## Scaling, Fractals and Wavelets (Google eBook)Patrice Abry, Paolo Goncalves, Jacques Levy Vehel 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

17 | |

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 |

503 | |

### Other editions - View all

Scaling Fractals And Wavelets Patrice Abry,Paulo Gonçalvès,Jacques Lévy Véhel No preview available - 2009 |

Scaling, Fractals and Wavelets Patrice Abry,Paolo Goncalves,Jacques Levy Vehel No preview available - 2009 |

### Common terms and phrases

algorithm applications approximation asymptotic attractor behavior Besov spaces calculate Chapter characterized coefﬁcients consider convergence covariance decomposition defined definition denoising derivative destination block distribution estimation example Figure filter finite fractal transformation fractional Brownian motion fractional calculus function f Gaussian Hausdorff dimension Hölder exponent Hölder function IEEE integral interval iterated iterated function systems JAFFARD Legendre transform Lévy process LÉVY VÉHEL linear log2 long memory long-range dependence MANDELBROT mathematical measure method multifractal analysis multifractal formalism multifractal spectrum multifractional function obtain oscillating parameter pointwise polynomial properties PROPOSITION regularity resolution result scale invariance scaling laws self-similar self-similar processes sequence signal singularity source block space spectra stationary increments statistical stochastic stock market structure theorem theory traffic variance variations vector verify wavelet coefficients Weierstrass function zero