Digital signal processing (DSP) Books

Book SynopsisThis title sets out to show that 2-D signal analysis has its own role to play alongside signal processing and image processing. Concentrating its coverage on those 2-D signals coming from physical sensors (such as radars and sonars), the discussion explores a 2-D spectral approach but develops the modeling of 2-D signals and proposes several data-oriented analysis techniques for dealing with them. Coverage is also given to potential future developments in this area.Table of ContentsIntroduction 13 Chapter 1. Basic Elements of 2-D Signal Processing 17Claude CARIOU, Olivier ALATA and Jean-Marc LE CAILLEC 1.1. Introduction 17 1.2. Deterministic 2-D signals 18 1.2.1. Definition 18 1.2.2. Particular 2-D signals 19 1.3. Random 2-D signals 22 1.3.1. Definition 22 1.3.2. Characterization up to the second order 23 1.3.3. Stationarity 24 1.3.4. Characterization of orders higher than two 26 1.3.5. Ergodicity 26 1.3.6. Specificities of random 2-D signals 27 1.3.7. Particular random signals 28 1.4. 2-D systems 31 1.4.1. Definition 31 1.4.2. Main 2-D operators 31 1.4.3. Main properties 32 1.4.4. Linear time-invariant (LTI) system 33 1.4.5. Example 34 1.4.6. Separable system 34 1.4.7. Stability of 2-D systems 36 1.4.8. Support of the impulse response – causality 37 1.5. Characterization of 2-D signals and systems 39 1.5.1. Frequency response of an LTI system 39 1.5.2. 2-D Fourier transform 41 1.5.3. Discrete 2-D Fourier transform 43 1.5.4. 2-D z transform 46 1.5.5. Frequency characterization of a random 2-D signal 55 1.5.6. Output of a 2-D system with random input 57 1.6. 2-D Wold decomposition 58 1.6.1. Innovation, determinism and regularity in the 2-D case 58 1.6.2. Total decomposition of three fields 60 1.6.3. Example of an outcome 61 1.7. Conclusion 63 1.8. Bibliography 63 Chapter 2. 2-D Linear Stochastic Modeling 65Olivier ALATA and Claude CARIOU 2.1. Introduction 65 2.2. 2-D ARMA models 66 2.2.1. Definition 66 2.2.2. 2-D ARMA models and prediction supports 67 2.3. L-Markovian fields 73 2.3.1. 2-D Markov fields and L-Markovian fields 73 2.3.2. 2-D L-Markovian fields and Gibbs fields 74 2.4. “Global” estimation methods 76 2.4.1. Maximum likelihood 76 2.4.2. Yule-Walker equations 79 2.4.3. 2-D Levinson algorithm (for the parametric 2-D AR estimation) 85 2.5. “Adaptive” or “recursive” estimation methods 93 2.5.1. Connectivity hypotheses for adaptive or recursive algorithms 93 2.5.2. Algorithms 93 2.6. Application: segmentation of textured images 100 2.6.1. Textured field and segmented field 100 2.6.2. Multiscale or hierarchical approach 103 2.6.3. Non-supervised estimation of the parameters 104 2.6.4. Examples of segmentation 108 2.7. Bibliography 109 Chapter 3. 2-D Spectral Analysis 115Claude CARIOU, Stéphanie ROUQUETTE and Olivier ALATA 3.1. Introduction 115 3.2. General concepts 116 3.3. Traditional 2-D spectral estimation 118 3.3.1. Periodogram technique 118 3.3.2. Correlogram technique 119 3.3.3. Limits of traditional spectral analysis 120 3.4. Parametric 2-D spectral estimation 121 3.4.1. Spectral estimation by linear stochastic models 122 3.4.2. Maximum entropy method 128 3.4.3. Minimum variance method 132 3.5. 2-D high resolution methods 134 3.5.1. 2-D MUSIC 135 3.5.2. Calculation of a pseudo-spectrum 135 3.5.3. Pseudo-spectrum estimation 137 3.6. Other techniques 138 3.7. Comparative study of some techniques 138 3.7.1. Analysis of 2-D harmonic components 139 3.7.2. Analysis of random fields 159 3.7.3. Conclusion 163 3.8. Application: spectral analysis of remote sensing images 165 3.8.1. Position of the problem 165 3.8.2. Stochastic modeling of a radar image 166 3.8.3. Example of application 167 3.9. Conclusion 169 3.10. Bibliography 170 Chapter 4. Bispectral Analysis of 2-D Signals 175Jean-Marc LE CAILLEC and René GARELLO 4.1. Introduction 175 4.1.1. Higher order moments and cumulants 175 4.1.2. Properties of moments and cumulants 179 4.1.3. Polyspectra of stationary signals 181 4.1.4. Polyspectra 185 4.1.5. Definition of the coherence of order p 185 4.2. Moments and spectra of order p for linear signals 185 4.2.1. Moments and cumulants of order p for linear signals 186 4.2.2. Spectrum of order p for a linear signal 187 4.2.3. General properties of the bispectra of linear signals 187 4.2.4. Polyspectrum of a linear signal 188 4.2.5. Coherence of order p for linear signals 189 4.3. Signals in quadratic phase coupling, non-linearity and the Volterra system 189 4.3.1. Bispectrum of a signal in quadratic phase coupling 190 4.3.2. Volterra models and decomposition of non-linear systems 192 4.4. Bispectral estimators for 2-D signals 195 4.4.1. Indirect method 196 4.4.2. Direct method 199 4.4.3. Autoregressive model 200 4.4.4. ARMA modeling 202 4.4.5. Measure of bias and variance of estimators 204 4.5. Hypothesis test for non-linearity and bicoherence tables 204 4.5.1. Hypothesis tests 204 4.5.2. Bicoherence tables 207 4.6. Applications 210 4.6.1. Image restoration 210 4.6.2. Artifact removal 210 4.7. Bibliography 211 Chapter 5. Time-frequency Representation of 2-D Signals 215Stéphane GRASSIN and René GARELLO 5.1. Introduction 215 5.1.1. Bilinear time-frequency representation 215 5.1.2. Four spaces of representation 216 5.1.3. Restriction to bilinear representation 217 5.1.4. Spectral description using bilinear representations 218 5.2. TFR application to sampled images 219 5.2.1. TFR expression of discrete images 219 5.2.2. Support of the sums 223 5.3. Minimum properties and constraints on the kernel 223 5.3.1. Compatibility with reversible linear transformations 224 5.3.2. Positivity 225 5.3.3. TFR with real values 225 5.3.4. Conservation of energy 225 5.3.5. Spectral estimation 226 5.3.6. Evolution of properties of a modified kernel 228 5.4. Notion of analytic images 230 5.4.1. Formulation of the problem for the images 230 5.4.2. Traditional solution 231 5.4.3. Symmetric solution with reference to a hyperplane 233 5.4.4. Solution with a non-symmetric half-plane 233 5.4.5. Choice of spectral division 237 5.5. Spectral analysis application of SAR images 241 5.5.1. Analysis of an internal waveform 243 5.5.2. Analysis of an internal wave field with superimposition 249 5.5.3. Analysis of a small area internal wave field 249 5.5.4. Prospects 250 5.6. Approximation of an internal wave train 252 5.6.1. Benefit of approximation of the frequency law 252 5.6.2. Problem resolution 252 5.6.3. Adequacy of bilinear modulation with instantaneous frequency estimation 255 5.7. Bibliography 257 Chapter 6. 2-D Wavelet Representation 259Philippe CARRÉ, Noël RICHARD and Christine FERNANDEZ 6.1. Introduction 259 6.2. Dyadic wavelet transform: from 1-D to 2-D 260 6.2.1. Multiresolution analysis 260 6.2.2. Wavelets and filter banks 262 6.2.3. Wavelet packets 264 6.2.4. 2-D extension by the simple product 266 6.2.5. Non-separable 2-D wavelets 272 6.2.6. Non-decimated decomposition 278 6.3. Trigonometric transform to adaptive windows 282 6.3.1. Malvar wavelets 282 6.3.2. Folding operator 284 6.3.3. Windowed orthonormal base 287 6.3.4. Extension of Malvar wavelets to 2-D 288 6.4. Transform by frequency slicing 292 6.4.1. Continuous theory of 1-D Meyer wavelets 293 6.4.2. Definition of Meyer wavelet packets 295 6.4.3. Numerical outcome of decomposition in 1-D Meyer wavelet packets 295 6.4.4. Extension of Meyer wavelet packets to 2-D 306 6.5. Conclusion 308 6.6. Bibliography 309 List of Authors 313 Index 315

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Book SynopsisThe purpose of this book is to provide graduate students and practitioners with traditional methods and more recent results for model-based approaches in signal processing. Firstly, discrete-time linear models such as AR, MA and ARMA models, their properties and their limitations are introduced. In addition, sinusoidal models are addressed. Secondly, estimation approaches based on least squares methods and instrumental variable techniques are presented. Finally, the book deals with optimal filters, i.e. Wiener and Kalman filtering, and adaptive filters such as the RLS, the LMS and their variants.Trade Review"This book provides the reader for the first time with a comprehensive collection of the significant results obtained to date in the field of parametric signal modeling and presents a number of new approaches." (Mathematical Reviews, 2010) Table of ContentsChapter 1. Introduction to Parametric Models. Chapter 2. Least-Squares Estimation of Linear Model Parameters. Chapter 3. Matched Filters and Wiener Filters. Chapter 4. Adaptive Filters. Chapter 5. Kalman Filters. Chapter 6. Kalman Filtering for Speech Enhancement. Chapter 7. Instrumental Variable Techniques. Chapter 8. H Infinity Techniques: An Alternative to Kalman filters? Chapter 9. Introduction to Particle Filtering. Appendix.

