Abry P., Goncalves P., Vehel J.L. Scaling, Fractals and Wavelets

Wiley-ISTE, 2009. — 464 p. — ISBN-10: 1848210728, ISBN-13: 978-1848210721.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.Preface.
Fractal and Multifractal Analysis in Signal Processing.
Introduction.
Dimensions of sets.
Minkowski-Bouligand dimension.
Packing dimension.
Covering dimension.
Methods for calculating dimensions.
Hölder exponents.
Hölder exponents related to a measure.
Theorems on set dimensions.
Hölder exponent related to a function.
Signal dimension theorem.
2-microlocal analysis.
An example: analysis of stock market price.
Multifractal analysis.
What is the purpose of multifractal analysis?
First ingredient: local regularity measures.
Second ingredient: the size of point sets of the same regularity.
Practical calculation of spectra.
Refinements: analysis of the sequence of capacities, mutual analysis and multisingularity.
The multifractal spectra of certain simple signals.
Two applications.
Scale Invariance and Wavelets.
Introduction.
Models for scale invariance.
Intuition.
Self-similarity.
Long-range dependence.
Local regularity.
Fractional Brownian motion: paradigm of scale invariance.
Beyond the paradigm of scale invariance.
Wavelet transform.
Continuous wavelet transform.
Discretewavelet transform.
Wavelet analysis of scale invariant processes.
Self-similarity.
Long-range dependence.
Local regularity.
Beyond second order.
Implementation: analysis, detection and estimation.
Estimation of the parameters of scale invariance.
Emphasis on scaling laws and determination of the scaling range.
Robustness of the wavelet approach.
Conclusion.
Wavelet Methods for Multifractal Analysis of Functions.
Introduction.
General points regarding multifractal functions.
Important definitions.
Wavelets and pointwise regularity.
Local oscillations.
Complements.
Random multifractal processes.
Lévy processes.
Burgers’ equation and Brownian motion.
Random wavelet series.
Multifractal formalisms.
Besov spaces and lacunarity.
Construction of formalisms.
Bounds of the spectrum.
Bounds according to the Besov domain.
Bounds deduced from histograms.
The grand-canonical multifractal formalism.
Multifractal Scaling: General Theory and Approach by Wavelets.
Introduction and summary.
Singularity exponents.
Hölder continuity.
Scaling of wavelet coefficients.
Other scaling exponents.
Multifractal analysis.
Dimension based spectra.
Grain based spectra.
Partition function and Legendre spectrum.
Deterministic envelopes.
Multifractal formalism.
Binomial multifractals.
Construction.
Wavelet decomposition.
Multifractal analysis of the binomial measure.
Examples.
Beyond dyadic structure.
Wavelet based analysis.
The binomial revisited with wavelets.
Multifractal properties of the derivative.
Self-similarity and LRD.
Multifractal processes.
Construction and simulation.
Global analysis.
Local analysis of warped FBM.
LRDand estimation ofwarped FBM.
Self-similar Processes.
Introduction.
Motivations.
Scalings.
Distributions of scale invariant masses.
Weierstrass functions.
Renormalization of sums of random variables.
A common structure for a stochastic (semi-)self-similar process.
Identifying Weierstrass functions.
The Gaussian case.
Self-similar Gaussian processes with r-stationary increments.
Elliptic processes.
Hyperbolic processes.
Parabolic processes.
Wavelet decomposition.
Renormalization of sums of correlated random variable.
Convergence towards fractional Brownian motion.
Non-Gaussian case.
Introduction.
Symmetric α-stable processes.
Censov and Takenaka processes.
Wavelet decomposition.
Process subordinated to Brownian measure.
Regularity and long-range dependence.
Introduction.
Two examples.
Locally Self-similar Fields.
Introduction.
Recap of two representations of fractional Brownian motion.
Reproducing kernel Hilbert space.
Harmonizable representation.
Two examples of locally self-similar fields.
Definition of the local asymptotic self-similarity (LASS).
Filtered white noise (FWN).
Elliptic Gaussian random fields (EGRP).
Multifractional fields and trajectorial regularity.
Two representations of theMBM.
Study of the regularity of the trajectories of the MBM.
Towards more irregularities: generalized multifractional Brownian motion (GMBM) and step fractional Brownian motion (SFBM).
Estimate of regularity.
General method: generalized quadratic variation.
Application to the examples.
An Introduction to Fractional Calculus.
Introduction.
Motivations.
Problems.
Outline.
Definitions.
Fractional integration.
Fractional derivatives within the framework of causal distributions.
Mild fractional derivatives, in the Caputo sense.
Fractional differential equations.
Example.
Framework of causal distributions.
Framework of functions expandable into fractional power series (α-FPSE).
Asymptotic behavior of fundamental solutions.
Controlled-and-observed linear dynamic systems of fractional order.
Diffusive structure of fractional differential systems.
Introduction to diffusive representations of pseudo-differential operators.
General decomposition result.
Connection with the concept of long memory.
