Maths for computer scientists Books

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Book SynopsisThis textbook is designed for undergraduate students taking a course on the mathematical foundations of computer science. It is written from an exclusively CS perspective rather than for a mixed-discipline audience, helping CS students see the ways that foundational mathematical material is central to the discipline of computer science.Trade Review'Finally! I've spent years struggling to find a textbook that makes the topic of Discrete Structures relevant to Computer Science students, David Liben-Nowell has put forth a book that will make CS students invested in the material. He not only connects every topic to Computer Science but does so in a clear and entertaining way.' Dan Arena, Vanderbilt University'Unlike most discrete math texts, here the computer science content and connections are woven extensively throughout, with “forward pointers” that can excite students about numerous computer science areas they will encounter in their future studies. In addition, the book is written TO students, not FOR faculty. It will be a joy to teach with!' Valerie Barr, Mount Holyoke College'By foregrounding the connections between the fields, this outstanding textbook makes a compelling case for why computer science students should embrace the study of discrete mathematics. This is an approachable yet rigorous book, written with wit and verve, that I look forward to teaching from!' Raghuram Ramanujan, Davidson College'David Liben-Nowell's Connecting Discrete Mathematics and Computer Science provides students with a beautifully motivated, clearly written, and accessible exploration of the mathematical foundations of computer science. The “Computer Science Connections” sections provide compelling applications of the mathematical content and the frequent “Taking in further” notes provide extra richness that add to the joy of the experience. This is a discrete math book that truly keeps the reader engaged!' Ran Libeskind-Hadas, Founding Chair of Integrated Sciences, Claremont McKenna College'An inspired approach to the introductory discrete math course, illuminating the aesthetic appeal of the subject together with the profound and inextricable links that connect it to the core ideas of computing.' Jon Kleinberg, Cornell UniversityTable of Contents1. On the point of this book; 2. Basic data types; 3. Logic; 4. Proofs; 5. Mathematical induction; 6. Analysis of algorithms; 7. Number theory; 8. Relations; 9. Counting; 10. Probability; 11. Graphs and trees; 12. Looking forward.

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Book SynopsisPublic key cryptography is a major interdisciplinary subject with many real-world applications. This book has been carefully written to communicate the major ideas and techniques in this subject to a wide readership. With numerous examples and exercises, it is suitable as a textbook for an advanced course or for self-study.Trade Review'… the book gathers the main mathematical topics related to public key cryptography and provides an excellent source of information for both students and researchers interested in the field.' Juan Tena Ayuso, Zentralblatt MATHTable of ContentsPreface; Acknowledgements; 1. Introduction; Part I. Background: 2. Basic algorithmic number theory; 3. Hash functions and MACs; Part II. Algebraic Groups: 4. Preliminary remarks on algebraic groups; 5. Varieties; 6. Tori, LUC and XTR; 7. Curves and divisor class groups; 8. Rational maps on curves and divisors; 9. Elliptic curves; 10. Hyperelliptic curves; Part III. Exponentiation, Factoring and Discrete Logarithms: 11. Basic algorithms for algebraic groups; 12. Primality testing and integer factorisation using algebraic groups; 13. Basic discrete logarithm algorithms; 14. Factoring and discrete logarithms using pseudorandom walks; 15. Factoring and discrete logarithms in subexponential time; Part IV. Lattices: 16. Lattices; 17. Lattice basis reduction; 18. Algorithms for the closest and shortest vector problems; 19. Coppersmith's method and related applications; Part V. Cryptography Related to Discrete Logarithms: 20. The Diffie–Hellman problem and cryptographic applications; 21. The Diffie–Hellman problem; 22. Digital signatures based on discrete logarithms; 23. Public key encryption based on discrete logarithms; Part VI. Cryptography Related to Integer Factorisation: 24. The RSA and Rabin cryptosystems; Part VII. Advanced Topics in Elliptic and Hyperelliptic Curves: 25. Isogenies of elliptic curves; 26. Pairings on elliptic curves; Appendix A. Background mathematics; References; Author index; Subject index.

