This book is organized around the notions of scaling phenomena and scale invariance. The various stochastic models commonly used to describe scaling are introduced: self-similarity, long-range dependence and multi-fractals. These models are compared and related one to the other. Second, they introduce fractional integration, a mathematical tool closely related to the notion of scale invariance. Also, they define stochastic processes with prescribed scaling properties (self-similar processes, locally self-similar processes, fractionally filtered processes, iterated function systems). 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.
1. Fractal and Multifractal Analysis in Signal Processing, Jacques Lévy Véhel and Claude Tricot.
2. Scale Invariance and Wavelets, Patrick Flandrin, Paulo Goncalvès and Patrice Abry.
3. Wavelet Methods for Multifractal Analysis of Functions, Stéphane Jaffard.
4. Multifractal Scaling: General Theory and Approach by Wavelets, Rudolf H Riedi.
5. Self-similar Processes, Albert Benassi and Jacques Istas.
6. Locally Self-similar Fields, Serge Cohen.
7. An Introduction to Fractional Calculus, Denis Matignon.
8. Fractional Synthesis, Fractional Filters, Liliane Bel, Georges Oppenheim, Luc Robbiano and Marie-Claude Viano.
9. Iterated Function Systems and Some Generalizations: Local Regularity Analysis and Multifractal Modeling of Signals, Khalid Daoudi.
10. Iterated Function Systems and Applications in Image Processing, Franck Davoine and Jean-Marc Chassery.
11. Local Regularity and Multifractal Methods for Image and Signal Analysis, Pierrick Legrand.
12. Scale Invariance in Computer Network Traffic, Darryl Veitch.
13. Research of Scaling Law on Stock Market Variations, Christian Walter.
14. Scale Relativity, Non-differentiability and Fractal Time-space, Laurent Nottale.
Patrice Abry is a Professor in the Laboratoire de Physique at the Ecole Normale Supérieure de Lyon, France.
Paulo Gonçalves is an Associate Researcher at INRIA, Lyon, France.
Jacques Lévy Véhel is a Research Director at INRIA, Orsay, France.