Scaling, Fractals and Wavelets

Patrice Abry is a Professor in the Laboratoire de Physique at the Ecole Normale Superieure de Lyon, France. His current research interests include wavelet-based analysis and modelling of scaling phenomena and related topics, stable processes, multi-fractal, long-range dependence, local regularity of processes, infinitely divisible cascades and departures from exact scale invariance. Paulo Goncalves graduated from the Signal Processing Department of ICPI, Lyon, France in 1993. He received the Masters (DEA) and Ph.D. degrees in signal processing from the Institut National Polytechnique, Grenoble, France, in 1990 and 1993 respectively. While working toward his Ph.D. degree, he was with Ecole Normale Superieure, Lyon. In 1994-96, he was a Postdoctoral Fellow at Rice University, Houston, TX. Since 1996, he is associate researcher at INRIA, first with Fractales (1996-99), and then with a research team at INRIA Rhone-Alpes (2000-2003). His research interests are in multiscale signal and image analysis, in wavelet-based statistical inference, with application to cardiovascular research and to remote sensing for land cover classification.Jacques Levy Vehel graduated from Ecole Polytechnique in 1983 and from Ecole Nationale Superieure des Telecommuncations in 1985. He holds a Ph.D in Applied Mathematics from Universite d'Orsay. He is currently a research director at INRIA, Rocquencourt, where he created the Fractales team, a research group devoted to the study of fractal analysis and its applications to signal/image processing. He also leads a research team at IRCCYN, Nantes, with the same scientific focus. His current research interests include (multi)fractal processes, 2-microlocal analysis and wavelets, with application to Internet traffic, image processing and financial data modelling.

• Preface 17Chapter 1. Fractal and Multifractal Analysis in Signal Processing 19Jacques LEVY VEHEL and Claude TRICOT1.1. Introduction 191.2.Dimensions of sets 201.2.1.Minkowski-Bouligand dimension 211.2.2. Packing dimension 251.2.3.Covering dimension 271.2.4. Methods for calculating dimensions 291.3. Holder exponents 331.3.1. Holder exponents related to a measure 331.3.2. Theorems on set dimensions 331.3.3. Holder exponent related to a function 361.3.4. Signal dimension theorem 421.3.5. 2-microlocal analysis 451.3.6. An example: analysis of stock market price 461.4. Multifractal analysis 481.4.1. What is the purpose of multifractal analysis? 481.4.2. First ingredient: local regularity measures 491.4.3. Second ingredient: the size of point sets of the same regularity 501.4.4. Practical calculation of spectra 521.4.5. Refinements: analysis of the sequence of capacities, mutual analysis and multisingularity 601.4.6. The multifractal spectra of certain simple signals 621.4.7.Two applications 661.5.Bibliography 68Chapter 2. Scale Invariance and Wavelets 71Patrick FLANDRIN, Paulo GONCALVES and Patrice ABRY2.1. Introduction 712.2. Models for scale invariance 722.2.1. Intuition 722.2.2. Self-similarity 732.2.3. Long-range dependence 752.2.4. Local regularity 762.2.5. Fractional Brownian motion: paradigm of scale invariance 772.2.6. Beyond the paradigm of scale invariance 792.3.Wavelet transform 812.3.1. Continuous wavelet transform 812.3.2.Discretewavelet transform 822.4. Wavelet analysis of scale invariant processes 852.4.1. Self-similarity 862.4.2. Long-range dependence 882.4.3. Local regularity 902.4.4. Beyond second order 922.5. Implementation: analysis, detection and estimation 922.5.1. Estimation of the parameters of scale invariance 932.5.2. Emphasis on scaling laws and determination of the scaling range 962.5.3. Robustness of the wavelet approach 982.6. Conclusion 1002.7.Bibliography 101Chapter 3.Wavelet Methods for Multifractal Analysis of Functions 103Stephane JAFFARD3.1. Introduction 1033.2. General