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Book SynopsisThis unique resource now includes coverage of Unix dates, Italian time, the Akan, Icelandic, Saudi Arabian Umm al-Qura, Babylonian, Samaritan, and Nepalese calendars, plus expanded treatments of Islamic and Hebrew calendars. The astronomical functions have been rewritten for more accurate results and include calculations of moonrise and moonset.Trade Review'It retains all the features that made the first edition … such a wonderful resource, while adding much new material … If you are at all interested in time and calendars, this book must find a place on your desk.' Victor J. Katz, Mathematical ReviewsTable of Contents1. Calendar basics; Part I. Arithmetical Calendars: 2. The Gregorian calendar; 3. The Julian calendar; 4. The Coptic and Ethiopic calendars; 5. The ISO calendar; 6. The Icelandic calendar; 7. The Islamic calendar; 8. The Hebrew calendar; 9. The Ecclesiastical calendars; 10. The old Hindu calendars; 11. The Mayan calendars; 12. The Balinese Pawukon calendar; 13. Generic Cyclical calendars; Part II. Astronomical Calendars: 14. Time and astronomy; 15. The Persian calendar; 16. The Bahá'í calendar; 17. The French Revolutionary calendar; 18. Astronomical Lunar calendars; 19. The Chinese calendar; 20. The modern Hindu calendars; 21. The Tibetan calendar; Part III. Appendices: A. Function, parameter, and constant types; B. Cross references; C. Sample data; D. Lisp implementation.

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Book SynopsisMany scientific, medical or engineering problems raise the issue of recovering some physical quantities from indirect measurements; for instance, detecting or quantifying flaws or cracks within a material from acoustic or electromagnetic measurements at its surface is an essential problem of non-destructive evaluation. The concept of inverse problems precisely originates from the idea of inverting the laws of physics to recover a quantity of interest from measurable data. Unfortunately, most inverse problems are ill-posed, which means that precise and stable solutions are not easy to devise. Regularization is the key concept to solve inverse problems. The goal of this book is to deal with inverse problems and regularized solutions using the Bayesian statistical tools, with a particular view to signal and image estimation. The first three chapters bring the theoretical notions that make it possible to cast inverse problems within a mathematical framework. The next three chapters address the fundamental inverse problem of deconvolution in a comprehensive manner. Chapters 7 and 8 deal with advanced statistical questions linked to image estimation. In the last five chapters, the main tools introduced in the previous chapters are put into a practical context in important applicative areas, such as astronomy or medical imaging.Table of ContentsIntroduction 15 Jérôme IDIER PART I. FUNDAMENTAL PROBLEMS AND TOOLS 23 Chapter 1. Inverse Problems, Ill-posed Problems 25 Guy DEMOMENT, Jérôme IDIER 1.1. Introduction 25 1.2. Basic example 26 1.3. Ill-posed problem 30 1.3.1. Case of discrete data 31 1.3.2. Continuous case 32 1.4. Generalized inversion 34 1.4.1. Pseudo-solutions 35 1.4.2. Generalized solutions 35 1.4.3. Example 35 1.5. Discretization and conditioning 36 1.6. Conclusion 38 1.7. Bibliography 39 Chapter 2. Main Approaches to the Regularization of Ill-posed Problems 41 Guy DEMOMENT, Jérôme IDIER 2.1. Regularization 41 2.1.1. Dimensionality control 42 2.1.2. Minimization of a composite criterion 44 2.2. Criterion descent methods 48 2.2.1.Criterion minimization for inversion 48 2.2.2. The quadratic case 49 2.2.3. The convex case 51 2.2.4. General case 52 2.3. Choice of regularization coefficient 53 2.3.1. Residual error energy control 53 2.3.2. “L-curve” method 53 2.3.3. Cross-validation 54 2.4. Bibliography 56 Chapter 3. Inversion within the Probabilistic Framework 59 Guy DEMOMENT, Yves GOUSSARD 3.1. Inversion and inference 59 3.2. Statistical inference 60 3.2.1. Noise law and direct distribution for data 61 3.2.2. Maximum likelihood estimation 63 3.3. Bayesian approach to inversion 64 3.4. Links with deterministic methods 66 3.5. Choice of hyperparameters 67 3.6. A priori model68 3.7. Choice of criteria 70 3.8. The linear, Gaussian case 71 3.8.1. Statistical properties of the solution 71 3.8.2. Calculation of marginal likelihood 73 3.8.3. Wiener filtering 74 3.9. Bibliography 