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Book SynopsisCovering a period of about 25 years, during which time-frequency has undergone significant developments, this book is principally addressed to researchers and engineers interested in non-stationary signal analysis and processing. It is written by recognized experts in the field.Table of ContentsPreface 13 FIRST PART. FUNDAMENTAL CONCEPTS AND METHODS 17 Chapter 1. Time-Frequency Energy Distributions: An Introduction 19 Patrick FLANDRIN 1.1. Introduction 19 1.2. Atoms 20 1.3. Energy 21 1.3.1. Distributions 22 1.3.2. Devices 22 1.3.3. Classes 23 1.4. Correlations 26 1.5. Probabilities 27 1.6. Mechanics 29 1.7. Measurements 29 1.8. Geometries 32 1.9. Conclusion 33 1.10.Bibliography 34 Chapter 2. Instantaneous Frequency of a Signal 37 Bernard PICINBONO 2.1. Introduction 37 2.2. Intuitive approaches 38 2.3. Mathematical definitions 40 2.3.1. Ambiguity of the problem 40 2.3.2. Analytic signal and Hilbert transform 40 2.3.3. Application to the definition of instantaneous frequency 42 2.3.4. Instantaneous methods 45 2.4. Critical comparison of the different definitions 46 2.4.1. Interest of linear filtering 46 2.4.2. Bounds of the quantities introduced 46 2.4.3. Instantaneous nature 47 2.4.4. Interpretation by the average 48 2.5. Canonical pairs 49 2.6. Phase signals 50 2.6.1. Blaschke factors 50 2.6.2. Oscillatory singularities 54 2.7. Asymptotic phase signals 57 2.7.1. Parabolic chirp 57 2.7.2. Cubic chirp 59 2.8. Conclusions 59 2.9. Bibliography 60 Chapter 3. Linear Time-Frequency Analysis I: Fourier-Type Representations 61 Remi GRIBONVAL 3.1. Introduction 61 3.2. Short-time Fourier analysis 62 3.2.1. Short-time Fourier transform 63 3.2.2. Time-frequency energy maps 64 3.2.3. Role of the window 66 3.2.4. Reconstruction/synthesis 71 3.2.5. Redundancy 71 3.3. Gabor transform; Weyl-Heisenberg and Wilson frames 71 3.3.1. Sampling of the short-time Fourier transform 71 3.3.2. Weyl-Heisenberg frames 72 3.3.3. Zak transform and “critical” Weyl-Heisenberg frames 74 3.3.4. Balian-Low theorem 75 3.3.5. Wilson bases and frames, local cosine bases 75 3.4. Dictionaries of time-frequency atoms; adaptive representations 77 3.4.1. Multi-scale dictionaries of time-frequency atoms 77 3.4.2. Pursuit algorithm 78 3.4.3. Time-frequency representation 79 3.5. Applications to audio signals 80 3.5.1. Analysis of superimposed structures 80 3.5.2. Analysis of instantaneous frequency variations 80 3.5.3. Transposition of an audio signal 82 3.6. Discrete algorithms 82 3.6.1. Fast Fourier transform 83 3.6.2. Filter banks: fast convolution 83 3.6.3. Discrete short-time Fourier transform 85 3.6.4. Discrete Gabor transform 86 3.7. Conclusion 86 3.8. Acknowledgements 87 3.9. Bibliography 87 Chapter 4. Linear Time-Frequency Analysis II: Wavelet-Type Representations 93 Thierry BLU and Jerome LEBRUN 4.1. Introduction: scale and frequency 94 4.2. Continuous wavelet transform 95 4.2.1. Analysis and synthesis 95 4.2.2. Multiscale properties 97 4.3. Discrete wavelet transform 98 4.3.1. Multi-resolution analysis 98 4.3.2. Mallat algorithm 104 4.3.3. Graphical representation 106 4.4. Filter banks and wavelets 107 4.4.1. Generation of regular scaling functions 108 4.4.2. Links with approximation theory 111 4.4.3. Orthonormality and bi-orthonormality/perfect reconstruction 112 4.4.4. Polyphase matrices and implementation 114 4.4.5. Design of wavelet filters with finite impulse response 114 4.5. Generalization: multi-wavelets 116 4.5.1. Multi-filter banks 116 4.5.2. Balancing and design of multi-filters 118 4.6. Other extensions 121 4.6.1. Wavelet packets 121 4.6.2. Redundant transformations: pyramids and frames 122 4.6.3. Multi-dimensional wavelets 123 4.7. Applications 124 4.7.1. Signal compression and denoising 124 4.7.2. Image alignment 125 4.8. Conclusion 125 4.9. Acknowledgments 126 4.10. Bibliography 126 Chapter 5. Quadratic Time-Frequency Analysis I: Cohen’s Class 131 Francois AUGER and Eric CHASSANDE-MOTTIN 5.1. Introduction 131 5.2. Signal representation in time or in frequency 132 5.2.1. Notion of signal representation 132 5.2.2. Temporal representations 133 5.2.3. Frequency representations 134 5.2.4. Notion of stationarity 135 5.2.5. Inadequacy of monodimensional representations 136 5.3. Representations in time and frequency 137 5.3.1. “Ideal” time-frequency representations 137 5.3.2. Inadequacy of the spectrogram 140 5.3.3. Drawbacks and benefits of the Rihaczek distribution 142 5.4. Cohen’s class 142 5.4.1. Quadratic representations covariant under translation 142 5.4.2. Definition of Cohen’s class 143 5.4.3. Equivalent parametrizations 144 5.4.4. Additional properties 145 5.4.5. Existence and localization of interference terms 148 5.5. Main elements 155 5.5.1. Wigner-Ville and its smoothed versions 155 5.5.2. Rihaczek and its smoothed versions 157 5.5.3. Spectrogram and S transform 158 5.5.4. Choi-Williams and reduced interference distributions 158 5.6. Conclusion 159 5.7. Bibliography 159 Chapter 6. Quadratic Time-Frequency Analysis II: Discretization of Cohen’s Class 165 Stephane GRASSIN 6.1. Quadratic TFRs of discrete signals 165 6.1.1. TFRs of continuous-time deterministic signals 167 6.1.2. Sampling equation 167 6.1.3. The autocorrelation functions of the discrete signal 168 6.1.4. TFR of a discrete signal as a function of its generalized ACF 169 6.1.5. Discussion 171 6.1.6. Corollary: ambiguity function of a discrete signal 172 6.2. Temporal support of TFRs 173 6.2.1. The characteristic temporal supports 173 6.2.2. Observations 175 6.3. Discretization of the TFR 176 6.3.1. Meaning of the frequency discretization of the TFR 176 6.3.2. Meaning of the temporal discretization of the TFR 176 6.3.3. Aliased discretization 177 6.3.4. “Non-aliased”discretization 179 6.4. Properties of discrete-time TFRs 180 6.4.1. Discrete-time TFRs 181 6.4.2. Effect of the discretization of the kernel 182 6.4.3. Temporal inversion 182 6.4.4. Complexcon jugation 183 6.4.5. Real-valued TFR 183 6.4.6. Temporal moment 183 6.4.7. Frequency moment 184 6.5. Relevance of the discretization to spectral analysis 185 6.5.1. Formulation of the problem 185 6.5.2. Trivial case of a sinusoid 187 6.5.3. Signal with linear frequency modulation 187 6.5.4. Spectral analysis with discretized TFRs 188 6.6. Conclusion 189 6.7. Bibliography 189 Chapter 7. Quadratic Time-Frequency Analysis III: The Affine Class and Other Covariant Classes 193 Paulo GONCALVES, Jean-Philippe OVARLEZ and Richard BARANIUK 7.1. Introduction 193 7.2. General construction of the affine class 194 7.2.1. Bilinearity of distributions 194 7.2.2. Covariance principle 195 7.2.3. Affine class of time-frequency representations 198 7.3. Properties of the affine class 201 7.3.1. Energy 201 7.3.2. Marginals 202 7.3.3. Unitarity 202 7.3.4. Localization 203 7.4. Affine Wigner distributions 206 7.4.1. Diagonal form of kernels 206 7.4.2. Covariance to the three-parameter affine group 209 7.4.3. Smoothed affine pseudo-Wigner distributions 211 7.5. Advanced considerations 216 7.5.1. Principle of tomography 216 7.5.2. Operators and groups 217 7.6. Conclusions 222 7.7. Bibliography 223 SECOND PART. ADVANCED CONCEPTS AND METHODS 227 Chapter 8. Higher-Order Time-Frequency Representations 229 Pierre-Olivier AMBLARD 8.1. Motivations 229 8.2. Construction of time-multifrequency representations 230 8.2.1. General form and desirable properties 230 8.2.2. General classes in the symmetric even case 231 8.2.3. Examples and interpretation 236 8.2.4. Desired properties and constraints on the kernel 237 8.2.5. Discussion 239 8.3. Multilinear time-frequency representations 240 8.3.1. Polynomial phase and perfect concentration 240 8.3.2. Multilinear time-frequency representations: general class 242 8.4. Towards affine multilinear representations 243 8.5. Conclusion 246 8.6. Bibliography 247 Chapter 9. Reassignment 249 Eric CHASSANDE-MOTTIN, Francois AUGER, and Patrick FLANDRIN 9.1. Introduction 249 9.2. The reassignment principle 250 9.2.1. Classical tradeoff in time-frequency and time-scale analysis 250 9.2.2. Spectrograms and scalograms re-examined and corrected by mechanics 252 9.2.3. Generalization to other representations 254 9.2.4. Link to similar approaches 257 9.3. Reassignment at work 257 9.3.1. Fast algorithms 258 9.3.2. Analysis of a few simple examples 259 9.4. Characterization of the reassignment vector fields 265 9.4.1. Statistics of the reassignment vectors of the spectrogram 265 9.4.2. Geometrical phase and gradient field 267 9.5. Two variations 269 9.5.1. Supervised reassignment 269 9.5.2. Differential reassignment 270 9.6. An application: partitioning the time-frequency plane 271 9.7. Conclusion 274 9.8. Bibliography 274 Chapter 10. Time-Frequency Methods for Non-stationary Statistical Signal Processing 279 Franz HLAWATSCH and Gerald MATZ 10.1. Introduction 279 10.2. Time-varying systems 281 10.3. Non-stationary processes 283 10.4. TF analysis of non-stationary processes – type I spectra 285 10.4.1. GeneralizedWigner-Ville spectrum 285 10.4.2. TF correlations and statistical cross-terms 286 10.4.3. TF smoothing and type I spectra 287 10.4.4. Properties of type I spectra 289 10.5. TF analysis of non-stationary processes – type II spectra 289 10.5.1. Generalized evolutionary spectrum 289 10.5.2. TF smoothing and type II spectra 291 10.6. Properties of the spectra of underspread processes 291 10.6.1. Approximate equivalences 292 10.6.2. Approximate properties 295 10.7. Estimation of time-varying spectra 296 10.7.1. A class of estimators 296 10.7.2. Bias-variance analysis 297 10.7.3. Designing an estimator 299 10.7.4. Numerical results 300 10.8. Estimation of non-stationary processes 302 10.8.1. TF formulation of the optimum filter 303 10.8.2. TF design of a quasi-optimum filter 304 10.8.3. Numerical results 305 10.9. Detection of non-stationary processes 