points regarding multifractal functions 1043.2.1. Important definitions 1043.2.2. Wavelets and pointwise regularity 1073.2.3. Local oscillations 1123.2.4. Complements 1163.3. Random multifractal processes 1173.3.1. Levy processes 1173.3.2. Burgers’ equation and Brownian motion 1203.3.3. Random wavelet series 1223.4. Multifractal formalisms 1233.4.1. Besov spaces and lacunarity 1233.4.2. Construction of formalisms 1263.5. Bounds of the spectrum 1293.5.1. Bounds according to the Besov domain 1293.5.2. Bounds deduced from histograms 1323.6. The grand-canonical multifractal formalism 1323.7.Bibliography 134Chapter 4. Multifractal Scaling: General Theory and Approach by Wavelets 139Rudolf RIEDI4.1. Introduction and summary 1394.2. Singularity exponents 1404.2.1.Holder continuity 1404.2.2. Scaling of wavelet coefficients 1424.2.3. Other scaling exponents 1444.3. Multifractal analysis 1454.3.1. Dimension based spectra 1454.3.2. Grain based spectra 1464.3.3. Partition function and Legendre spectrum 1474.3.4. Deterministic envelopes 1494.4. Multifractal formalism 1514.5. Binomial multifractals 1544.5.1.Construction 1544.5.2. Wavelet decomposition 1574.5.3. Multifractal analysis of the binomial measure 1584.5.4. Examples 1604.5.5. Beyond dyadic structure 1624.6. Wavelet based analysis 1634.6.1. The binomial revisited with wavelets 1634.6.2. Multifractal properties of the derivative 1654.7. Self-similarity and LRD 1674.8. Multifractal processes 1684.8.1.Construction and simulation 1694.8.2. Global analysis 1704.8.3. Local analysis of warped FBM 1704.8.4.LRDand estimation ofwarped FBM 1734.9.Bibliography 173Chapter 5. Self-similar Processes 179Albert BENASSI and Jacques ISTAS5.1. Introduction 1795.1.1.Motivations 1795.1.2. Scalings 1825.1.3. Distributions of scale invariant masses 1845.1.4. Weierstrass functions 1855.1.5. Renormalization of sums of random variables 1865.1.6. A common structure for a stochastic (semi-)self-similar process 1875.1.7. Identifying Weierstrass functions 1885.2. The Gaussian case 1895.2.1. Self-similar Gaussian processes with r-stationary increments 1895.2.2. Elliptic processes 1905.2.3. Hyperbolic processes 1915.2.4. Parabolic processes 1925.2.5. Wavelet decomposition 1925.2.6. Renormalization of sums of correlated random variable 1935.2.7. Convergence towards fractional Brownian motion 1935.3. Non-Gaussian case 1955.3.1. Introduction 1955.3.2. Symmetric α-stable processes 1965.3.3. Censov and Takenaka processes 1985.3.4. Wavelet decomposition 1985.3.5. Process subordinated to Brownian measure 1995.4. Regularity and long-range dependence 2005.4.1. Introduction 2005.4.2. Two examples 2015.5.Bibliography 202Chapter 6. Locally Self-similar Fields 205Serge COHEN6.1. Introduction 2056.2. Recap of two representations of fractional Brownian motion 2076.2.1. Reproducing kernel Hilbert space 2076.2.2. Harmonizable representation 2086.3. Two examples of locally self-similar fields 2136.3.1. Definition of the local asymptotic self-similarity (LASS) 2136.3.2. Filtered white noise (FWN) 2146.3.3. Elliptic Gaussian random fields (EGRP) 2156.4. Multifractional fields and trajectorial regularity 2186.4.1.Two representations of theMBM 2196.4.2. Study of the regularity of the trajectories of the MBM 2216.4.3. Towards more irregularities: generalized multifractional Brownian motion (GMBM) and step fractional Brownian motion (SFBM) 2226.5. Estimate of regularity 2266.5.1. General method: generalized quadratic variation 2266.5.2. Application to the examples 2286.6.Bibliography 235Chapter 7. An Introduction to Fractional Calculus 237Denis MATIGNON7.1. Introduction 2377.1.1.Motivations 2377.1.2. Problems 2387.1.3. Outline 2397.2. Definitions 2407.2.1. Fractional integration 2407.2.2. Fractional derivatives within the framework of causal distributions 2427.2.3. Mild fractional derivatives, in the Caputo sense 