76 PART II. DECONVOLUTION 79 Chapter 4. Inverse Filtering and Other Linear Methods 81 Guy LE BESNERAIS, Jean-François GIOVANNELLI, Guy DEMOMENT 4.1. Introduction 81 4.2. Continuous-time deconvolution 82 4.2.1. Inverse filtering 82 4.2.2. Wiener filtering 84 4.3. Discretization of the problem 85 4.3.1. Choice of a quadrature method 85 4.3.2. Structure of observation matrix H 87 4.3.3. Usual boundary conditions 89 4.3.4. Problem conditioning 89 4.3.5.Generalized inversion 91 4.4. Batch deconvolution 92 4.4.1. Preliminary choices 92 4.4.2. Matrix form of the estimate 93 4.4.3. Hunt’s method (periodic boundary hypothesis) 94 4.4.4. Exact inversion methods in the stationary case 96 4.4.5. Case of non-stationary signals 98 4.4.6. Results and discussion on examples 98 4.5. Recursive deconvolution 102 4.5.1. Kalman filtering 102 4.5.2. Degenerate state model and recursive least squares 104 4.5.3. Autoregressive state model 105 4.5.4. Fast Kalman filtering 108 4.5.5. Asymptotic techniques in the stationary case 110 4.5.6. ARMA model and non-standard Kalman filtering 111 4.5.7. Case of non-stationary signals 111 4.5.8. On-lineprocessing: 2Dcase 112 4.6. Conclusion 112 4.7. Bibliography 113 Chapter 5. Deconvolution of Spike Trains 117 Frédéric CHAMPAGNAT, Yves GOUSSARD, Stéphane GAUTIER, Jérôme IDIER 5.1. Introduction 117 5.2. Penalization of reflectivities, L2LP/L2Hy deconvolutions 119 5.2.1. Quadratic regularization 121 5.2.2. Non-quadratic regularization 122 5.2.3. L2LPorL2Hy deconvolution 123 5.3. Bernoulli-Gaussian deconvolution 124 5.3.1. Compound BG model 124 5.3.2. Various strategies for estimation 124 5.3.3. General expression for marginal likelihood 125 5.3.4. An iterative method for BG deconvolution 126 5.3.5. Other methods 128 5.4. Examples of processing and discussion 130 5.4.1. Nature of the solutions 130 5.4.2. Setting the parameters 132 5.4.3. Numerical complexity 133 5.5. Extensions 133 5.5.1. Generalization of structures of R and H 134 5.5.2. Estimation of the impulse response . . . 134 5.6. Conclusion 136 5.7. Bibliography 137 Chapter 6. Deconvolution of Images 141 Jérôme IDIER, Laure BLANC-FÉRAUD 6.1. Introduction 141 6.2. Regularization in the Tikhonov sense 142 6.2.1. Principle 142 6.2.2. Connection with image processing by linear PDE 144 6.2.3. Limits of Tikhonov’s approach 145 6.3. Detection-estimation 148 6.3.1. Principle 148 6.3.2. Disadvantages 149 6.4. Non-quadratic approach 150 6.4.1. Detection-estimation and non-convex penalization 154 6.4.2. Anisotropic diffusion by PDE 155 6.5. Half-quadratic augmented criteria 156 6.5.1. Duality between non-quadratic criteria and HQ criteria 157 6.5.2. Minimization of HQ criteria 158 6.6. Application in image deconvolution 159 6.6.1. Calculation of the solution 159 6.6.2. Example 161 6.7. Conclusion 164 6.8. Bibliography 165 PART III. ADVANCED PROBLEMS AND TOOLS 169 Chapter 7. Gibbs-Markov Image Models 171 Jérôme IDIER 7.1. Introduction 171 7.2. Bayesian statistical framework 172 7.3. Gibbs-Markov fields 173 7.3.1. Gibbs fields 174 7.3.2. Gibbs-Markov equivalence 177 7.3.3. Posterior law of a GMRF 180 7.3.4. Gibbs-Markov models for images 181 7.4. Statistical tools, stochastic sampling 185 7.4.1. Statistical tools 185 7.4.2. Stochastic sampling 188 7.5. Conclusion 194 7.6. Bibliography 195 Chapter 8. Unsupervised Problems 197 Xavier DESCOMBES, Yves GOUSSARD 8.1. Introduction and statement of problem 197 8.2. Directly observed field 199 8.2.1. Likelihood properties 199 8.2.2. Optimization 200 8.2.3. Approximations 202 8.3. Indirectly observed field 205 8.3.1. Statement of problem 205 8.3.2. EM algorithm 206 8.3.3. Application to estimation of the parameters of a GMRF 207 8.3.4. EM algorithm and gradient 208 8.3.5. Linear GMRF relative to hyperparameters 210 8.3.6. Extensions and approximations 212 8.4. Conclusion 215 8.5. Bibliography 216 PART IV. SOME APPLICATIONS 219 Chapter 9. Deconvolution Applied to Ultrasonic Non-destructive Evaluation 221 Stéphane GAUTIER, Frédéric CHAMPAGNAT, Jérôme IDIER 9.1. Introduction 221 9.2. Example of evaluation and difficulties of interpretation 222 9.2.1. Description of the part to be inspected 222 9.2.2. Evaluation principle 222 9.2.3. Evaluation results and interpretation 223 9.2.4. Help with interpretation by restoration of discontinuities 224 9.3. Definition of direct convolution model 225 9.4. Blind deconvolution 226 9.4.1. Overview of approaches for blind deconvolution 226 9.4.2. DL2Hy/DBGd