Scaling, Fractals And Wavelets

Scaling 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.

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.
Chapter 1. Fractal and Multifractal Analysis in Signal Processing (Jacques LÉVY VÉHEL and Claude TRICOT).
1.1. Introduction.
1.2.Dimensions of sets.
1.3. Hölder exponents.
1.4. Multifractal analysis.
1.5.Bibliography.
Chapter 2. Scale Invariance and Wavelets (Patrick FLANDRIN, Paulo GONCALVES and Patrice ABRY).
2.1. Introduction.
2.2. Models for scale invariance.
2.3.Wavelet transform.
2.4. Wavelet analysis of scale invariant processes.
2.5. Implementation: analysis, detection and estimation.
2.6. Conclusion.
2.7.Bibliography.
Chapter 3.Wavelet Methods for Multifractal Analysis of Functions (Stéphane JAFFARD).
3.1. Introduction.
3.2. General points regarding multifractal functions.
3.3. Random multifractal processes.
3.4. Multifractal formalisms.
3.5. Bounds of the spectrum.
3.6. The grand-canonical multifractal formalism.
3.7.Bibliography.
Chapter 4. Multifractal Scaling: General Theory and Approach by Wavelets (Rudolf RIEDI).
4.1. Introduction and summary.
4.2. Singularity exponents.
4.3. Multifractal analysis.
4.4. Multifractal formalism.
4.5. Binomial multifractals.
4.6. Wavelet based analysis.
4.7. Self-similarity and LRD.
4.8. Multifractal processes.
4.9.Bibliography.
Chapter 5. Self-similar Processes (Albert BENASSI and Jacques ISTAS).
5.1. Introduction.
5.2. The Gaussian case.
5.3. Non-Gaussian case.
5.4. Regularity and long-range dependence.
5.5.Bibliography.
Chapter 6. Locally Self-similar Fields (Serge COHEN).
6.1. Introduction.
6.2. Recap of two representations of fractional Brownian motion.
6.3. Two examples of locally self-similar fields.
6.4. Multifractional fields and trajectorial regularity.
6.5. Estimate of regularity.
6.6.Bibliography.
Chapter 7. An Introduction to Fractional Calculus (Denis MATIGNON).
7.1. Introduction.
7.2. Definitions.
7.3. Fractional differential equations.
7.4. Diffusive structure of fractional differential systems.
7.5. Example of a fractional partial differential equation.
7.6. Conclusion.
7.7.Bibliography.
Chapter 8. Fractional Synthesis, Fractional Filters (Liliane BEL, Georges OPPENHEIM, Luc ROBBIANO and Marie-Claude VIANO).
8.1. Traditional and less traditional questions about fractionals.
8.2. Fractional filters.
8.3. Discrete time fractional processes.
8.4. Continuous time fractional processes.
8.5. Distribution processes.
8.6.Bibliography.
Chapter 9. Iterated Function Systems and Some Generalizations: Local Regularity Analysis and Multifractal Modeling of Signals (Khalid DAOUDI).
9.1. Introduction.
9.2. Definition of the Hölder exponent.
9.3. Iterated function systems (IFS).
9.4. Generalization of iterated function systems.
9.5. Estimation of pointwise Hölder exponent by GIFS.
9.6. Weak self-similar functions and multifractal formalism.
9.7. Signal representation by WSA functions.
9.8. Segmentation of signals by weak self-similar functions.
9.9. Estimation of the multifractal spectrum.
9.10. Experiments.
9.11.Bibliography.
Chapter 10. Iterated Function Systems and Applications in Image Processing (Franck DAVOINE and Jean-Marc CHASSERY).
10.1. Introduction.
10.2. Iterated transformation systems.
10.3. Application to natural image processing: image coding.
10.4.Bibliography.
Chapter 11. Local Regularity and Multifractal Methods for Image and Signal Analysis (Pierrick LEGRAND).
11.1. Introduction.
11.2.Basic tools.
11.3. Hölderian regularity estimation.
11.4. Denoising.
11.5. Hölderian regularity based interpolation.
11.6. Biomedical signal analysis.
11.7. Texture segmentation.
11.8. Edge detection.
11.9. Change detection in image sequences using multifractal analysis.
11.10. Image reconstruction.
11.11.Bibliography.
Chapter 12. Scale Invariance in Computer Network Traffic (Darryl VEITCH).
12.1. Teletraffic – a new natural phenomenon.
12.2. From a wealth of scales arise scaling laws.
12.3. Sources as the source of the laws.
12.4. New models, new behaviors.
12.5. Perspectives.
12.6.Bibliography.
Chapter 13. Research of Scaling Law on Stock Market Variations (Christian WALTER).
13.1. Introduction: fractals in finance.
13.2. Presence of scales in the study of stock market variations.
13.3. Modeling postulating independence on stock market returns.
13.4. Research of dependency and memory of markets.
13.5. Towards a rediscovery of scaling laws in finance.
13.6.Bibliography.
Chapter 14. Scale Relativity, Non-differentiability and Fractal Space-time (Laurent NOTTALE).
14.1. Introduction.
14.2. Abandonment of the hypothesis of space-time differentiability.
14.3. Towards a fractal space-time.
14.4. Relativity and scale covariance.
14.5. Scale differential equations.
14.6. Quantum-like induced dynamics.
14.7. Conclusion.
14.8.Bibliography.
List of Authors.
Index.