2467.3. Fractional differential equations 2517.3.1. Example 2517.3.2. Framework of causal distributions 2547.3.3. Framework of functions expandable into fractional power series (α-FPSE) 2557.3.4. Asymptotic behavior of fundamental solutions 2577.3.5. Controlled-and-observed linear dynamic systems of fractional order 2617.4. Diffusive structure of fractional differential systems 2627.4.1. Introduction to diffusive representations of pseudo-differential operators 2637.4.2. General decomposition result 2647.4.3. Connection with the concept of long memory 2657.4.4. Particular case of fractional differential systems of commensurate orders 2657.5. Example of a fractional partial differential equation 2667.5.1. Physical problem considered 2677.5.2. Spectral consequences 2687.5.3. Time-domain consequences 2687.5.4. Free problem 2727.6. Conclusion 2737.7.Bibliography 273Chapter 8. Fractional Synthesis, Fractional Filters 279Liliane BEL, Georges OPPENHEIM, Luc ROBBIANO and Marie-Claude VIANO8.1. Traditional and less traditional questions about fractionals 2798.1.1.Notes on terminology 2798.1.2. Short and long memory 2798.1.3. From integer to non-integer powers: filter based sample path design 2808.1.4. Local and global properties 2818.2. Fractional filters 2828.2.1. Desired general properties: association 2828.2.2. Construction and approximation techniques 2828.3. Discrete time fractional processes 2848.3.1. Filters: impulse responses and corresponding processes 2848.3.2. Mixing and memory properties 2868.3.3. Parameter estimation 2878.3.4. Simulated example 2898.4. Continuous time fractional processes 2918.4.1. A non-self-similar family: fractional processes designed from fractional filters 2918.4.2. Sample path properties: local and global regularity, memory 2938.5. Distribution processes 2948.5.1. Motivation and generalization of distribution processes 2948.5.2. The family of linear distribution processes 2948.5.3. Fractional distribution processes 2958.5.4. Mixing and memory properties 2968.6.Bibliography 297Chapter 9. Iterated Function Systems and Some Generalizations: Local Regularity Analysis and Multifractal Modeling of Signals 301Khalid DAOUDI9.1. Introduction 3019.2. Definition of the Holder exponent 3039.3. Iterated function systems (IFS) 3049.4. Generalization of iterated function systems 3069.4.1. Semi-generalized iterated function systems 3079.4.2. Generalized iterated function systems 3089.5. Estimation of pointwise Holder exponent by GIFS 3119.5.1. Principles of themethod 3129.5.2. Algorithm 3149.5.3.Application 3159.6. Weak self-similar functions and multifractal formalism 3189.7. Signal representation by WSA functions 3209.8. Segmentation of signals by weak self-similar functions 3249.9. Estimation of the multifractal spectrum 3269.10. Experiments 3279.11.Bibliography 329Chapter 10. Iterated Function Systems and Applications in Image Processing 333Franck DAVOINE and Jean-Marc CHASSERY10.1. Introduction 33310.2. Iterated transformation systems 33310.2.1. Contracting transformations and iterated transformation systems 33410.2.2.Attractor of an iterated transformation system 33510.2.3. Collage theorem 33610.2.4. Finally contracting transformation 33810.2.5. Attractor and invariant measures 33910.2.6. Inverse problem 34010.3. Application to natural image processing: image coding 34010.3.1. Introduction 34010.3.2. Coding of natural images by fractals 34210.3.3. Algebraic formulation of the fractal transformation 34510.3.4. Experimentation on triangular partitions 35110.3.5. Coding and decoding acceleration 35210.3.6. Other optimization diagrams: hybrid methods 36010.4.Bibliography 362Chapter 11. Local Regularity and Multifractal Methods for Image and Signal Analysis 367Pierrick LEGRAND11.1. Introduction 36711.2.Basic tools 36811.2.1. Holder regularity analysis 36811.2.2. Reminders on multifractal