Analysis of a fractal boundary: the graph of the Knopp function

A usual classification tool to study a fractal interface is the computation of its fractal dimension. But a recent method developed by Y. Heurteaux and S. Jaffard proposes to compute either weak and strong accessibility exponents or local Lp regularity exponents (the so-called p-exponent). These exponents describe locally the behavior of the interface. We apply this method to the graph of the Knopp function. The Knopp function itself has everywhere the same p-exponent. Nevertheless, using the characterization of the maxima and minima done by B. Dubuc and S. Dubuc, we will compute the p-exponent of the characteristic function of domain under the graph of F at each point (x,F(x)) and show that p-exponents, weak and strong accessibility exponents change from point to point. Furthermore we will derive a characterization of the local extrema of the function according to the values of these exponents

A family of functions with two different spectra of singularities

International audienceOur goal is to study the multifractal properties of functions of a given family which have few non vanishing wavelet coefficients. We compute at each point the pointwise Holder exponent of these functions and also their local Lp regularity, computing the so-called p-exponent. We prove that in the general case the Holder and p exponent are different at each point. We also compute the dimension of the sets where the functions have a given pointwise regularity and prove that these functions are multifractal both from the point of view of Holder and Lp local regularity with different spectra of singularities. Furthermore, we check that multifractal formalism type formulas hold for functions in that family

Some proximal methods for Poisson intensity CBCT and PET

International audienceCone-Beam Computerized Tomography (CBCT) and Positron Emission Tomography (PET) are two complementary medical imaging modalities providing respectively anatomic and metabolic information on a patient. In the context of public health, one must address the problem of dose reduction of the potentially harmful quantities related to each exam protocol : X-rays for CBCT and radiotracer for PET. Two demonstrators based on a technological breakthrough (acquisition devices work in photon-counting mode) have been developed. It turns out that in this low-dose context, i.e. for low intensity signals acquired by photon counting devices, noise should not be approximated anymore by a Gaussian distribution, but is following a Poisson distribution. We investigate in this paper the two related tomographic reconstruction problems. We formulate separately the CBCT and the PET problems in two general frameworks that encompass the physics of the acquisition devices and the specific discretization of the object to reconstruct. We propose various fast numerical schemes based on proximal methods to compute the solution of each problem. In particular, we show that primal-dual approaches are well suited in the PET case when considering non differentiable regularizations such as Total Variation. Experiments on numerical simulations and real data are in favor of the proposed algorithms when compared with well-established methods

Oscillating singularities in Besov spaces

International audienceThe purpose of multifractal analysis is to evaluate the Hausdorff dimensions d(h) of the sets Sh of points where the pointwise Hölder exponent of a function, a signal or an image has a given value h∈[h0,h1]. Inside the realm of mathematics this makes good sense but for most signals or images such calculations are out of reach. That is why Uriel Frisch and Giorgio Parisi proposed an algorithm which relates these dimensions d(h) to some averaged increments. Averaged increments are named structure functions in fluid dynamics and can be easily computed. The Frisch and Parisi algorithm is called multifractal formalism. Unfortunately multifractal formalism is not valid in full generality and one should know when it holds. A general answer is supplied by "Baire-type" results. These results show that in many function spaces, quasi-all functions (in the sense of Baire's categories) do not obey the multifractal formalism if the Hölder exponent is large. Our purpose is to understand this phenomenon. We will prove that a cause of the failure of the multifractal formalism is the presence of oscillating singularities, which was guessed by A. Arnéodo and his collaborators

Sur les singularités oscillantes et le formalisme multifractal

The purpose of multifractal analysis is to compute the dimension of the sets where a function has a given pointwise Holder regularity. This computation can't be done directly on real signals (signal of the velocity in fully developped turbulence...) and a formula called multifractal formalism has been introduced to compute these dimensions from quantities obtained directly by signal processing. It is not true in all generality and we study in these PHD several situations where the multifractal formalism doesn't hold.L'objectif de l'analyse multifractale (introduite dans le cadre de la turbulence pleinement developpee) est de déterminer la dimension des ensembles de points où une fonction a une régularité hölderienne fixée. Cette information ne peut être calculée directement sur les signaux réels et une formule appelée formalisme multifractal a été introduite pour calculer ces dimensions à partir de quantités obtenues directement par traitement du signal. Elle n'est pas vraie en toute généralité et nous étudions dans cette thèse différentes situations dans lesquelles le formalisme multifractal n'est pas valide.Des résultats de type " Baire " démontrent que le formalisme multifractal est vrai quasi-sûrement pour de petites valeurs de l'exposant de Hölder et faux pour les autres valeurs. Nous montrons que cela est dû à la présence de singularités oscillantes.D'autre part le formalisme multifractal ne s'applique qu'aux fonctions continues. Nous montrons qu'il est possible de généraliser la formule, en passant d'un critère de régularité ponctuelle hölderienne à un critère plus faible, à des fonctions qui peuvent ne plus être continues.Enfin nous étudions un cas particulier de phénomène oscillant en dimension 2 qui n'est pas caractérisé par les critères de régularité ponctuelle précédents. Nous proposons une méthode d'analyse de ce comportement à base d'un algorithme de traitement de l'image

Wavelet Analysis of Fractal Boundaries. Part 2: Multifractal Analysis

International audienceThis second part deals with the global analysis of the boundary of domains Omega ⊂ Rd . We develop methods for determining the dimensions of the sets where the local behaviors introduced in Part 1 occur. These methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains

Wavelet analysis of fractal Boundaries, Part 1: Local regularity

International audienceLet Ω be a domain of Rd. In Part 1 of this paper, we introduce new tools in order to analyse the local behavior of the boundary of Ω. Classifica- tions based on geometric accessibility conditions are introduced and compared; they are related to analytic criteria based either on local Lp regularity of the characteristic function 1Ω, or on its wavelet coefficients. Part 2 deals with the global analysis of the boundary of Ω. We develop methods for determining the dimensions of the sets where the local behaviors previously introduced occur. These methods are based on analogies with the thermodynamic formalism in statistical physics and lead to new classification tools for fractal domains

Sur les singularités oscillantes et le formalisme multifractal

PARIS12-CRETEIL BU Multidisc. (940282102) / SudocSudocFranceF

Exponential moments of self-intersection local times of stable random walks in subcritical dimensions

Let $(X_t, t \geq 0)$ be an $\alpha$-stable random walk with values in $\Z^d$. Let $l_t(x) = \int_0^t \delta_x(X_s) ds$ be its local time. For $p>1$, not necessarily integer, $I_t = \sum_x l_t^p(x)$ is the so-called $p$-fold self- intersection local time of the random walk. When $p(d -\alpha) < d$, we derive precise logarithmic asymptotics of the probability $P(I_t \geq r_t)$ for all scales r_t \gg \E(I_t). Our result extends previous works by Chen, Li and Rosen 2005, Becker and König 2010, and Laurent 2012

Approximate and exact solutions of intertwining equations through random spanning forests

International audienceFor different reversible Markov kernels on finite state spaces, we look for families of probability measures for which the time evolution almost remains in their convex hull. Motivated by signal processing problems and metastability studies we are interested in the case when the size of such families is smaller than the size of the state space, and we want such distributions to be with small overlap among them. To this aim we introduce a squeezing function to measure the common overlap of such families, and we use random forests to build random approximate solutions of the associated intertwining equations for which we can bound from above the expected values of both squeezing and total variation errors. We also explain how to modify some of these approximate solutions into exact solutions by using those eigenvalues of the associated Laplacian with the largest absolute values