306 10.9.1. TF formulation of the optimum detector 309 10.9.2. TF design of a quasi-optimum detector 310 10.9.3. Numerical results 311 10.10. Conclusion 313 10.11. Acknowledgements 315 10.12. Bibliography 315 Chapter 11. Non-stationary Parametric Modeling 321 Corinne MAILHES and Francis CASTANIE 11.1. Introduction 321 11.2. Evolutionary spectra 322 11.2.1. Definition of the “evolutionary spectrum”322 11.2.2. Properties of the evolutionary spectrum 324 11.3. Postulate of local stationarity 325 11.3.1. Sliding methods 325 11.3.2. Adaptive and recursive methods 326 11.3.3. Application to time-frequency analysis 328 11.4. Suppression of a stationarity condition 329 11.4.1. Unstable models 329 11.4.2. Models with time-varying parameters 332 11.4.3. Models with non-stationary input 340 11.4.4. Application to time-frequency analysis 346 11.5. Conclusion 348 11.6. Bibliography 349 Chapter 12. Time-Frequency Representations in Biomedical Signal Processing 353 Lotfi SENHADJI and Mohammad Bagher SHAMSOLLAHI 12.1. Introduction 353 12.2. Physiological signals linked to cerebral activity 356 12.2.1. Electroencephalographic (EEG) signals 356 12.2.2. Electrocorticographic (ECoG) signals 359 12.2.3. Stereoelectroencephalographic (SEEG) signals 359 12.2.4. Evoked potentials (EP) 362 12.3. Physiological signals related to the cardiac system 363 12.3.1. Electrocardiographic (ECG) signals 363 12.3.2. R-R sequences 365 12.3.3. Late ventricular potentials (LVP) 367 12.3.4. Phonocardiographic (PCG) signals 369 12.3.5. Doppler signals 372 12.4. Other physiological signals 372 12.4.1. Electrogastrographic (EGG) signals 372 12.4.2. Electromyographic (EMG) signals 373 12.4.3. Signals related to respiratory sounds (RS) 374 12.4.4. Signals related to muscle vibrations 374 12.5. Conclusion 375 12.6. Bibliography 376 Chapter 13. Application of Time-Frequency Techniques to Sound Signals: Recognition and Diagnosis 383 Manuel DAVY 13.1. Introduction 383 13.1.1. 384 13.1.2. Sound signals 384 13.1.3. Time-frequency analysis as a privileged decision-making tool 384 13.2. Loudspeaker fault detection 386 13.2.1. Existing tests 386 13.2.2. A test signal 388 13.2.3. A processing procedure 389 13.2.4. Application and results 391 13.2.5. Use of optimized kernels 395 13.2.6. Conclusion 399 13.3. Speaker verification 399 13.3.1. Speaker identification: the standard approach 399 13.3.2. Speaker verification: a time-frequency approach 403 13.4. Conclusion 405 13.5. Bibliography 406 List of Authors 409 Index 413

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Book SynopsisThis book describes the optimization methods most commonly encountered in signal and image processing: artificial evolution and Parisian approach; wavelets and fractals; information criteria; training and quadratic programming; Bayesian formalism; probabilistic modeling; Markovian approach; hidden Markov models; and metaheuristics (genetic algorithms, ant colony algorithms, cross-entropy, particle swarm optimization, estimation of distribution algorithms, and artificial immune systems).Table of ContentsIntroduction xiii Chapter 1. Modeling and Optimization in Image Analysis 1 Jean Louchet 1.1. Modeling at the source of image analysis and synthesis 1 1.2. From image synthesis to analysis 2 1.3. Scene geometric modeling and image synthesis 3 1.4. Direct model inversion and the Hough transform 4 1.5. Optimization and physical modeling 9 1.6. Conclusion 12 1.7. Acknowledgements 13 1.8. Bibliography 13 Chapter 2. Artificial Evolution and the Parisian Approach. Applications in the Processing of Signals and Images 15 Pierre Collet and Jean Louchet 2.1. Introduction 15 2.2. The Parisian approach for evolutionary algorithms 15 2.3. Applying the Parisian approach to inverse IFS problems 17 2.4. Results obtained on the inverse problems of IFS 20 2.5. Conclusion on the usage of the Parisian approach for inverse IFS problems 22 2.6. Collective representation: the Parisian approach and the Fly algorithm 23 2.7. Conclusion 40 2.8. Acknowledgements 41 2.9.Bibliography 41 Chapter 3. Wavelets and Fractals for Signal and Image Analysis 45 Abdeldjalil Ouahabi and Djedjiga Ait Aouit 3.1. Introduction 45 3.2. Some general points on fractals 46 3.3. Multifractal analysis of signals 54 3.4. Distribution of singularities based on wavelets 60 3.5. Experiments 70 3.6. Conclusion 76 3.7. Bibliography 76 Chapter 4. Information Criteria: Examples of Applications in Signal and Image Processing 79 Christian Oliver and Olivier Alata 4.1. Introduction and context 79 4.2. Overview of the different criteria 81 4.3. The case of auto-regressive (AR) models 83 4.4. Applying the process to unsupervised clustering 95 4.5. Law approximation with the help of histograms 98 4.6. Other applications 103 4.7. Conclusion 106 4.8. Appendix 106 4.9. Bibliography 107 Chapter 5. Quadratic Programming and Machine Learning – Large Scale Problems and Sparsity 111 Gaëlle Looslil, Stéphane Canu 5.1. Introduction 111 5.2. Learning processes and optimization 112 5.3. From learning methods to quadratic programming 117 5.4. Methods and resolution 119 5.5. Experiments 128 5.6. Conclusion 132 5.7. Bibliography 133 Chapter 6. Probabilistic Modeling of Policies and Application to Optimal Sensor Management 137 Frédéric Dambreville, Francis Celeste and Cécile Simonin 6.1. Continuum, a path toward oblivion 137 6.2. The cross-entropy (CE) method 138 6.3. Examples of implementation of CE for surveillance 146 6.4. Example of implementation of CE for exploration 153 6.5. Optimal control under partial observation 158 6.6. Conclusion 166 6.7. Bibliography 166 Chapter 7. Optimizing Emissions for Tracking and Pursuit of Mobile Targets 169 Jean-Pierre Le Cadre 7.1. Introduction 169 7.2. Elementary modeling of the problem (deterministic case) 170 7.3. Application to the optimization of emissions (deterministic case) 175 7.4. The case of a target with a Markov trajectory 181 7.5. Conclusion 189 7.6. Appendix: monotonous functional matrices 189 7.7. Bibliography 192 Chapter 8. Bayesian Inference and Markov Models 195 Christophe Collet 8.1. Introduction and application framework 195 8.2. Detection, segmentation and classification 196 8.3. General modeling 199 8.4. Segmentation using the causal-in-scale Markov model 201 8.5. Segmentation into three classes 203 8.6. The classification of objects 206 8.7. The classification of seabeds 212 8.8. Conclusion and perspectives 214 8.9. Bibliography 215 Chapter 9. The Use of Hidden Markov Models for Image Recognition: Learning with Artificial Ants, Genetic Algorithms and Particle Swarm Optimization 219 Sébastien Aupetit, Nicolas Monmarchè and Mohamed Slimane 9.1. Introduction 219 9.2. Hidden Markov models (HMMs) 220 9.3. Using metaheuristics to learn HMMs 223 9.4. Description, parameter setting and evaluation of the six approaches that are used to train HMMs 226 9.5. Conclusion 240 9.6. Bibliography 240 Chapter 10. Biological Metaheuristics for Road Sign Detection 245 Guillaume Dutilleux and Pierre Charbonnier 10.1. Introduction 245 10.2. Relationship to existing works 246 10.3. Template and deformations 248 10.4. Estimation problem 248 10.5. Three biological metaheuristics 252 10.6. Experimental results 259 10.7. Conclusion 265 10.8. Bibliography 266 Chapter 11. Metaheuristics for Continuous Variables. The Registration of Retinal Angiogram Images 269 Johann Drèo, Jean-Claude Nunes and Patrick Siarry 11.1. Introduction 269 11.2. Metaheuristics for difficult optimization problems 270 11.3. Image registration of retinal angiograms 275 11.4. Optimizing the image registration process 279 11.5. Results 288 11.6. Analysis of the results 295 11.7. Conclusion 296 11.8. Acknowledgements 296 11.9. Bibliography 296 Chapter 12. Joint Estimation of the Dynamics and Shape of Physiological Signals through Genetic Algorithms 301 Amine Naït-Ali and Patrick Siarry 12.1. Introduction 301 12.2. Brainstem Auditory Evoked Potentials (BAEPs) 302 12.3. Processing BAEPs 303 12.4. Genetic algorithms 305 12.5. BAEP dynamics 307 12.6. The non-stationarity of the shape of the BAEPs 324 12.7. Conclusion 327 12.8. Bibliography 327 Chapter 13. Using Interactive Evolutionary Algorithms to Help Fit Cochlear Implants 329 Pierre Collet, Pierrick Legrand, Claire Bourgeois-République, Vincent Péan and Bruno Frachet 13.1. Introduction 329 13.2. Choosing an optimization algorithm 333 13.3. Adapting an evolutionary algorithm to the interactive fitting of cochlear implants 335 13.4. Evaluation 338 13.5. Experiments 339 13.6. Medical issues which were raised during the experiments 350 13.7. Algorithmic conclusions for patient A 352 13.8. Conclusion 354 13.9. Bibliography 354 List of Authors 357 Index 359

