Stochastic 2-microlocal analysis

A lot is known about the H\"older regularity of stochastic processes, in particular in the case of Gaussian processes. Recently, a finer analysis of the local regularity of functions, termed 2-microlocal analysis, has been introduced in a deterministic frame: through the computation of the so-called 2-microlocal frontier, it allows in particular to predict the evolution of regularity under the action of (pseudo-) differential operators. In this work, we develop a 2-microlocal analysis for the study of certain stochastic processes. We show that moments of the increments allow, under fairly general conditions, to obtain almost sure lower bounds for the 2-microlocal frontier. In the case of Gaussian processes, more precise results may be obtained: the incremental covariance yields the almost sure value of the 2-microlocal frontier. As an application, we obtain new and refined regularity properties of fractional Brownian motion, multifractional Brownian motion, stochastic generalized Weierstrass functions, Wiener and stable integrals.Comment: 35 page

Estimation locale : compromis régression-variance

International audienceWe present a method allowing under some hypothesis to decrease the variance of an optimal estimator without causing damage to its risk, using many estimators jointly, thanks to a very basic technique leading to a hybrid estimator that takes advantage of the best of them. This method can be applied in various fields, typically on estimators that were obtained using linear regressions

Localisable moving average stable and multistable processes

We study a particular class of moving average processes which possess a property called localisability. This means that, at any given point, they admit a ``tangent process'', in a suitable sense. We give general conditions on the kernel g defining the moving average which ensures that the process is localisable and we characterize the nature of the associated tangent processes. Examples include the reverse Ornstein-Uhlenbeck process and the multistable reverse Ornstein-Uhlenbeck process. In the latter case, the tangent process is, at each time t, a L\'evy stable motion with stability index possibly varying with t. We also consider the problem of path synthesis, for which we give both theoretical results and numerical simulations

On various multifractal spectra

International audienceWe introduce two classes of multifractal spectra, called respectively dimension and continuous spectra. Dimension spectra offer an interesting alternative to the classical Hausdorff spectrum: They are much easier to estimate yet still give relevant information about the geometry of the H¨older function. Continuous spectra are a generalization of the large deviation spectrum that allow to obtain partition free results. Both classes of spectra allow to perform efficient multifractal analysis in an experimental framewor

The local Hölder function of a continuous function

AbstractThis work focuses on the local Hölder exponent as a measure of the regularity of a function around a given point. We investigate in detail the structure and the main properties of the local Hölder function (i.e., the function that associates to each point its local Hölder exponent). We prove that it is possible to construct a continuous function with prescribed local and pointwise Hölder functions outside a set of Hausdorff dimension 0

Wavelet packet based digital watermarking

International audienceWe present a method for digital image watermarking based on the modification of certain subsets of the wavelet packet decomposition. These subsets are determined both from a secret key and an image dependent procedure that chooses a best basis from an energy criterion. The mark is set by imposing a parity constraint at each level of the decomposition. We elaborate on the choice of some of the parameters of the model, showing how they can be tuned so as to obtain good resistance to attacks. Examples are displayed to assess the validity of our approach

Fractal and integral geometry tools for texture deformation measurement

In this work, we investigate the use of two techniques for segmentation of different states of one texture (e.g. deformations of an homogeneous texture): - fractal geometry, that deals with the analysis of complex irregular shapes which cannot well be described by the classical Euclidean geometry - Integral geometry, that treats sets globally and allows to introduce robust measures.We focus on the study of two parameters, Lacunarity and Favard length, and proove a theoritical link between them. As an application,we are able to achieve with an excellent accuracy automatic classification of Lung diseases on the basis on SPECT images. Classical techniques tried on those images give poor results

Analyse Fractale : une nouvelle génération d'outils pour le Traitement du Signal

International audienceRecently, a number of important progresses in fractal analysis have had a major impact in signal processing applications. We review briefly IFS theory, multifractal analysis and stable fractional processes theory; we indicate how these theoretical tools lead to new methods for image processing (segmentation, denoising, compression), financial analysis, and internet traffic modeling. Among others, these applications show that fractal analysis is no longer restricted to a descriptive role, but has entered an operational phase. .Récemment, plusieurs développements importants en analyse fractale ont eu un impact majeur sur les applications en traitement du signal. Nous abordons brièvement la théorie des systèmes de fonctions itérées, l'analyse multifractale, et les processus stables fractionnaires, en expliquant comment des progrès dans ces divers champs ont conduit à de nouvelles méthodes en traitement des images (compression, watermarking, segmentation, débruitage, . . . ), analyse financière, et modélisation du trafic sur les réseaux informatiques. Ces applications, parmi d'autres, montrent que l'analyse fractale est résolument passée depuis quelques années du “stade descriptif” au “stade opérationnel”