Scaling, Fractals and Wavelets (Google eBook)

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Patrice Abry, Paolo Goncalves, Jacques Levy Vehel
John Wiley & Sons, Jan 5, 2010 - Mathematics - 464 pages
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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.
  

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Contents

Preface
17
Chapter 1 Fractal and Multifractal Analysis in Signal Processing
19
Chapter 2 Scale Invariance and Wavelets
71
Chapter 3 Wavelet Methods for Multifractal Analysis of Functions
103
Chapter 4 Multifractal Scaling General Theory and Approach by Wavelets
139
Chapter 5 Selfsimilar Processes
179
Chapter 6 Locally Selfsimilar Fields
205
Chapter 7 An Introduction to Fractional Calculus
237
Chapter 9 Iterated Function Systems and Some Generalizations Local Regularity Analysis and Multifractal Modeling of Signals
301
Chapter 10 Iterated Function Systems and Applications in Image Processing
333
Chapter 11 Local Regularity and Multifractal Methods for Image and Signal Analysis
367
Chapter 12 Scale Invariance in Computer Network Traffic
413
Chapter 13 Research of Scaling Law on Stock Market Variations
437
Chapter 14 Scale Relativity Nondifferentiability and Fractal Spacetime
465
List of Authors
499
Index
503

Chapter 8 Fractional Synthesis Fractional Filters
279

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About the author (2010)

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.

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