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Book SynopsisScaling is a mathematical transformation that enlarges or diminishes objects. The technique is used in a variety of areas, including finance and image processing. This book is organized around the notions of scaling phenomena and scale invariance. The various stochastic models commonly used to describe scaling — self-similarity, long-range dependence and multi-fractals — are introduced. These models are compared and related to one another. Next, fractional integration, a mathematical tool closely related to the notion of scale invariance, is discussed, and stochastic processes with prescribed scaling properties (self-similar processes, locally self-similar processes, fractionally filtered processes, iterated function systems) are defined. A number of applications where the scaling paradigm proved fruitful are detailed: image processing, financial and stock market fluctuations, geophysics, scale relativity, and fractal time-space.Table of ContentsPreface 17 Chapter 1. Fractal and Multifractal Analysis in Signal Processing 19 Jacques LEVY VEHEL and Claude TRICOT 1.1. Introduction 19 1.2.Dimensions of sets 20 1.2.1.Minkowski-Bouligand dimension 21 1.2.2. Packing dimension 25 1.2.3.Covering dimension 27 1.2.4. Methods for calculating dimensions 29 1.3. Holder exponents 33 1.3.1. Holder exponents related to a measure 33 1.3.2. Theorems on set dimensions 33 1.3.3. Holder exponent related to a function 36 1.3.4. Signal dimension theorem 42 1.3.5. 2-microlocal analysis 45 1.3.6. An example: analysis of stock market price 46 1.4. Multifractal analysis 48 1.4.1. What is the purpose of multifractal analysis? 48 1.4.2. First ingredient: local regularity measures 49 1.4.3. Second ingredient: the size of point sets of the same regularity 50 1.4.4. Practical calculation of spectra 52 1.4.5. Refinements: analysis of the sequence of capacities, mutual analysis and multisingularity 60 1.4.6. The multifractal spectra of certain simple signals 62 1.4.7.Two applications 66 1.5.Bibliography 68 Chapter 2. Scale Invariance and Wavelets 71 Patrick FLANDRIN, Paulo GONCALVES and Patrice ABRY 2.1. Introduction 71 2.2. Models for scale invariance 72 2.2.1. Intuition 72 2.2.2. Self-similarity 73 2.2.3. Long-range dependence 75 2.2.4. Local regularity 76 2.2.5. Fractional Brownian motion: paradigm of scale invariance 77 2.2.6. Beyond the paradigm of scale invariance 79 2.3.Wavelet transform 81 2.3.1. Continuous wavelet transform 81 2.3.2.Discretewavelet transform 82 2.4. Wavelet analysis of scale invariant processes 85 2.4.1. Self-similarity 86 2.4.2. Long-range dependence 88 2.4.3. Local regularity 90 2.4.4. Beyond second order 92 2.5. Implementation: analysis, detection and estimation 92 2.5.1. Estimation of the parameters of scale invariance 93 2.5.2. Emphasis on scaling laws and determination of the scaling range 96 2.5.3. Robustness of the wavelet approach 98 2.6. Conclusion 100 2.7.Bibliography 101 Chapter 3.Wavelet Methods for Multifractal Analysis of Functions 103 Stephane JAFFARD 3.1. Introduction 103 3.2. General points regarding multifractal functions 104 3.2.1. Important definitions 104 3.2.2. Wavelets and pointwise regularity 107 3.2.3. Local oscillations 112 3.2.4. Complements 116 3.3. Random multifractal processes 117 3.3.1. Levy processes 117 3.3.2. Burgers’ equation and Brownian motion 120 3.3.3. Random wavelet series 122 3.4. Multifractal formalisms 123 3.4.1. Besov spaces and lacunarity 123 3.4.2. Construction of formalisms 126 3.5. Bounds of the spectrum 129 3.5.1. Bounds according to the Besov domain 129 3.5.2. Bounds deduced from histograms 132 3.6. The grand-canonical multifractal formalism 132 3.7.Bibliography 134 Chapter 4. Multifractal Scaling: General Theory and Approach by Wavelets 139 Rudolf RIEDI 4.1. Introduction and summary 139 4.2. Singularity exponents 140 4.2.1.Holder continuity 140 4.2.2. Scaling of wavelet coefficients 142 4.2.3. Other scaling exponents 144 4.3. Multifractal analysis 145 4.3.1. Dimension based spectra 145 4.3.2. Grain based spectra 146 4.3.3. Partition function and Legendre spectrum 147 4.3.4. Deterministic envelopes 149 4.4. Multifractal formalism 151 4.5. Binomial multifractals 154 4.5.1.Construction 154 4.5.2. Wavelet decomposition 157 4.5.3. Multifractal analysis of the binomial measure 158 4.5.4. Examples 160 4.5.5. Beyond dyadic structure 162 4.6. Wavelet based analysis 163 4.6.1. The binomial revisited with wavelets 163 4.6.2. Multifractal properties of the derivative 165 4.7. Self-similarity and LRD 167 4.8. Multifractal processes 168 4.8.1.Construction and simulation 169 4.8.2. Global analysis 170 4.8.3. Local analysis of warped FBM 170 4.8.4.LRDand estimation ofwarped FBM 173 4.9.Bibliography 173 Chapter 5. Self-similar Processes 179 Albert BENASSI and Jacques ISTAS 5.1. Introduction 179 5.1.1.Motivations 179 5.1.2. Scalings 182 5.1.3. Distributions of scale invariant masses 184 5.1.4. Weierstrass functions 185 5.1.5. Renormalization of sums of random variables 186 5.1.6. A common structure for a stochastic (semi-)self-similar process 187 5.1.7. Identifying Weierstrass functions 188 5.2. The Gaussian case 189 5.2.1. Self-similar Gaussian processes with r-stationary increments 189 5.2.2. Elliptic processes 190 5.2.3. Hyperbolic processes 191 5.2.4. Parabolic processes 192 5.2.5. Wavelet decomposition 192 5.2.6. Renormalization of sums of correlated random variable 193 5.2.7. Convergence towards fractional Brownian motion 193 5.3. Non-Gaussian case 195 5.3.1. Introduction 195 5.3.2. Symmetric α-stable processes 196 5.3.3. Censov and Takenaka processes 198 5.3.4. Wavelet decomposition 198 5.3.5. Process subordinated to Brownian measure 199 5.4. Regularity and long-range dependence 200 5.4.1. Introduction 200 5.4.2. Two examples 201 5.5.Bibliography 202 Chapter 6. Locally Self-similar Fields 205 Serge COHEN 6.1. Introduction 205 6.2. Recap of two representations of fractional Brownian motion 207 6.2.1. Reproducing kernel Hilbert space 207 6.2.2. Harmonizable representation 208 6.3. Two examples of locally self-similar fields 213 6.3.1. Definition of the local asymptotic self-similarity (LASS) 213 6.3.2. Filtered white noise (FWN) 214 6.3.3. Elliptic Gaussian random fields (EGRP) 215 6.4. Multifractional fields and trajectorial regularity 218 6.4.1.Two representations of theMBM 219 6.4.2. Study of the regularity of the trajectories of the MBM 221 6.4.3. Towards more irregularities: generalized multifractional Brownian motion (GMBM) and step fractional Brownian motion (SFBM) 222 6.5. Estimate of regularity 226 6.5.1. General method: generalized quadratic variation 226 6.5.2. Application to the examples 228 6.6.Bibliography 235 Chapter 7. An Introduction to Fractional Calculus 237 Denis MATIGNON 7.1. Introduction 237 7.1.1.Motivations 237 7.1.2. Problems 238 7.1.3. Outline 239 7.2. Definitions 240 7.2.1. Fractional integration 240 7.2.2. Fractional derivatives within the framework of causal distributions 242 7.2.3. Mild fractional derivatives, in the Caputo sense 246 7.3. Fractional differential equations 251 7.3.1. Example 251 7.3.2. Framework of causal distributions 254 7.3.3. Framework of functions expandable into fractional power series (α-FPSE) 255 7.3.4. Asymptotic behavior of fundamental solutions 257 7.3.5. Controlled-and-observed linear dynamic systems of fractional order 261 7.4. Diffusive structure of fractional differential systems 262 7.4.1. Introduction to diffusive representations of pseudo-differential operators 263 7.4.2. General decomposition result 264 7.4.3. Connection with the concept of long memory 265 7.4.4. Particular case of fractional differential systems of commensurate orders 265 7.5. Example of a fractional partial differential equation 266 7.5.1. Physical problem considered 267 7.5.2. Spectral consequences 268 7.5.3. Time-domain consequences 268 7.5.4. Free problem 272 7.6. Conclusion 273 7.7.Bibliography 273 Chapter 8. Fractional Synthesis, Fractional Filters 279 Liliane BEL, Georges OPPENHEIM, Luc ROBBIANO and Marie-Claude VIANO 8.1. Traditional and less traditional questions about fractionals 279 8.1.1.Notes on terminology 279 8.1.2. Short and long memory 279 8.1.3. From integer to non-integer powers: filter based sample path design 280 8.1.4. Local and global properties 281 8.2. Fractional filters 282 8.2.1. Desired general properties: association 282 8.2.2. Construction and approximation techniques 282 8.3. Discrete time fractional processes 284 8.3.1. Filters: impulse responses and corresponding processes 284 8.3.2. Mixing and memory properties 286 8.3.3. Parameter estimation 287 8.3.4. Simulated example 289 8.4. Continuous time fractional processes 291 8.4.1. A non-self-similar family: fractional processes designed from fractional filters 291 8.4.2. Sample path properties: local and global regularity, memory 293 8.5. Distribution processes 294 8.5.1. Motivation and generalization of distribution processes 294 8.5.2. The family of linear distribution processes 294 8.5.3. Fractional distribution processes 295 8.5.4. Mixing and memory properties 296 8.6.Bibliography 297 Chapter 9. Iterated Function Systems and Some Generalizations: Local Regularity Analysis and Multifractal Modeling of Signals 301 Khalid DAOUDI 9.1. Introduction 301 9.2. Definition of the Holder exponent 303 9.3. Iterated function systems (IFS) 304 9.4. Generalization of iterated function systems 306 9.4.1. Semi-generalized iterated function systems 307 9.4.2. Generalized iterated function systems 308 9.5. Estimation of pointwise Holder exponent by GIFS 311 9.5.1. Principles of themethod 312 9.5.2. Algorithm 314 9.5.3.Application 315 9.6. Weak self-similar functions and multifractal formalism 318 9.7. Signal representation by WSA functions 320 9.8. Segmentation of signals by weak self-similar functions 324 9.9. Estimation of the multifractal spectrum 326 9.10. Experiments 327 9.11.Bibliography 329 Chapter 10. Iterated Function Systems and Applications in Image Processing 333 Franck DAVOINE and Jean-Marc CHASSERY 10.1. Introduction 333 10.2. Iterated transformation systems 333 10.2.1. Contracting transformations and iterated transformation systems 334 10.2.2.Attractor of an iterated transformation system 335 10.2.3. Collage theorem 336 10.2.4. Finally contracting transformation 338 10.2.5. Attractor and invariant measures 339 10.2.6. Inverse problem 340 10.3. Application to natural image processing: image coding 340 10.3.1. Introduction 340 10.3.2. Coding of natural images by fractals 342 10.3.3. Algebraic formulation of the fractal transformation 345 10.3.4. Experimentation on triangular partitions 351 10.3.5. Coding and decoding acceleration 352 10.3.6. Other optimization diagrams: hybrid methods 360 10.4.Bibliography 362 Chapter 11. Local Regularity and Multifractal Methods for Image and Signal Analysis 367 Pierrick LEGRAND 11.1. Introduction 367 11.2.Basic tools 368 11.2.1. Holder regularity analysis 368 11.2.2. Reminders on multifractal analysis 