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.

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.