UMR 5672

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Invariance d'échelle, multifractales et ondelettes

L'analyse et la modélisation des phénomènes d'invariance d'échelle constituent des thèmes de recherche au long court de l'équipe SiSyPhe. Les efforts portent d'abord depuis longtemps sur la définition, la conception et la synthèse numérique de processus stochastiques aux propriétés d'invariance d'échelle prescrites a priori, conjointement soit avec d'autres propriétés statistiques (distributions, dépendances,...), soit avec des propriétés géométriques (divergence nulle, vorticité prescrite,..) pour ajuster des applications particulière (la turbulence hydrodynamique en particulier). Des outils statistiques sont également construits et étudiés, théoriquement et pratiquement, visant à tester la présence d'invariance d'échelle, à estimer les paramètres de scaling ou les bornes de la gamme d'échelle. Les efforts récents portent notamment sur l'étude des signaux multivariés et l'étude des champs (multidimensionnels) compliqués par une éventuelle anisotropie.

Scale invariance for multivariate signals and fields: theory and applications

Multivariate scaling signals. (V. Pipiras, UNC, USA) Theoretical definitions and practical synthesis of multivariate non Gaussian processes whose marginal distributions and covariance function are a priori and jointly prescribed has been achieved both via non linear pointwise transformations f of a suitable Gaussian process, whose covariance function depends both on the targeted covariance and on the Hermite polynomial expansion of f, and via optimal transport, a technique borrowed from image processing, that displaces alternatively and iteratively the time and frequency contents of a well chosen Gaussian seed. This has notably been used to obtain long range dependent non Gaussian processes, with same covariance and marginals with yet different joint distributions. It has also been extended to multivariate fields. Multifractal Random Walk, a close relative, yet with additional multifractal properties, has also been thoroughly studied theoretically, aiming at defining the range of parameters within which the process is well defined.

Multifractal and anisotropic image textures. (ANR AMATIS, S. Jaffard, Paris Est, H. Wendt, IRIT Toulouse, B. Vedel, Bretagne Sud, M. Clausel, UJF, Grenoble). Multifractal Analysis, based on wavelet Leaders, has been extended to isotropic fields. This required notably careful analyses and understanding of the role of Hölder global regularity and of the use of fractional integration. The interplay between self-similarity and anisotropy in image textures has been carefully studied, yielding an accurate estimate of the selfsimilar parameter despite anisotropy. This disentangling of selfsimilarity from anisotropy has been made possible by the use of the 2D Hyperbolic Wavelet Transform, that permits anisotropic dilations.

Multifractal vector fields. (ANR CHAMU, V. Vargas, ENS Paris, C. Garban, ENS Lyon, R. Rhodes, Paris 7) Motivated by the analysis the physical mechanisms for 3D fluid turbulence, modeled by Euler or Navier-Stokes equations (energy cascade and vorticity stretching,...), random vector fields that combines scaling (multifractal) properties and geo- metrical constraints have been defined and studied. Their exact statistical characterization is chal- lenging as it amounts to generalizing multiplicative chaos. A first step has been achieved that showed that such random vector fields defined from exponential of long range dependent processes are well defined mathematical objects, whose covariance and higher order moments are analytically tractable.

Further advances in multifractal analysis. (ANR AMATIS, S. Jaffard, Paris Est, H. Wendt, IRIT Toulouse). Have also been studied: The estimation of the Long Memory parameter for non Gaussian processes, the multifractal properties of non Gaussian self similar processes, and the relevance of a bayesian framework for estimation of the multifractality parameter. The segmentation of image textures into pieces with homogeneous local regularity (as measured from wavelet leaders) has been investigated; it relies on the use of proximal methods for functional minimization.

Fractal analysis in Applications.

Ionosphere. (CNRS PICS, P. Sauli, Atmospheric Phys. Dept., Prague) The Ionosphere electron con- centration fluctuations measured across several mid- latitude European stations have been shown to have correlation both in seasonal trends and within scaling behaviors, at short time scales, and they were related to the Geomagnetic activity.

Astrophysics. The 2D Wavelet Transform Modulus Maxima Method has been used to detect and extract coronal loops in ultraviolet images of the solar corona and to disentangle in solar magnetogram data the multifractal properties in active regions from the surrounding monofractal quiet-Sun field.

Art Investigations. (MoMA, NYC, Van Gogh Museum, Amsterdam) Scaling analysis in image textures was used for art work investigations, tending to show that copies, replica and forgeries show lesser irregularities (at very fine scales, below the millimeter) than originals. The extend to which this betrays creation processes will be investigated.

Contacts

Patrice Abry, Laurent Chevillard, Nelly Pustelnik, Stéphane Roux

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