369 11.3. Holderian regularity estimation 371 11.3.1. Oscillations (OSC) 371 11.3.2. Wavelet coefficient regression (WCR) 372 11.3.3. Wavelet leaders regression (WL) 372 11.3.4.Limit inf and limit sup regressions 373 11.3.5. Numerical experiments 374 11.4. Denoising 376 11.4.1. Introduction 376 11.4.2. Minimax risk, optimal convergence rate and adaptivity 377 11.4.3. Wavelet based denoising 378 11.4.4. Non-linear wavelet coefficients pumping 380 11.4.5. Denoising using exponent between scales 383 11.4.6. Bayesian multifractal denoising 386 11.5. Holderian regularity based interpolation 393 11.5.1. Introduction 393 11.5.2.Themethod 393 11.5.3. Regularity and asymptotic properties 394 11.5.4. Numerical experiments 394 11.6. Biomedical signal analysis 394 11.7. Texture segmentation 401 11.8. Edge detection 403 11.8.1. Introduction 403 11.8.1.1. Edge detection 406 11.9. Change detection in image sequences using multifractal analysis 407 11.10. Image reconstruction 408 11.11.Bibliography 409 Chapter 12. Scale Invariance in Computer Network Traffic 413 Darryl VEITCH 12.1. Teletraffic – a new natural phenomenon 413 12.1.1. A phenomenon of scales 413 12.1.2. An experimental science of “man-made atoms” 415 12.1.3. A random current 416 12.1.4. Two fundamental approaches 417 12.2. From a wealth of scales arise scaling laws 419 12.2.1. First discoveries 419 12.2.2.Laws reign 420 12.2.3. Beyond the revolution 424 12.3. Sources as the source of the laws 426 12.3.1.The sumor its parts 426 12.3.2.The on/off paradigm 427 12.3.3. Chemistry 428 12.3.4. Mechanisms 429 12.4. New models, new behaviors 430 12.4.1. Character of a model 430 12.4.2. The fractional Brownian motion family 431 12.4.3. Greedy sources 432 12.4.4. Never-ending calls 432 12.5. Perspectives 433 12.6.Bibliography 434 Chapter 13. Research of Scaling Law on Stock Market Variations 437 Christian WALTER 13.1. Introduction: fractals in finance 437 13.2. Presence of scales in the study of stock market variations 439 13.2.1. Modeling of stock market variations 439 13.2.2. Time scales in financial modeling 445 13.3. Modeling postulating independence on stock market returns 446 13.3.1. 1960-1970: from Pareto’s law to Levy’s distributions 446 13.3.2. 1970–1990: experimental difficulties of iid-α-stable model 448 13.3.3. Unstable iid models in partial scaling invariance 452 13.4. Research of dependency and memory of markets 454 13.4.1. Linear dependence: testing of H-correlative models on returns 454 13.4.2. Non-linear dependence: validating H-correlative model on volatilities 456 13.5. Towards a rediscovery of scaling laws in finance 457 13.6.Bibliography 458 Chapter 14. Scale Relativity, Non-differentiability and Fractal Space-time 465 Laurent NOTTALE 14.1. Introduction 465 14.2. Abandonment of the hypothesis of space-time differentiability 466 14.3. Towards a fractal space-time 466 14.3.1. Explicit dependence of coordinates on spatio-temporal resolutions 467 14.3.2. From continuity and non-differentiability to fractality 467 14.3.3. Description of non-differentiable process by differential equations 469 14.3.4. Differential dilation operator 471 14.4. Relativity and scale covariance 472 14.5. Scale differential equations 472 14.5.1. Constant fractal dimension: “Galilean” scale relativity 473 14.5.2. Breaking scale invariance: transition scales 474 14.5.3. Non-linear scale laws: second order equations, discrete scale invariance, log-periodic laws 475 14.5.4. Variable fractal dimension: Euler-Lagrange scale equations 476 14.5.5. Scale dynamics and scale force 478 14.5.6. Special scale relativity – log-Lorentzian dilation laws, invariant scale limit under dilations 481 14.5.7. Generalized scale relativity and scale-motion coupling 482 14.6. Quantum-like induced dynamics 488 14.6.1. Generalized Schrodinger equation 488 14.6.2. Application in gravitational structure formation 492 14.7. Conclusion 493 14.8.Bibliography 495 List of Authors 499 Index 503

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Book SynopsisOptimal filtering applied to stationary and non-stationary signals provides the most efficient means of dealing with problems arising from the extraction of noise signals. Moreover, it is a fundamental feature in a range of applications, such as in navigation in aerospace and aeronautics, filter processing in the telecommunications industry, etc. This book provides a comprehensive overview of this area, discussing random and Gaussian vectors, outlining the results necessary for the creation of Wiener and adaptive filters used for stationary signals, as well as examining Kalman filters which are used in relation to non-stationary signals. Exercises with solutions feature in each chapter to demonstrate the practical application of these ideas using MATLAB.Table of ContentsPreface ix Introduction xi Chapter 1. Random Vectors 1 1.1. Definitions and general properties. 1 1.2. Spaces L1 (dP) and L2 (dP) 20 1.3. Mathematical expectation and applications 23 1.4. Second order random variables and vectors. 39 1.5. Linear independence of vectors of L2 (dP) 46 1.6. Conditional expectation (concerning random vectors with density function) 51 1.7. Exercises for Chapter 1 56 Chapter 2. Gaussian Vectors 63 2.1. Some reminders regarding random Gaussian vectors 63 2.2. Definition and characterization of Gaussian vectors 66 2.3. Results relative to independence 68 2.4. Affine transformation of a Gaussian vector 72 2.5. The existence of Gaussian vectors. 74 2.6. Exercises for Chapter 2 84 Chapter 3. Introduction to Discrete Time Processes 93 3.1. Definition 93 3.2. WSS processes and spectral measure 105 3.3. Spectral representation of a WSS process 109 3.4. Introduction to digital filtering 114 3.5. Important example: autoregressive process 127 3.6. Exercises for Chapter 3 132 Chapter 4. Estimation 139 4.1. Position of the problem 139 4.2. Linear estimation 142 4.3. Best estimate – conditional expectation 154 4.4. Example: prediction of an autoregressive process AR (1) 162 4.5. Multivariate processes 163 4.6. Exercises for Chapter 4 172 Chapter 5. The Wiener Filter 177 5.1. Introduction 177 5.2. Resolution and calculation of the FIR filter 179 5.3. Evaluation of the least error 181 5.4. Resolution and calculation of the IIR filter 183 5.5. Evaluation of least mean square error 187 5.6. Exercises for Chapter 5 188 Chapter 6. Adaptive Filtering: Algorithm of the Gradient and the LMS 195 6.1. Introduction 195 6.2. Position of problem 198 6.3. Data representation 200 6.4. Minimization of the cost function 202 6.5. Gradient algorithm 209 6.6. Geometric interpretation 212 6.7. Stability and convergence 216 6.8. Estimation of gradient and LMS algorithm 221 6.9. Example of the application of the LMS algorithm 224 6.10. Exercises for Chapter 6 233 Chapter 7. The Kalman Filter 235 7.1. Position of problem 235 7.2. Approach to estimation 239 7.3. Kalman filtering 243 7.4. Exercises for Chapter 7 261 7.5. Appendices 267 7.6. Examples treated using Matlab software 273 Table of Symbols and Notations 281 Bibliography 283 Index 285

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Book SynopsisMultiresolution analysis using the wavelet transform has received considerable attention in recent years by researchers in various fields. It is a powerful tool for efficiently representing signals and images at multiple levels of detail with many inherent advantages, including compression, level-of-detail display, progressive transmission, level-of-detail editing, filtering, modeling, fractals and multifractals, etc. This book aims to provide a simple formalization and new clarity on multiresolution analysis, rendering accessible obscure techniques, and merging, unifying or completing the technique with encoding, feature extraction, compressive sensing, multifractal analysis and texture analysis. It is aimed at industrial engineers, medical researchers, university lab attendants, lecturer-researchers and researchers from various specializations. It is also intended to contribute to the studies of graduate students in engineering, particularly in the fields of medical imaging, intelligent instrumentation, telecommunications, and signal and image processing. Given the diversity of the problems posed and addressed, this book paves the way for the development of new research themes, such as brain–computer interface (BCI), compressive sensing, functional magnetic resonance imaging (fMRI), tissue characterization (bones, skin, etc.) and the analysis of complex phenomena in general. Throughout the chapters, informative illustrations assist the uninitiated reader in better conceptualizing certain concepts, taking the form of numerous figures and recent applications in biomedical engineering, communication, multimedia, finance, etc.Table of ContentsIntroduction xi Chapter 1. Introduction to Multiresolution Analysis 1 1.1. Introduction 1 1.2. Wavelet transforms: an introductory review 3 1.2.1. Brief history 3 1.2.2. Continuous wavelet transforms 6 1.2.2.1. Wavelet transform modulus maxima 9 1.2.2.2. Reconstruction 13 1.2.3. Discrete wavelet transforms 14 1.3. Multiresolution 16 1.3.1. Multiresolution analysis and wavelet bases 17 1.3.1.1. Approximation spaces 17 1.3.1.2. Detail spaces 19 1.3.2. Multiresolution analysis: points to remember 21 1.3.3. Decomposition and reconstruction 22 1.3.3.1. Calculation of coefficients 22 1.3.3.2. Implementation of MRA: Mallat algorithm 24 1.3.3.3. Extension to images 26 1.3.4. Wavelet packets 28 1.3.5. Multiresolution analysis summarized 30 1.4. Which wavelets to choose? 33 1.4.1. Number of vanishing moments, regularity, support (compactness), symmetry, etc 33 1.4.2. Well-known wavelets, scale functions and associated filters 34 1.4.2.1. Haar wavelet 34 vi Signal and Image Multiresolution Analysis 1.4.2.2. Daubechies wavelets 36 1.4.2.3. Symlets 38 1.4.2.4. Coiflets 39 1.4.2.5. Meyer wavelets 41 1.4.2.6. Polynomial spline wavelets 43 1.5. Multiresolution analysis and biorthogonal wavelet bases 48 1.5.1. Why biorthogonal bases? 