweldens and more 157 2.3. Factorization theorems for matrix functions 158 2.3.1. The case of polynomial functions [the polyphase matrix, joint work with Ola Bratteli] 160 2.3.2. General results in mathematics on matrix functions 162 2.3.3. Connection between matrix functions and wavelets 164 2.3.3.1. Multiresolution wavelets 165 2.3.3.2. Generalized multiresolutions [joint work with L. Baggett, K. Merrill, and J. Packer] 166 2.3.4. Matrix completion 168 2.3.5. Connections between matrix functions and signal processing 170 Acknowledgments 172 References 173 Unitary Systems, Wavelet Sets, and Operator-Theoretic Interpolation of Wavelets and Frames David R. Larson 179 1. Introduction 179 1.1. Talks and abstracts 180 1.2. Some background 180 1.2.1. Interpolation 182 1.2.2. Some basic terminology 182 1.2.3. Acknowledgements 184 2. Unitary Systems and Wavelet Sets 184 2.1. The one-dimensional wavelet system 184 2.1.1. Dyadic wavelets 184 2.1.2. The dyadic unitary system 185 2.1.3. Non-dyadic wavelets in one dimension 185 2.2. N dimensions 186 2.2.1. The expansive-dilation case 186 2.2.2. The non-expansive dilation case 186 2.3. Abstract systems 187 2.3.1. Restrictions on wandering vectors 187 2.3.2. Group systems 187 2.4. The local commutant 188 2.4.1. The local commutant of the system UD;T 188 2.4.2. The local commutant of an abstract unitary system 189 2.4.3. Operator-theoretic interpolation 190 2.4.4. Normalizing the commutant 191 2.4.5. An elementary interpolation result 192 2.4.6. Interpolation pairs of wandering vectors 193 2.4.7. A test for interpolation pairs 193 2.4.8. Connectedness 195 2.5. Wavelet sets 195 2.5.1. The Fourier transform 195 2.5.2. The commutant of (D, T) 196 2.5.3. Wavelets of computationally elementary form 197 2.5.4. De nition of wavelet set 199 2.5.5. The spectral set condition 200 2.5.6. Translation and dilation congruence 200 2.5.7. A criterion 202 2.6. Phases 202 2.6.1. Some examples of one-dimensional wavelet sets 203 2.7. Operator-theoretic interpolation of wavelets: The special case of wavelet sets 206 2.7.1. The interpolation map 206 2.7.2. An algorithm for the interpolation unitary 207 2.8. The interpolation unitary normalizes the commutant 208 2.8.1. C (D; T) is nonabelian 209 2.8.2. The coe cient criterion 209 2.9. Interpolation pairs of wavelet sets 210 2.10. Journe family interpolation pairs 210 3. Unitary Systems and Frames 211 3.1. Basics on frames 212 3.2. Dilation of frames: The discrete version of Naimark's theorem 213 3.3. Complements of frames 213 3.4. Super-frames, super-wavelets, and multiplexing 215 3.5. Frame vectors for unitary systems 216 3.6. An operator model 217 3.7. Group representations 217 4. Decompositions of Operators and Operator-Valued Frames 218 4.1. Ellipsoidal frames 219 4.2. A problem in operator theory 221 References 223 Includes exposition articles by the tutorial speakers on the foundations of Gabor analysis, subband filters and wavelet algorithms, and operator-theoretic interpolation of wavelets and frames. This volume presents research papers on Gabor analysis, written by specialists in their respective areas.