Particular case of fractional differential systems of commensurate orders.
Example of a fractional partial differential equation.
Physical problem considered.
Spectral consequences.
Time-domain consequences.
Free problem.
Conclusion.
Fractional Synthesis, Fractional Filters.
Traditional and less traditional questions about fractionals.
Notes on terminology.
Short and long memory.
From integer to non-integer powers: filter based sample path design.
Local and global properties.
Fractional filters.
Desired general properties: association.
Construction and approximation techniques.
Discrete time fractional processes.
Filters: impulse responses and corresponding processes.
Mixing and memory properties.
Parameter estimation.
Simulated example.
Continuous time fractional processes.
A non-self-similar family: fractional processes designed from fractional filters.
Sample path properties: local and global regularity, memory.
Distribution processes.
Motivation and generalization of distribution processes.
The family of linear distribution processes.
Fractional distribution processes.
Mixing and memory properties.
Iterated Function Systems and Some Generalizations: Local Regularity Analysis and Multifractal Modeling of Signals.
Introduction.
Definition of the Hölder exponent.
Iterated function systems (IFS).
Generalization of iterated function systems.
Semi-generalized iterated function systems.
Generalized iterated function systems.
Estimation of pointwise Hölder exponent by GIFS.
Principles of themethod.
Algorithm.
Application.
Weak self-similar functions and multifractal formalism.
Signal representation by WSA functions.
Segmentation of signals by weak self-similar functions.
Estimation of the multifractal spectrum.
Experiments.
Iterated Function Systems and Applications in Image Processing.
Introduction.
Iterated transformation systems.
Contracting transformations and iterated transformation systems.
Attractor of an iterated transformation system.
Collage theorem.
Finally contracting transformation.
Attractor and invariant measures.
Inverse problem.
Application to natural image processing: image coding.
Introduction.
Coding of natural images by fractals.
Algebraic formulation of the fractal transformation.
Experimentation on triangular partitions.
Coding and decoding acceleration.
Other optimization diagrams: hybrid methods.
Local Regularity and Multifractal Methods for Image and Signal Analysis.
Introduction.
Basic tools.
Hölder regularity analysis.
Reminders on multifractal analysis.
Hölderian regularity estimation.
Oscillations (OSC).
Wavelet coefficient regression (WCR).
Wavelet leaders regression (WL).
Limit inf and limit sup regressions.
Numerical experiments.
Denoising.
Introduction.
Minimax risk, optimal convergence rate and adaptivity.
Wavelet based denoising.
Non-linear wavelet coefficients pumping.
Denoising using exponent between scales.
Bayesian multifractal denoising.
Hölderian regularity based interpolation.
Introduction.
Themethod.
Regularity and asymptotic properties.
Numerical experiments.
Biomedical signal analysis.
Texture segmentation.
Edge detection.
Introduction.
Change detection in image sequences using multifractal analysis.
Image reconstruction.
Scale Invariance in Computer Network Traffic.
Teletraffic – a new natural phenomenon.
A phenomenon of scales.
An experimental science of “man-made atoms”.
A random current.
Two fundamental approaches.
From a wealth of scales arise scaling laws.
First discoveries.
Laws reign.
Beyond the revolution.
Sources as the source of the laws.
The sumor its parts.
The on/off paradigm.
Chemistry.
Mechanisms.
New models, new behaviors.
Character of a model.
The fractional Brownian motion family.
Greedy sources.
Never-ending calls.
Perspectives.
Research of Scaling Law on Stock Market Variations.
Introduction: fractals in finance.
Presence of scales in the study of stock market variations.
Modeling of stock market variations.
Time scales in financial modeling.
Modeling postulating independence on stock market returns.
1960-1970: from Pareto’s law to Lévy’s distributions.
1970–1990: experimental difficulties of iid-α-stable model.
Unstable iid models in partial scaling invariance.
Research of dependency and memory of markets.
Linear dependence: testing of H-correlative models on returns.
Non-linear dependence: validating H-correlative model on volatilities.
Towards a rediscovery of scaling laws in finance.
Scale Relativity, Non-differentiability and Fractal Space-time.
Introduction.
Abandonment of the hypothesis of space-time differentiability.
Towards a fractal space-time.
Explicit dependence of coordinates on spatio-temporal resolutions.
From continuity and non-differentiability to fractality.
Description of non-differentiable process by differential equations.
Differential dilation operator.
Relativity and scale covariance.
Scale differential equations.
Constant fractal dimension: “Galilean” scale relativity.
Breaking scale invariance: transition scales.
Non-linear scale laws: second order equations, discrete scale invariance, log-periodic laws.
Variable fractal dimension: Euler-Lagrange scale equations.
Scale dynamics and scale force.
Special scale relativity – log-Lorentzian dilation laws, invariant scale limit under dilations.
Generalized scale relativity and scale-motion coupling.
Quantum-like induced dynamics.
Generalized Schrödinger equation.
Application in gravitational structure formation.
Conclusion.
List of Authors.
Index.