48 1.5.2. Multiresolution context 48 1.5.3. Example of biorthogonal wavelets, scaling functions and associated filters 49 1.5.4. The concept of wavelet lifting 51 1.5.4.1. The notion of lifting 51 1.5.4.2. Significance of structure lifting 52 1.6. Wavelet choice at a glance 54 1.6.1. Regularity 54 1.6.2. Vanishing moments 54 1.6.3. Other criteria 55 1.6.4. Conclusion 55 1.7. Worked examples 55 1.7.1. Examples of multiresolution analysis 55 1.7.2. Compression 58 1.7.3. Denoising (reduction of noise) 64 1.8. Some applications 74 1.8.1. Discovery and contributions of wavelets 74 1.8.2. Biomedical engineering 76 1.8.2.1. ECG, EEG and BCI 77 1.8.2.2. Medical imaging 97 1.8.3. Telecommunications 110 1.8.3.1. Adaptive compression for sensor networks 110 1.8.3.2. Masking image encoding and transmission errors 114 1.8.3.3. Suppression of correlated noise 118 1.8.4. “Compressive sensing”, ICA, PCA and MRA 119 1.8.4.1. Principal component analysis 120 1.8.4.2. Independent component analysis 121 1.8.4.3. Compressive sensing 122 1.8.5. Conclusion 128 1.9. Bibliography 129 Chapter 2. Discrete Wavelet Transform-Based Multifractal Analysis 135 2.1. Introduction 135 2.1.1. Fractals and wavelets: a happy marriage? 135 2.1.2. Background 136 2.1.3. Mono/multifractal processes 137 2.1.4. Chapter outline 138 2.2. Fractality, variability and complexity 139 2.2.1. System complexity 139 2.2.2. Complex phenomena properties 141 2.2.2.1. Tendency of autonomous agents to self-organize 141 2.2.2.2. Variability and adaptability 142 2.2.2.3. Bifurcation concept and chaotic model 143 2.2.2.4. Hierarchy and scale invariance 146 2.2.2.5. Self-organized critical phenomena 146 2.2.2.6. Highly optimized tolerance 147 2.2.3. Fractality 148 2.3. Multifractal analysis 150 2.3.1. Point-wise regularity 150 2.3.2. Hölder exponent 150 2.3.3. Signal classification according to the regularity properties 152 2.3.3.1. Monofractal signal 152 2.3.3.2. Multifractal signal 152 2.3.4. Hausdorff dimension 154 2.3.4.1. Theoretic approach 155 2.3.4.2. Qualitative approach and multifractal spectrum 155 2.4. Multifractal formalism 156 2.4.1. Reminder on wavelet decomposition 156 2.4.2. Point-wise regularity characterization 157 2.4.3. Structure function and power law behavior 158 2.4.4. Link between scaling exponents and singularity spectrum 159 2.4.5. Use of wavelet leaders 160 2.4.5.1. Indexing a dyadic square and wavelet leaders 161 2.4.5.2. Polynomial expansion and log-cumulants 162 2.4.6. Wavelet leaders variant: “maximum” coefficients 165 2.5. Algorithm and performances 165 2.5.1. Singularity spectrum estimation algorithm 165 2.5.2. Analysis of a few widely used processes 167 2.5.2.1. fBm: a monofractal process 167 2.5.2.2. CMC: a multifractal process 170 2.5.2.3. BMC: another class of multifractal processes 173 2.5.3. Estimation performances 176 2.5.3.1. fBm and CMC-LN and CMC-LP simulation 176 2.5.3.2. Results 177 2.5.3.3. Interpretation and recommendations 182 2.6. Applications 186 2.6.1. Turbulence 186 viii Signal and Image Multiresolution Analysis 2.6.1.1. From Leonardo da Vinci to Kolmogorov 186 2.6.1.2. Multifractal process in turbulence 191 2.6.2. Multifractal process in finance 194 2.6.2.1. Stock market complexity modeling 194 2.6.2.2. Market turbulence 198 2.6.3. Internet traffic 203 2.6.3.1. The Internet revolution 203 2.6.3.2. Multifractal nature of Internet traffic? 204 2.6.4. Biomedical field 205 2.6.4.1. Analyzing image texture through a multifractal approach 205 2.6.4.2. Multifractality in medical imaging 211 2.7. Conclusion 219 2.8. Bibliography 220 Chapter 3. Multimodal Compression Using JPEG 2000: Supervised Insertion Approach 225 3.1. Introduction 225 3.2. The JPEG 2000 standard 226 3.3. Multimodal compression by unsupervised insertion 227 3.3.1. Principle of insertion in the wavelet transform domain 228 3.3.2. Principle of insertion in the spatial domain 229 3.4. Multimodal compression by supervised insertion 231 3.4.1. Choice of insertion zone 232 3.4.2. Insertion and separation function 233 3.4.2.1. Insertion function 233 3.4.2.2. Separation function 235 3.5. Criteria for quality evaluation 236 3.5.1. Peak signal-to-noise ratio 236 3.5.2. Percent residual difference 237 3.6. Some preliminary results 238 3.7. Conclusion 242 3.8. Bibliography 243 Chapter 4. Cerebral Microembolism Synchronous Detection with Wavelet Packets 245 4.1. Issue and stakes 245 4.2. Prior information research 247 4.2.1. Doppler ultrasound blood flow signal 247 4.2.2. Embolic Doppler ultrasound signal 250 4.3. Doppler ultrasound blood emboli signal modeling 251 4.3.1. “Physical” model 251 4.3.2. “Signal” model 255 4.3.3. Statistical tests 258 4.3.3.1. Stationarity test 259 4.3.3.2. Cyclostationarity test 261 4.3.3.3. “Gaussian” test and cardiac cycle regularity 262 4.4. Energy detection 263 4.4.1. State-of-the-art and standard detection 263 4.4.2. Synchronous detection 266 4.5. Wavelet packet energy detection 269 4.5.1. Introduction 269 4.5.2. Multiresolution analysis 271 4.5.3. Wavelet packet subband detection 274 4.5.4. Wavelet packet synchronous detection 277 4.6. Results and discussions 279 4.6.1. In simulation 279 4.6.2. In vivo 282 4.7. Conclusion 285 4.8. Bibliography 285 List of Authors 289 Index 291

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Book SynopsisThis book deals with these parametric methods, first discussing those based on time series models, Capon’s method and its variants, and then estimators based on the notions of sub-spaces. However, the book also deals with the traditional “analog” methods, now called non-parametric methods, which are still the most widely used in practical spectral analysis.Table of ContentsPreface 9 Specific Notations 13 PART I. Tools and Spectral Analysis 15 Chapter 1. Fundamentals 17 Francis CASTANIÉ 1.1. Classes of signals 17 1.1.1. Deterministic signals 17 1.1.2. Random signals 20 1.2. Representations of signals 23 1.2.1. Representations of deterministic signals 23 1.2.1.1. Complete representations 23 1.2.1.2. Partial representations 25 1.2.2. Representations of random signals 27 1.2.2.1. General approach 27 1.2.2.2. 2nd order representations 28 1.2.2.3. Higher order representations 32 1.3. Spectral analysis: position of the problem 33 1.4. Bibliography 35 Chapter 2. Digital Signal Processing 37 Éric LE CARPENTIER 2.1. Introduction 37 2.2. Transform properties 38 2.2.1. Some useful functions and series 38 2.2.2. Fourier transform 43 2.2.3. Fundamental properties 47 2.2.4. Convolution sum 48 2.2.5. Energy conservation (Parseval’s theorem) 50 2.2.6. Other properties 51 2.2.7. Examples 53 2.2.8. Sampling 55 2.2.9. Practical calculation, FFT 59 2.3. Windows 62 2.4. Examples of application 71 2.4.1. LTI systems identification 71 2.4.2. Monitoring spectral lines 75 2.4.3. Spectral analysis of the coefficient of tide fluctuation 76 2.5. Bibliography 78 Chapter 3. Estimation in Spectral Analysis 79 Olivier BESSON and André FERRARI 3.1. Introduction to estimation 79 3.1.1. Formalization of the problem 79 3.1.2. Cramér-Rao bounds 81 3.1.3. Sequence of estimators 86 3.1.4. Maximum likelihood estimation 89 3.2. Estimation of 1st and 2nd order moments 92 3.3. Periodogram analysis 97 3.4. Analysis of estimators based on cˆxx m?n101 3.4.1. Estimation of parameters of an AR model 103 3.4.2. Estimation of a noisy cisoid by MUSIC 106 3.5. Conclusion 108 3.6. Bibliography 108 Chapter 4. Time-Series Models 111 Francis CASTANIÉ 4.1. Introduction 111 4.2. Linear models 113 4.2.1. Stationary linear models 113 4.2.2. Properties 116 4.2.2.1. Stationarity 116 4.2.2.2. Moments and spectra 117 4.2.2.3. Relation with Wold’s decomposition 119 4.2.3. Non-stationary linear models 120 4.3. Exponential models 123 4.3.1. Deterministic model 123 4.3.2. Noisy deterministic model 124 4.3.3. Models of random stationary signals 125 4.4. Non-linear models 126 4.5. Bibliography 126 PART II. Non-Parametric Methods 129 Chapter 5. Non-Parametric Methods 131 Éric LE CARPENTIER 5.1. Introduction 131 5.2. Estimation of the power spectral density 136 5.2.1. Filter bank method 136 5.2.2. Periodogram method 139 5.2.3. Periodogram variants 142 5.3. Generalization to higher order spectra 146 5.4. Bibliography 148 PART III. Parametric Methods 149 Chapter 6. Spectral Analysis by Stationary Time Series Modeling 151 Corinne MAILHES and Francis CASTANIÉ 6.1. Parametric models 151 6.2. Estimation of model parameters 153 6.2.1. Estimation of AR parameters 153 6.2.2. Estimation of ARMA parameters 160 6.2.3. Estimation of Prony parameters 161 6.2.4. Order selection criteria 164 6.3. Properties of spectral estimators produced 167 6.4. Bibliography 172 Chapter 7. Minimum Variance 175 Nadine MARTIN 7.1. Principle of the MV method 179 7.2. Properties of the MV estimator 182 7.2.1. Expressions of the MV filter 182 7.2.2. Probability density of the MV estimator 186 7.2.3. Frequency resolution of the MV estimator 192 7.3. Link with the Fourier estimators 193 7.4. Link with a maximum likelihood estimator 196 7.5. Lagunas methods: normalized and generalized MV 198 7.5.1. Principle of normalized MV 198 7.5.2. Spectral refinement of the NMV estimator 200 7.5.3. Convergence of the NMV estimator 202 7.5.4. Generalized MV estimator 204 7.6. The CAPNORM estimator 206 7.7. Bibliography 209 Chapter 8. Subspace-based Estimators 213 Sylvie MARCOS 8.1. Model, concept of subspace, definition of high resolution 213 8.1.1. Model of signals 213 8.1.2. Concept of subspaces 214 8.1.3. Definition of high-resolution 216 8.1.4. Link with spatial analysis or array processing 217 8.2. MUSIC 217 8.2.1. Pseudo-spectral version of MUSIC 220 8.2.2. Polynomial version of MUSIC 221 8.3. Determination criteria of the number of complex sine waves 223 8.4. The MinNorm method 224 8.5. “Linear” subspace methods 226 8.5.1. The linear methods 226 8.5.2. The propagator method 226 8.5.2.1. Propagator estimation using least squares technique 228 8.5.2.2. Determination of the propagator in the presence of a white noise 229 8.6. The ESPRIT method 232 8.7. Illustration of subspace-based methods performance 235 8.8. Adaptive research of subspaces 236 8.9. Bibliography 242 Chapter 9. Introduction to Spectral Analysis of Non-Stationary Random Signals 245 Corinne MAILHES and Francis CASTANIÉ 9.1. Evolutive spectra 246 9.1.1. Definition of the “evolutive spectrum” 246 9.1.2. Evolutive spectrum properties 247 9.2. Non-parametric spectral estimation 248 9.3. Parametric spectral estimation 249 9.3.1. Local stationary postulate 250 9.3.2. Elimination of a stationary condition 251 9.3.3. Application to spectral analysis 254 9.4. Bibliography 255 List of Authors 259 Index 261

