Excursions in Harmonic Analysis, Volume 5: The February Fourier Talks at the Norbert Wiener Center (Applied and Numerical Harmonic Analysis)

This volume consists of contributions spanning a wide spectrum of harmonic analysis and its applications written by speakers at the February Fourier Talks from 2002 - 2016. Containing cutting-edge results by an impressive array of mathematicians, engineers, and scientists in academia, industry and government, it will be an excellent reference for graduate students, researchers, and professionals in pure and applied mathematics, physics, and engineering. Topics covered include: Theoretical harmonic analysis Image and signal processing Quantization Algorithms and representations The February Fourier Talks are held annually at the Norbert Wiener Center for Harmonic Analysis and Applications. Located at the University of Maryland, College Park, the Norbert Wiener Center provides a state-of- the-art research venue for the broad emerging area of mathematical engineering.-- Provided by publisher ANHA Series Preface 7 Preface 10 The February Fourier Talks (FFT) 10 The Norbert Wiener Center 12 The Structure of the Volumes 12 Acknowledgments 14 Contents 15 Part XVII Theoretical Harmonic Analysis 17 Time-Frequency Analysis and Representations of the Discrete Heisenberg Group 18 1 Introduction 18 2 Direct Integrals 20 3 The Rational Case 21 4 The Irrational Case 27 References 31 Fractional Differentiation: Leibniz Meets Hölder 32 1 Introduction 32 2 The counterexample 35 3 The sharp Kato-Ponce inequalities and preliminaries 38 4 The proof of the homogeneous inequality (7) 42 5 Final remarks 47 References 48 Wavelets and Graph C*-Algebras 49 1 Introduction 50 2 C-Algebras and Work by Bratteli and Jorgensen and Dutkay and Jorgensen on Representations of ON 53 3 Marcolli-Paolucci Wavelets 62 4 C*-Algebras Corresponding to Directed Graphs and Higher-Rank Graphs 74 4.1 Directed Graphs, Higher-Rank Graphs, and C*-Algebras 74 4.2 -Semibranching Function Systems and Representations of C*() 77 5 Wavelets on L2(∞, M) 79 6 Traffic Analysis Wavelets on 2(0) for a Finite Strongly Connected k-Graph , and Wavelets from Spectral Graph Theory 91 6.1 Wavelets for Spatial Traffic Analysis 92 6.2 Wavelets on 2(0) Coming from Spectral Graph Theory 94 References 98 Part XVIII Image and Signal Processing 101 Precise State Tracking Using Three-Dimensional Edge Detection 103 1 Introduction 103 1.1 Previous Work in Tracking 104 1.2 Previous Work in Edge Detection 105 1.3 Outline and Contributions 107 2 The Data 108 3 3D Edge Detectors 110 3.1 3D Canny Edge Detection 110 3.2 3D Wavelet Edge Detection 111 3.3 3D Shearlet Edge Detector 112 3.4 3D Hybrid Wavelet and Shearlet Edge Detectors 114 3.5 Performance of the Edge Detectors 116 4 From Edge Detection to Tracking 118 5 Experimental Results 120 6 Conclusions 121 References 123 Approaches for Characterizing Nonlinear Mixtures in Hyperspectral Imagery 126 1 Introduction 126 2 Methodology 128 2.1 Fully Constrained Least Squares 128 2.2 Proposed Method 1: Fully Constrained Least Squares (FCLS) Applied to Single Scattering Albedo Spectra 129 2.3 Proposed Method 2: Generalized Kernel Fully Constrained Least Squares 129 3 Description of Experiment 131 4 Results 134 5 Concluding Remarks 139 References 140 An Application of Spectral Regularization to Machine Learning and Cancer Classification 142 1 Introduction 142 1.1 Machine Learning 143 1.2 Approach 144 1.3 Prior Work 144 1.4 Paper Contents 145 2 Denoising Theorems 146 2.1 Statements of Theorems 147 2.1.1 Method 1: Local averaging on a graph 148 2.1.2 Method 2: Support vector regression/regularization on a graph 152 3 Application: Using Prior Information to Form Graphs 159 3.1 Gene Expression 159 4 Conclusion 163 References 164 Part XIX Quantization 166 Embedding-Based Representation of Signal Geometry 168 1 Introduction 168 1.1 Notation 169 1.2 Outline 169 2 Preserving Distances 170 2.1 Randomized Linear Embeddings 170 2.2 Embedding Map Design 172 2.3 Distance-preserving properties of the map 174 2.4 Learning the Embedding Map 176 3 Preserving Inner Products, Angles, and Correlations 178 3.1 Inner Product Embeddings 178 3.2 Angle Embeddings 180 4 Quantized Embeddings 182 4.1 Quantization of Continuous Embeddings 182 4.2 Universal Quantization and Embeddings 185 5 Discussion 188 References 189 Distributed Noise-Shaping Quantization: II. Classical Frames 192 1 Introduction 192 1.1 Statement of the Main Results 194 2 Background and Review of Methodology 195 2.1 Basics of Noise Shaping for Frames 195 2.2 Distributed Beta Encoding and Beta Duals of Frames 196 3 Warm up: Beta Duals of Finite Fourier Frames 199 4 Generalization: Unitarily Generated Frames 201 4.1 Unitary frame paths 201 4.2 Beta Duals of Unitarily Generated Frames 203 