analysis 36911.3. Holderian regularity estimation 37111.3.1. Oscillations (OSC) 37111.3.2. Wavelet coefficient regression (WCR) 37211.3.3. Wavelet leaders regression (WL) 37211.3.4.Limit inf and limit sup regressions 37311.3.5. Numerical experiments 37411.4. Denoising 37611.4.1. Introduction 37611.4.2. Minimax risk, optimal convergence rate and adaptivity 37711.4.3. Wavelet based denoising 37811.4.4. Non-linear wavelet coefficients pumping 38011.4.5. Denoising using exponent between scales 38311.4.6. Bayesian multifractal denoising 38611.5. Holderian regularity based interpolation 39311.5.1. Introduction 39311.5.2.Themethod 39311.5.3. Regularity and asymptotic properties 39411.5.4. Numerical experiments 39411.6. Biomedical signal analysis 39411.7. Texture segmentation 40111.8. Edge detection 40311.8.1. Introduction 40311.8.1.1. Edge detection 40611.9. Change detection in image sequences using multifractal analysis 40711.10. Image reconstruction 40811.11.Bibliography 409Chapter 12. Scale Invariance in Computer Network Traffic 413Darryl VEITCH12.1. Teletraffic – a new natural phenomenon 41312.1.1. A phenomenon of scales 41312.1.2. An experimental science of “man-made atoms” 41512.1.3. A random current 41612.1.4. Two fundamental approaches 41712.2. From a wealth of scales arise scaling laws 41912.2.1. First discoveries 41912.2.2.Laws reign 42012.2.3. Beyond the revolution 42412.3. Sources as the source of the laws 42612.3.1.The sumor its parts 42612.3.2.The on/off paradigm 42712.3.3. Chemistry 42812.3.4. Mechanisms 42912.4. New models, new behaviors 43012.4.1. Character of a model 43012.4.2. The fractional Brownian motion family 43112.4.3. Greedy sources 43212.4.4. Never-ending calls 43212.5. Perspectives 43312.6.Bibliography 434Chapter 13. Research of Scaling Law on Stock Market Variations 437Christian WALTER13.1. Introduction: fractals in finance 43713.2. Presence of scales in the study of stock market variations 43913.2.1. Modeling of stock market variations 43913.2.2. Time scales in financial modeling 44513.3. Modeling postulating independence on stock market returns 44613.3.1. 1960-1970: from Pareto’s law to Levy’s distributions 44613.3.2. 1970–1990: experimental difficulties of iid-α-stable model 44813.3.3. Unstable iid models in partial scaling invariance 45213.4. Research of dependency and memory of markets 45413.4.1. Linear dependence: testing of H-correlative models on returns 45413.4.2. Non-linear dependence: validating H-correlative model on volatilities 45613.5. Towards a rediscovery of scaling laws in finance 45713.6.Bibliography 458Chapter 14. Scale Relativity, Non-differentiability and Fractal Space-time 465Laurent NOTTALE14.1. Introduction 46514.2. Abandonment of the hypothesis of space-time differentiability 46614.3. Towards a fractal space-time 46614.3.1. Explicit dependence of coordinates on spatio-temporal resolutions 46714.3.2. From continuity and non-differentiability to fractality 46714.3.3. Description of non-differentiable process by differential equations 46914.3.4. Differential dilation operator 47114.4. Relativity and scale covariance 47214.5. Scale differential equations 47214.5.1. Constant fractal dimension: “Galilean” scale relativity 47314.5.2. Breaking scale invariance: transition scales 47414.5.3. Non-linear scale laws: second order equations, discrete scale invariance, log-periodic laws 47514.5.4. Variable fractal dimension: Euler-Lagrange scale equations 47614.5.5. Scale dynamics and scale force 47814.5.6. Special scale relativity – log-Lorentzian dilation laws, invariant scale limit under dilations 48114.5.7. Generalized scale relativity and scale-motion coupling 48214.6. Quantum-like induced dynamics 48814.6.1. Generalized Schrodinger equation 48814.6.2. Application in gravitational structure formation 49214.7. Conclusion 49314.8.Bibliography 495List of Authors 499Index 503