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Book SynopsisThe last 15 years have seen an explosion of interest in wavelets with applications in fields such as image compression, turbulence, human vision, radar and earthquake prediction. Wavelets represent an area that combines signal in image processing, mathematics, physics and electrical engineering. As such, this title is intended for the wide audience that is interested in mastering the basic techniques in this subject area, such as decomposition and compression.Table of ContentsNotations xiii Introduction xvii Chapter 1. A Guided Tour 1 1.1. Introduction 1 1.2. Wavelets 2 1.2.1. General aspects 2 1.2.2. A wavelet 6 1.2.3. Organization of wavelets 8 1.2.4. The wavelet tree for a signal 10 1.3. An electrical consumption signal analyzed by wavelets 12 1.4. Denoising by wavelets: before and afterwards 14 1.5. A Doppler signal analyzed by wavelets 16 1.6. A Doppler signal denoised by wavelets 17 1.7. An electrical signal denoised by wavelets 19 1.8. An image decomposed by wavelets 21 1.8.1. Decomposition in tree form 21 1.8.2. Decomposition in compact form 22 1.9. An image compressed by wavelets 24 1.10. A signal compressed by wavelets 25 1.11. A fingerprint compressed using wavelet packets 27 Chapter 2. Mathematical Framework 29 2.1. Introduction 29 2.2. From the Fourier transform to the Gabor transform 30 2.2.1. Continuous Fourier transform 30 2.2.2. The Gabor transform 35 2.3. The continuous transform in wavelets 37 2.4. Orthonormal wavelet bases 41 2.4.1. From continuous to discrete transform 41 2.4.2. Multi-resolution analysis and orthonormal wavelet bases 42 2.4.3. The scaling function and the wavelet 46 2.5. Wavelet packets 50 2.5.1. Construction of wavelet packets 50 2.5.2. Atoms of wavelet packets 52 2.5.3. Organization of wavelet packets 53 2.6. Biorthogonal wavelet bases 55 2.6.1. Orthogonality and biorthogonality 55 2.6.2. The duality raises several questions 56 2.6.3. Properties of biorthogonal wavelets 57 2.6.4. Semi-orthogonal wavelets 60 Chapter 3. From Wavelet Bases to the Fast Algorithm 63 3.1. Introduction. 63 3.2. From orthonormal bases to the Mallat algorithm 64 3.3. Four filters 65 3.4. Efficient calculation of the coefficients 67 3.5. Justification: projections and twin scales 68 3.5.1. The decomposition phase 69 3.5.2. The reconstruction phase 72 3.5.3. Decompositions and reconstructions of a higher order 75 3.6. Implementation of the algorithm 75 3.6.1. Initialization of the algorithm 76 3.6.2. Calculation on finite sequences 77 3.6.3. Extra coefficients 77 3.7. Complexity of the algorithm 78 3.8. From 1D to 2D 79 3.9. Translation invariant transform 81 3.9.1. e-decimated DWT 83 3.9.2. Calculation of the SWT 83 3.9.3. Inverse SWT 87 Chapter 4. Wavelet Families 89 4.1. Introduction 89 4.2. What could we want from a wavelet? 90 4.3. Synoptic table of the common families 91 4.4. Some well known families 92 4.4.1. Orthogonal wavelets with compact support 93 4.4.2. Biorthogonal wavelets with compact support: bior 99 4.4.3. Orthogonal wavelets with non-compact support 101 4.4.4. Real wavelets without filters 104 4.4.5. Complex wavelets without filters 106 4.5. Cascade algorithm 109 4.5.1. The algorithm and its justification 110 4.5.2. An application 112 4.5.3. Quality of the approximation 113 Chapter 5. Finding and Designing a Wavelet 115 5.1. Introduction 115 5.2. Construction of wavelets for continuous analysis 116 5.2.1. Construction of a new wavelet 116 5.2.2. Application to pattern detection 124 5.3. Construction of wavelets for discrete analysis 131 5.3.1. Filter banks 132 5.3.2. Lifting 140 5.3.3. Lifting and biorthogonal wavelets 146 5.3.4. Construction examples 149 Chapter 6. A Short 1D Illustrated Handbook 159 6.1. Introduction 159 6.2. Discrete 1D illustrated handbook 160 6.2.1. The analyzed signals 160 6.2.2. Processing carried out 161 6.2.3. Commented examples 162 6.3. The contribution of analysis by wavelet packets 178 6.3.1. Example 1: linear and quadratic chirp 178 6.3.2. Example 2: a sine181 6.3.3. Example 3: a composite signal 182 6.4. “Continuous” 1D illustrated handbook 183 6.4.1. Time resolution 183 6.4.2. Regularity analysis 187 6.4.3. Analysis of a self-similar signal 193 Chapter 7. Signal Denoising and Compression 197 7.1. Introduction 197 7.2. Principle of denoising by wavelets 198 7.2.1. The model 198 7.2.2. Denoising: before and after 198 7.2.3. The algorithm 199 7.2.4. Why does it work? 200 7.3. Wavelets and statistics 200 7.3.1. Kernel estimators and estimators by orthogonal projection 201 7.3.2. Estimators by wavelets 201 7.4. Denoising methods 202 7.4.1. A first estimator 203 7.4.2. From coefficient selection to thresholding coefficients 204 7.4.3. Universal thresholding 206 7.4.4. Estimating the noise standard deviation 206 7.4.5. Minimax risk 207 7.4.6. Further information on thresholding rules 208 7.5. Example of denoising with stationary noise 209 7.6. Example of denoising with non-stationary noise 212 7.6.1. The model with ruptures of variance 213 7.6.2. Thresholding adapted to the noise level change-points 214 7.7. Example of denoising of a real signal 216 7.7.1. Noise unknown but “homogenous” in variance by level 216 7.7.2. Noise unknown and “non-homogenous” in variance by level 217 7.8. Contribution of the translation invariant transform 218 7.9. Density and regression estimation 221 7.9.1. Density estimation 221 7.9.2. Regression estimation 224 7.10. Principle of compression by wavelets 225 7.10.1. The problem 225 7.10.2. The basic algorithm 225 7.10.3. Why does it work? 226 7.11. Compression methods 226 7.11.1. Thresholding of the coefficients 226 7.11.2. Selection of coefficients 228 7.12. Examples of compression 229 7.12.1. Global thresholding 229 7.12.2. A comparison of the two compression strategies 230 7.13. Denoising and compression by wavelet packets 233 7.14. Bibliographical comments 234 Chapter 8. Image Processing with Wavelets 235 8.1. Introduction 235 8.2. Wavelets for the image 236 8.2.1. 2D wavelet decomposition 237 8.2.2. Approximation and detail coefficients 238 8.2.3. Approximations and details 241 8.3. Edge detection and textures 243 8.3.1. A simple geometric example 243 8.3.2. Two real life examples 245 8.4. Fusion of images 247 8.4.1. The problem through a simple example 247 8.4.2. Fusion of fuzzy images 250 8.4.3. Mixing of images 252 8.5. Denoising of images 256 8.5.1. An artificially noisy image 257 8.5.2. A real image 260 8.6. Image compression 262 8.6.1. Principles of compression 262 8.6.2. Compression and wavelets 263 8.6.3. “True” compression 269 Chapter 9. An Overview of Applications 279 9.1. Introduction 279 9.1.1. Why does it work? 279 9.1.2. A classification of the applications 281 9.1.3. Two problems in which the wavelets are competitive 283 9.1.4. Presentation of applications 283 9.2. Wind gusts 285 9.3. Detection of seismic jolts 287 9.4. Bathymetric study of the marine floor 290 9.5. Turbulence analysis 291 9.6. Electrocardiogram (ECG): coding and moment of the maximum 294 9.7. Eating behavior 295 9.8. Fractional wavelets and fMRI 297 9.9. Wavelets and biomedical sciences 298 9.9.1. Analysis of 1D biomedical signals 300 9.9.2. 2D biomedical signal analysis 301 9.10. Statistical process control 302 9.11. Online compression of industrial information 304 9.12. Transitories in underwater signals 306 9.13. Some applications at random 308 9.13.1. Video coding 308 9.13.2. Computer-assisted tomography 309 9.13.3. Producing and analyzing irregular signals or images 309 9.13.4. Forecasting 310 9.13.5. Interpolation by kriging 310 Appendix. The EZW Algorithm 313 A.1. Coding 313 A.1.1. Detailed description of the EZW algorithm (coding phase) 313 A.1.2. Example of application of the EZW algorithm (coding phase) 314 A.2. Decoding 317 A.2.1. Detailed description of the EZW algorithm (decoding phase) 317 A.2.2. Example of application of the EZW algorithm (decoding phase) 318 A.3. Visualization on a real image of the algorithm’s decoding phase 318 Bibliography 321 Index 329

