UMR 5672

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Scale invariance, multifractals and wavelets

Scale invariance analysis and modeling has been a long standing research topic within SiSyPh team. Significant efforts have been and still are devoted to the definition, design and practical synthesis of stochastic processes with prescribed scaling properties, combined either to other statistic properties (distributions, dependencies) or to geometrical properties (free divergence, prescribed curl,...) to match a given application field (mainly hydrodynamic turbulence). Statistical signal processing tools are also designed and studied theoretically and practically, aiming to test the existence of scale invariance, to estimate scaling parameters or the scaling range bounds. The recent focus is on multivariate signals, and multidimensional fields with possible anisotropy.

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.


Patrice Abry, Laurent Chevillard, Nelly Pustelnik, Stéphane Roux

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