econvolution 230 9.4.3. Blind DL2Hy/DBG deconvolution 232 9.5. Processing real data 232 9.5.1. Processing by blind deconvolution 233 9.5.2. Deconvolution with a measured wave 234 9.5.3. Comparison between DL2Hy and DBG 237 9.5.4. Summary 240 9.6. Conclusion 240 9.7. Bibliography 241 Chapter 10. Inversion in Optical Imaging through Atmospheric Turbulence 243 Laurent MUGNIER, Guy LE BESNERAIS, Serge MEIMON 10.1. Optical imaging through turbulence 243 10.1.1. Introduction 243 10.1.2. Image formation 244 10.1.4. Imaging techniques 249 10.2. Inversion approach and regularization criteria used 253 10.3. Measurement of aberrations 254 10.3.1. Introduction 254 10.3.2. Hartmann-Shack sensor 255 10.3.3. Phase retrieval and phase diversity 257 10.4. Myopic restoration in imaging 258 10.4.1. Motivation and noise statistic 258 10.4.2. Data processing in deconvolution from wavefront sensing 259 10.4.3. Restoration of images corrected by adaptive optics 263 10.4.4. Conclusion 267 10.5. Image reconstruction in optical interferometry (OI) 268 10.5.1. Observation model 268 10.5.2. Traditional Bayesian approach 271 10.5.3. Myopic modeling 272 10.5.4. Results 274 10.6. Bibliography 277 Chapter 11. Spectral Characterization in Ultrasonic Doppler Velocimetry 285 Jean-François GIOVANNELLI, Alain HERMENT 11.1. Velocity measurement in medical imaging 285 11.1.1. Principle of velocity measurement in ultrasound imaging 286 11.1.2. Information carried by Doppler signals 286 11.1.3.Some characteristics and limitations 288 11.1.4. Data and problems treated 288 11.2. Adaptive spectral analysis 290 11.2.1. Least squares and traditional extensions 290 11.2.2. Long AR models – spectral smoothness – spatial continuity 291 11.2.3. Kalman smoothing 293 11.2.4. Estimation of hyperparameters 294 11.2.5. Processing results and comparisons 296 11.3. Tracking spectral moments 297 11.3.1. Proposed method 298 11.3.2. Likelihood of the hyperparameters 302 11.3.3. Processing results and comparisons 304 11.4. Conclusion 306 11.5. Bibliography 307 Chapter 12. Tomographic Reconstruction from Few Projections 311 Ali MOHAMMAD-DJAFARI, Jean-Marc DINTEN 12.1. Introduction 311 12.2. Projection generation model 312 12.3. 2D analytical methods 313 12.4. 3D analytical methods 317 12.5. Limitations of analytical methods 317 12.6. Discrete approach to reconstruction 319 12.7. Choice of criterion and reconstruction methods 321 12.8. Reconstruction algorithms 323 12.8.1. Optimization algorithms for convex criteria 323 12.8.2. Optimization or integration algorithms 327 12.9. Specific models for binary objects 328 12.10. Illustrations 328 12.10.1.2D reconstruction 328 12.10.2.3Dreconstruction 329 12.11. Conclusions 331 12.12. Bibliography 332 Chapter 13. Diffraction Tomography 335 Hervé CARFANTAN, Ali MOHAMMAD-DJAFARI 13.1. Introduction 335 13.2. Modeling the problem 336 13.2.1. Examples of diffraction tomography applications 336 13.2.2. Modeling the direct problem 338 13.3. Discretization of the direct problem 340 13.3.1. Choice of algebraic framework 340 13.3.2. Method of moments 341 13.3.3. Discretization by the method of moments 342 13.4. Construction of criteria for solving the inverse problem 343 13.4.1. First formulation: estimation of x 344 13.4.2. Second formulation: simultaneous estimation of x and φ 345 13.4.3. Properties of the criteria 347 13.5. Solving the inverse problem 347 13.5.1. Successive linearizations 348 13.5.2. Joint minimization 350 13.5.3. Minimizing MAP criterion 351 13.6. Conclusion 353 13.7. Bibliography 354 Chapter 14. Imaging from Low-intensity Data 357 Ken SAUER, Jean-Baptiste THIBAULT 14.1. Introduction 357 14.2. Statistical properties of common low-intensity image data 359 14.2.1. Likelihood functions and limiting behavior 359 14.2.2. Purely Poisson measurements 360 14.2.3. Inclusion of background counting noise 362 14.2.4. Compound noise models with Poisson information 362 14.3. Quantum-limited measurements in inverse problems 363 14.3.1. Maximum likelihood properties 363 14.3.2. Bayesian estimation 366 14.4. Implementation and calculation of Bayesian estimates 368 14.4.1. Implementation for pure Poisson model 368 14.4.2. Bayesian implementation for a compound data model 370 14.5. Conclusion 372 14.6. Bibliography 372 List of Authors 375 Index 377

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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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