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Book SynopsisDealing with digital filtering methods for 1-D and 2-D signals, this book provides the theoretical background in signal processing, covering topics such as the z-transform, Shannon sampling theorem and fast Fourier transform. An entire chapter is devoted to the design of time-continuous filters which provides a useful preliminary step for analog-to-digital filter conversion. Attention is also given to the main methods of designing finite impulse response (FIR) and infinite impulse response (IIR) filters. Bi-dimensional digital filtering (image filtering) is investigated and a study on stability analysis, a very useful tool when implementing IIR filters, is also carried out. As such, it will provide a practical and useful guide to those engaged in signal processing.Table of ContentsIntroduction xiii Chapter 1. Introduction to Signals and Systems 1 Yannick BERTHOUMIEU, Eric GRIVEL and Mohamed NAJIM 1.1. Introduction 1 1.2. Signals: categories, representations and characterizations 1 1.2.1. Definition of continuous-time and discrete-time signals 1 1.2.2. Deterministic and random signals 6 1.2.3. Periodic signals 8 1.2.4. Mean, energy and power 9 1.2.5. Autocorrelation function 12 1.3. Systems 15 1.4. Properties of discrete-time systems 16 1.4.1. Invariant linear systems 16 1.4.2. Impulse responses and convolution products 16 1.4.3. Causality 17 1.4.4. Interconnections of discrete-time systems 18 1.5. Bibliography 19 Chapter 2. Discrete System Analysis 21 Mohamed NAJIM and Eric GRIVEL 2.1. Introduction 21 2.2. The z-transform 21 2.2.1. Representations and summaries 21 2.2.2. Properties of the z-transform 28 2.2.2.1. Linearity 28 2.2.2.2. Advanced and delayed operators 29 2.2.2.3. Convolution 30 2.2.2.4. Changing the z-scale 31 2.2.2.5. Contrasted signal development 31 2.2.2.6. Derivation of the z-transform 31 2.2.2.7. The sum theorem 32 2.2.2.8. The final-value theorem 32 2.2.2.9. Complex conjugation 32 2.2.2.10. Parseval’s theorem 33 2.2.3. Table of standard transform 33 2.3. The inverse z-transform 34 2.3.1. Introduction 34 2.3.2. Methods of determining inverse z-transforms 35 2.3.2.1. Cauchy’s theorem: a case of complex variables 35 2.3.2.2. Development in rational fractions 37 2.3.2.3. Development by algebraic division of polynomials 38 2.4. Transfer functions and difference equations 39 2.4.1. The transfer function of a continuous system 39 2.4.2. Transfer functions of discrete systems 41 2.5. Z-transforms of the autocorrelation and intercorrelation functions 44 2.6. Stability 45 2.6.1. Bounded input, bounded output (BIBO) stability 46 2.6.2. Regions of convergence 46 2.6.2.1. Routh’s criterion 48 2.6.2.2. Jury’s criterion 49 Chapter 3. Frequential Characterization of Signals and Filters 51 Eric GRIVEL and Yannick BERTHOUMIEU 3.1. Introduction 51 3.2. The Fourier transform of continuous signals 51 3.2.1. Summary of the Fourier series decomposition of continuous signals 51 3.2.1.1. Decomposition of finite energy signals using an orthonormal base 51 3.2.1.2. Fourier series development of periodic signals 52 3.2.2. Fourier transforms and continuous signals 57 3.2.2.1. Representations 57 3.2.2.2. Properties 58 3.2.2.3. The duality theorem 59 3.2.2.4. The quick method of calculating the Fourier transform 59 3.2.2.5. The Wiener-Khintchine theorem 63 3.2.2.6. The Fourier transform of a Dirac comb 63 3.2.2.7. Another method of calculating the Fourier series development of a periodic signal 66 3.2.2.8. The Fourier series development and the Fourier transform 68 3.2.2.9. Applying the Fourier transform: Shannon’s sampling theorem 75 3.3. The discrete Fourier transform (DFT) 78 3.3.1. Expressing the Fourier transform of a discrete sequence 78 3.3.2. Relations between the Laplace and Fourier z-transforms 80 3.3.3. The inverse Fourier transform 81 3.3.4. The discrete Fourier transform 82 3.4. The fast Fourier transform (FFT) 86 3.5. The fast Fourier transform for a time/frequency/energy representation of a non-stationary signal 90 3.6. Frequential characterization of a continuous-time system 91 3.6.1. First and second order filters 91 3.6.1.1. 1st order system 91 3.6.1.2. 2nd order system 93 3.7. Frequential characterization of discrete-time system 95 3.7.1. Amplitude and phase frequential diagrams 95 3.7.2. Application 96 Chapter 4. Continuous-Time and Analog Filters 99 Daniel BASTARD and Eric GRIVEL 4.1. Introduction 99 4.2. Different types of filters and filter specifications 99 4.3. Butterworth filters and the maximally flat approximation 104 4.3.1. Maximally flat functions (MFM) 104 4.3.2. A specific example of MFM functions: Butterworth polynomial filters 106 4.3.2.1. Amplitude-squared expression 106 4.3.2.2. Localization of poles 107 4.3.2.3. Determining the cut-off frequency at –3 dB and filter orders 110 4.3.2.4. Application 111 4.3.2.5. Realization of a Butterworth filter 112 4.4. Equiripple filters and the Chebyshev approximation 113 4.4.1. Characteristics of the Chebyshev approximation 113 4.4.2. Type I Chebyshev filters 114 4.4.2.1. The Chebyshev polynomial 114 4.4.2.2. Type I Chebyshev filters 115 4.4.2.3. Pole determination 116 4.4.2.4. Determining the cut-off frequency at –3 dB and the filter order 118 4.4.2.5. Application 121 4.4.2.6. Realization of a Chebyshev filter 121 4.4.2.7. Asymptotic behavior 122 4.4.3. Type II Chebyshev filter 123 4.4.3.1. Determining the filter order and the cut-off frequency 123 4.4.3.2. Application 124 4.5. Elliptic filters: the Cauer approximation 125 4.6. Summary of four types of low-pass filter: Butterworth, Chebyshev type I, Chebyshev type II and Cauer 125 4.7. Linear phase filters (maximally flat delay or MFD): Bessel and Thomson filters 126 4.7.1. Reminders on continuous linear phase filters 126 4.7.2. Properties of Bessel-Thomson filters 128 4.7.3. Bessel and Bessel-Thomson filters 130 4.8. Papoulis filters (optimum (On)) 132 4.8.1. General characteristics 132 4.8.2. Determining the poles of the transfer function 135 4.9. Bibliography 135 Chapter 5. Finite Impulse Response Filters 137 Yannick BERTHOUMIEU, Eric GRIVEL and Mohamed NAJIM 5.1. Introduction to finite impulse response filters 137 5.1.1. Difference equations and FIR filters 137 5.1.2. Linear phase FIR filters 142 5.1.2.1. Representation 142 5.1.2.2. Different forms of FIR linear phase filters 147 5.1.2.3. Position of zeros in FIR filters 150 5.1.3. Summary of the properties of FIR filters 152 5.2. Synthesizing FIR filters using frequential specifications 152 5.2.1. Windows 152 5.2.2. Synthesizing FIR filters using the windowing method 159 5.2.2.1. Low-pass filters 159 5.2.2.2. High-pass filters 164 5.3. Optimal approach of equal ripple in the stop-band and passband 165 5.4. Bibliography 172 Chapter 6. Infinite Impulse Response Filters 173 Eric GRIVEL and Mohamed NAJIM 6.1. Introduction to infinite impulse response filters 173 6.1.1. Examples of IIR filters 174 6.1.2. Zero-loss and all-pass filters 178 6.1.3. Minimum-phase filters180 6.1.3.1. Problem 180 6.1.3.2. Stabilizing inverse filters 181 6.2. Synthesizing IIR filters 183 6.2.1. Impulse invariance method for analog to digital filter conversion 183 6.2.2. The invariance method of the indicial response 185 6.2.3. Bilinear transformations 185 6.2.4. Frequency transformations for filter synthesis using low-pass filters 188 6.3. Bibliography 189 Chapter 7. Structures of FIR and IIR Filters 191 Mohamed NAJIM and Eric GRIVEL 7.1. Introduction 191 7.2. Structure of FIR filters 192 7.3. Structure of IIR filters 192 7.3.1. Direct structures 192 7.32. The cascade structure 209 7.3.3. Parallel structures 211 7.4. Realizing finite precision filters 211 7.4.1. Introduction 211 7.4.2. Examples of FIR filters 212 7.4.3. IIR filters 213 7.4.3.1. Introduction 213 7.4.3.2. The influence of quantification on filter stability 221 7.4.3.3. Introduction to scale factors 224 7.4.3.4. Decomposing the transfer function into first- and second-order cells 226 7.5. Bibliography 231 Chapter 8. Two-Dimensional Linear Filtering 233 Philippe BOLON 8.1. Introduction 233 8.2. Continuous models 233 8.2.1. Representation of 2-D signals 233 8.2.2. Analog filtering 235 8.3. Discrete models 236 8.3.1. 2-D sampling 236 8.3.2. The aliasing phenomenon and Shannon’s theorem 240 8.3.2.1. Reconstruction by linear filtering (Shannon’s theorem) 240 8.3.2.2. Aliasing effect 240 8.4. Filtering in the spatial domain 242 8.4.1. 2-D discrete convolution 242 8.4.2. Separable filters 244 8.4.3. Separable recursive filtering 246 8.4.4. Processing of side effects 249 8.4.4.1. Prolonging the image by pixels of null intensity 250 8.4.4.2. Prolonging by duplicating the border pixels 251 8.4.4.3. Other approaches 252 8.5. Filtering in the frequency domain 253 8.5.1. 2-D discrete Fourier transform (DFT) 253 8.5.2. The circular convolution effect 255 8.6. Bibliography 259 Chapter 9. Two-Dimensional Finite Impulse Response Filter Design 261 Yannick BERTHOUMIEU 9.1. Introduction 261 9.2. Introduction to 2-D FIR filters 262 9.3. Synthesizing with the two-dimensional windowing method 263 9.3.1. Principles of method 263 9.3.2. Theoretical 2-D frequency shape 264 9.3.2.1. Rectangular frequency shape 264 9.3.2.2. Circular shape 266 9.3.3. Digital 2-D filter design by windowing 271 9.3.4. Applying filters based on rectangular and circular shapes 271 9.3.5. 2-D Gaussian filters 274 9.3.6. 1-D and 2-D representations in a continuous space 274 9.3.6.1. 2-D specifications 276 9.3.7. Approximation for FIR filters 277 9.3.7.1. Truncation of the Gaussian profile 277 9.3.7.2. Rectangular windows and convolution 279 9.3.8. An example based on exploiting a modulated Gaussian filter 280 9.4. Appendix: spatial window functions and their implementation 286 9.5. Bibliography 291 Chapter 10. Filter Stability 293 Michel BARRET 10.1. Introduction 293 10.2. The Schur-Cohn criterion 298 10.3. Appendix: resultant of two polynomials 314 10.4. Bibliography 319 Chapter 11. The Two-Dimensional Domain 321 Michel BARRET 11.1. Recursive filters 321 11.1.1. Transfer functions 321 11.1.2. The 2-D z-transform 322 11.1.3. Stability, causality and semi-causality 324 11.2. Stability criteria 328 11.2.1. Causal filters 329 11.2.2. Semi-causal filters 332 11.3. Algorithms used in stability tests 334 11.3.1. The jury Table 334 11.3.2. Algorithms based on calculating the Bezout resultant 339 11.3.2.1. First algorithm 340 11.3.2.2. Second algorithm 343 11.3.3. Algorithms and rounding-off errors 347 11.4. Linear predictive coding 351 11.5. Appendix A: demonstration of the Schur-Cohn criterion 355 11.6. Appendix B: optimum 2-D stability criteria 358 11.7. Bibliography 362 List of Authors 365 Index 367

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