5 An Infinite-Dimensional Case: Bandlimited Functions on R 204 6 Concluding Remarks 208 Appendix: Greedy Quantizer for Complex Measurements 208 References 211 Consistent Reconstruction: Error Moments and Sampling Distributions 212 1 Introduction 212 1.1 Worst case error 213 1.2 Background 213 1.3 Overview and main results 214 2 Error moments for consistent reconstruction: unit-norm distributions 215 2.1 Consistent reconstruction and coverage problems 215 2.2 Error moment bounds 217 3 Error moments for consistent reconstruction: general distributions 222 3.1 General admissibility condition 222 3.2 Coverage problems revisited 223 3.2.1 Conditioning and a bound by caps with an=1 223 3.2.2 Covering and discretization 225 3.3 Moment bounds for general distributions 226 3.4 Numerical experiment 231 References 233 Part XX Algorithms and Representations 235 Frame Theory for Signal Processing in Psychoacoustics 236 1 Introduction 236 2 The auditory analysis of sounds 240 2.1 Ear's anatomy 240 2.2 The auditory filters concept 242 2.3 Auditory masking 244 2.3.1 Spectral masking 244 2.3.2 Temporal masking 245 2.3.3 Time-frequency masking 246 2.4 Computational auditory scene analysis 248 3 Frame theory 249 3.1 Frames: A Mathematical viewpoint 251 3.1.1 Frame-related operators 253 3.1.2 Perfect reconstruction via frames 255 3.2 Frame multipliers 258 3.2.1 Implementation 259 4 Filter bank frames: a signal processing viewpoint 260 4.1 Basics of filter banks 260 4.2 The equivalent uniform filter bank 263 4.3 Connection to Frame Theory 265 5 Frame Theory: Psychoacoustics-motivated Applications 270 5.1 A perfectly invertible, perceptually motivated filter bank 270 5.2 Perceptual Sparsity 272 6 Conclusion 274 References 275 A Flexible Scheme for Constructing (Quasi-)Invariant Signal Representations 280 1 Introduction 280 1.1 Path Descriptions under Deformations 282 1.2 The Importance of Invariants 283 1.3 Pairwise Relations in Signal Modeling 284 2 Invariance, Uniqueness, and Completeness 286 2.1 Definitions and Examples 286 2.2 General Case 289 3 Invariant Relations in Line Drawings 290 3.1 Uniqueness 291 3.2 Intersection of feasible sets 292 3.3 Completeness 293 4 A Continuous Trace Model 293 4.1 Definitions 294 4.2 Profile Trace 295 4.3 Invariance to Dynamic Warps 297 4.4 Invariance to Monotonic Illumination Change 298 4.5 Generalized Texture Trace 299 5 Plugin for Metric Pairwise Relations 300 6 Discrete Approximation 302 6.1 Discretization of Curve γ 303 6.2 Quantization of Image Function 304 6.3 Discretized Textured Trace 305 6.4 Patch Model 306 7 Results 307 7.1 Matching Image Patches 308 7.2 Tracking 309 7.3 One-Shot Tracking 310 8 Conclusion and Outlook 311 References 312 Use of Quillen-Suslin Theorem for Laurent Polynomials in Wavelet Filter Bank Design 314 1 Introduction 314 2 Wavelet Filter Bank Design via Laurent Polynomial Matrices 315 2.1 Polyphase Representation and Wavelet FB Design 316 2.2 Quillen-Suslin Theorem and Wavelet FB Design 316 3 New Quillen-Suslin based Method for Designing Wavelet FBs 318 3.1 Motivation for the theory 318 3.2 Main ingredients of the theory 319 3.3 Main ingredients of the algorithms 321 4 Conclusion 322 References 323 A Fast Fourier Transform for Fractal Approximations 325 1 Introduction 325 1.1 Diţǎ's Construction of Large Hadamard Matrices 329 1.2 Complexity of Matrix Multiplication in Diţǎ's Construction 330 2 A Fast Fourier Transform on SN 331 2.1 Computational Complexity of Theorems 9 and 11 337 2.2 The Diagonal Matrices 337 References 339 Index 340 Applied and Numerical Harmonic Analysis (77 volumes) 343 Front Matter....Pages i-xviii Front Matter....Pages 1-1 Time-Frequency Analysis and Representations of the Discrete Heisenberg Group....Pages 3-16 Fractional Differentiation: Leibniz Meets Hölder....Pages 17-33 Wavelets and Graph C ∗-Algebras....Pages 35-86 Front Matter....Pages 87-88 Precise State Tracking Using Three-Dimensional Edge Detection....Pages 89-111 Approaches for Characterizing Nonlinear Mixtures in Hyperspectral Imagery....Pages 113-128 An Application of Spectral Regularization to Machine Learning and Cancer Classification....Pages 129-152 Front Matter....Pages 153-154 Embedding-Based Representation of Signal Geometry....Pages 155-178 Distributed Noise-Shaping Quantization: II. Classical Frames....Pages 179-198 Consistent Reconstruction: Error Moments and Sampling Distributions....Pages 199-221 Front Matter....Pages 223-223 Frame Theory for Signal Processing in Psychoacoustics....Pages 225-268 A Flexible Scheme for Constructing (Quasi-)Invariant Signal Representations....Pages 269-302 Use of Quillen-Suslin Theorem for Laurent Polynomials in Wavelet Filter Bank Design....Pages 303-313 A Fast Fourier Transform for Fractal Approximations....Pages 315-329 Back Matter....Pages 331-338