Bootstrap for log Wavelet Leaders Cumulant based Multifractal Analysis.

Multifractal analysis, which mostly consists of estimating scaling exponents related to the power law behaviors of the moments of wavelet coefficients, is becoming a popular tool for empirical data analysis. However, little is known about the statistical performance of such procedures. Notably, despite their being of major practical importance, no confidence intervals are available. Here, we choose to replace wavelet coefficients with wavelet Leaders and to use a log-cumulant based multifractal analysis. We investigate the potential use of bootstrap to derive confidence intervals for wavelet Leaders log-cumulant multifractal estimation procedures. From numerical simulations involving well-known and well-controlled synthetic multifractal processes, we obtain two results of major importance for practical multifractal analysis : we demonstrate that the use of Leaders instead of wavelet coefficients brings significant improvements in log-cumulant based multifractal estimation, we show that accurate bootstrap designed confidence intervals can be obtained for a single finite length time series

Multiscale and Anisotropic Characterization of Images Based on Complexity: an Application to Turbulence

This article presents a multiscale, non-linear and directional statistical characterization of images based on the estimation of the skewness, flatness, entropy and distance from Gaussianity of the spatial increments. These increments are characterized by their magnitude and direction; they allow us to characterize the multiscale properties directionally and to explore anisotropy. To describe the evolution of the probability density function of the increments with their magnitude and direction, we use the skewness to probe the symmetry, the entropy to measure the complexity, and both the flatness and distance from Gaussianity to describe the shape. These four quantities allow us to explore the anisotropy of the linear correlations and non-linear dependencies of the field across scales. First, we validate the methodology on two-dimensional synthetic scale-invariant fields with different multiscale properties and anisotropic characteristics. Then, we apply it on two synthetic turbulent velocity fields: a perfectly isotropic and homogeneous one, and a channel flow where boundaries induce inhomogeneity and anisotropy. Our characterization unambiguously detects the anisotropy in the second case, where our quantities report scaling properties that depend on the direction of analysis. Furthermore, we show in both cases that turbulent velocity fluctuations are always isotropic, when the mean velocity profile is adequately removed

La dynamique fait son cinéma : De l'apport de l'imagerie et des mesures de champs cinématiques pour l'analyse du comportement dynamique des matériaux

National audienceDepuis de nombreuses décennies, l'imagerie rapide a permis d'observer des phénomènes se produisant sur des échelles de temps très petites (de l'ordre de la milliseconde voire de la microseconde). Avec l'avènement plus récent des caméras numériques, de nouvelles applications sont possibles (p.ex. la tomographie rapide). L'utilisation quantitative d'images est également possible, notamment grâce aux techniques de corrélation et de stéréocorrélation d'images. Différentes applications seront présentées afin d'illustrer les apports pour l'analyse du comportement mécanique des matériaux sous sollicitations dynamiques

Multifractal Analysis for Images : The wavelet Leaders contribution

1. Motivation Scale invariance has been observed in numerous applications involving data of various and very different natures. It can be operationally defined as the power law behavior with respect to scale of the structure functions, which are given by the empirical moments of the absolute value of the multiresolution coefficients of the data at a given scale (cf. Eq. (1)). The estimation of the exponents characterizing these power laws – termed scaling exponents – constitutes the ultimate goal of the practical analysis of scale invariance (also called scaling analysis). These scaling exponents are then commonly involved in standard signal processing tasks, such as detection, identification, or classification. In practice, scaling analysis is often conducted within the theoretical framework of multifractal analysis. In a nutshell, multifractal analysis aims at characterizing the fluctuation (in time or space) of the local regularity of the process under analysis through analysis of the (power law) behavior of the structure functions in the limit of fine scales. Though multifractal analysis can theoretically be extended to dimensions higher than 1 without technical difficulties, most practical implementations remain restricted to one dimensional signals. This is mainly due to the fact that multifractal analysis requires the use of a range of both positive and negative empirical moments, hence demanding for multiresolution quantities with adequate properties. To date, the only practically available procedure for the multifractal analysis of 2D signals, hence images, is the so called Wavelet Transform Modulus Maxima (WTMM) procedure (based on the skeleton of a continuous wavelet transform (CWT)). Yet, the WTMM procedure suffers from a number of theoretical and practical drawbacks: It has a high computational cost; The calculation of the 2D CWT skeleton requires involved theoretical definitions as well as a cumbersome practical procedure; It is still lacking a theoretical support. Therefore, in numerous applications where the data are naturally images, multifractal analysis remains restricted to 1D slices of the data and hence incomplete. In the present contribution, elaborating on previous results obtained for (1D) signals, we propose a practical multifractal analysis method for (2D) images based on two key features: The use of a 2D Discrete Wavelet Transform (DWT) (instead of a 2D CWT); The replacement of wavelet coefficients with wavelet Leaders. This yields two major benefits: The computation cost is very low; Wavelet Leaders have been shown to yield a complete and rigorous analysis of the multifractal analysis of bounded functions. This is because wavelet Leaders consist of monotonous increasing quantities that finely account for the irregularities of the analyzed function. The aims of the present contribution are twofold: First, studying the necessary theoretical elements, validity and limitations of a wavelet Leader based multifractal analysis of images, and second, the evaluation of its practical statistical performance....Nous nous intéressons à la réalisation pratique d’une procédure permettant d’effectuer une analyse multifractale, c’est-à-dire des fluctuations de régularité locale, de champs scalaires bidimensionnels, d’images notamment. L’originalité de la procédure réside dans la construction, à partir des coefficients d’une transformée discrète bidimensionnelle en ondelettes, de coefficients dominants, impliqués ensuite dans l’estimation des attributs multifractals. Nous donnons des éléments mathématiques relatifs aux problèmes théoriques liés à la validité du formalisme multifractal ainsi construit, et à son application à des images réelles. Nous indiquons comment l’utiliser pour détecter la présence éventuelle de singularités oscillantes. Pour étudier les performances des procédures construites, ces estimateurs sont mis en oeuvre sur un grand nombre de réalisations de processus synthétiques, dont les propriétés multifractales sont connues théoriquement. Nous validons le fait que l’analyse multifractale 2D, construite sur les coefficients dominants, permet une mesure effective et complète des propriétés multifractales des images analysées. De plus, comparant les résultats obtenus d’images mono-fractales à ceux produits sur des images multi-fractales, nous commentons de façon détaillée l’apport des coefficients dominants par rapport à l’usage des coefficients d’ondelettes. Les attributs multifractals ainsi estimés peuvent ensuite être impliqués dans des tâches de classification, par exemple

Influence de la dynamique de la rupture sur l'exposant de rugosité dans les ractures 1D

Nous déterminons l'exposant de rugosité de fractures en utilisant plusieurs estimateurs de lois d'échelle existant dans la littérature dont un récent : les coefficients dominants [1]. A partir d'un large jeu de réalisations d'une expérience de rupture d'une feuille de papier (102 fronts), nous comparons les exposants estimés dans deux régimes de croissance différents. Les résultats montrent un écart significatif à l'invariance d'échelle et une différence significative entre la valeur des exposants estimés dans le régime de croissance lente (sous-critique) et le régime de croissance rapide

A joint time-scale representation methodology or the detection o acoustic gravity wave induced by solar eclipses

Nous proposons l'utilisation de la transformée en ondelettes complexes pour détecter et caractériser la propagation d'ondes gravito-acoustiques dans l'ionosphère, à partir de données représentant les fluctuations temporelles de concentrations électroniques à différentes altitudes. Nous détectons d'abord, à chaque altitude, les maxima locaux des transformées. Les maxima qui existent simultanément à différentes altitudes dans un même voisinage temps-fréquence sont ensuite rassemblés dans une même structure. La dérivation de la phase des coefficients en ondelettes le long de ces lignes de maxima nous permet d'extraire les paramètres de propagation. Cet outil est utilisé pour étudier trois éclipses solaires différentes. Cela nous permet de mettre en évidence l'occurrence d'ondes gravito-acoustiques durant ces événements

Shearing of loose granular materials: A statistical mesoscopic model

A two-dimensional lattice model for the formation and evolution of shear bands in granular media is proposed. Each lattice site is assigned a random variable which reflects the local density. At every time step, the strain is localized along a single shear-band which is a spanning path on the lattice chosen through an extremum condition. The dynamics consists of randomly changing the `density' of the sites only along the shear band, and then repeating the procedure of locating the extremal path and changing it. Starting from an initially uncorrelated density field, it is found that this dynamics leads to a slow compaction along with a non-trivial patterning of the system, with high density regions forming which shelter long-lived low-density valleys. Further, as a result of these large density fluctuations, the shear band which was initially equally likely to be found anywhere on the lattice, gets progressively trapped for longer and longer periods of time. This state is however meta-stable, and the system continues to evolve slowly in a manner reminiscent of glassy dynamics. Several quantities have been studied numerically which support this picture and elucidate the unusual system-size effects at play.Comment: 11 pages, 15 figures revtex, submitted to PRE, See also: cond-mat/020921

Conformal Mapping on Rough Boundaries I: Applications to harmonic problems

The aim of this study is to analyze the properties of harmonic fields in the vicinity of rough boundaries where either a constant potential or a zero flux is imposed, while a constant field is prescribed at an infinite distance from this boundary. We introduce a conformal mapping technique that is tailored to this problem in two dimensions. An efficient algorithm is introduced to compute the conformal map for arbitrarily chosen boundaries. Harmonic fields can then simply be read from the conformal map. We discuss applications to "equivalent" smooth interfaces. We study the correlations between the topography and the field at the surface. Finally we apply the conformal map to the computation of inhomogeneous harmonic fields such as the derivation of Green function for localized flux on the surface of a rough boundary

Slow relaxation due to optimization and restructuring: Solution on a hierarchical lattice

Motivated by the large strain shear of loose granular materials we introduced a model which consists of consecutive optimization and restructuring steps leading to a self organization of a density field. The extensive connections to other models of statistical phyics are discussed. We investigate our model on a hierarchical lattice which allows an exact asymptotic renormalization treatment. A surprisingly close analogy is observed between the simulation results on the regular and the hierarchical lattices. The dynamics is characterized by the breakdown of ergodicity, by unusual system size effects in the development of the average density as well as by the age distribution, the latter showing multifractal properties.Comment: 11 pages, 7 figures revtex, submitted to PRE see also: cond-mat/020920