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This paper provides representation theorems for classes of nonstationary covariance functions on spheres across time. Isotropy on spheres comes out naturally in virtue of the addition theorem, and we show how to escape from this assumption, in order to obtain kernels that can be anisotropic. The same criterion is used in order to escape from stationarity in the temporal component. Our findings are then illustrated through a simulation study.

Let X be a real-valued stationary Gaussian random field defined on $R^d$ (d ≥ 1), with almost every realization of class $C^2$. This paper is concerned with the random variable giving the number of points in $T$ (a compact set of $R^d$) where the gradient $X'$ takes a fixed value $v\in R^d$, $N_{X'}(T, v) = \{t \in T : X'(t) = v\}$. More precisely, it deals with the finiteness of the variance of $N_{X'} (T, v)$, under some non-degeneracy hypothesis on $X$. For d = 1, the so-called " Geman condition " has been proved to be a sufficient condition for $N_{X'} (T, v)$ to admit a finite second moment. This condition on the fourth derivative $r^{(4)}$ of the covariance function of $X$ does not depend on $v$ and requires $t \mapsto \frac{ r ^{(4)}(0)−r ^{(4)}(t)}{t} $ to be integrable in a neighbourhood of zero. We prove that for d ≥ 1, a generalization of the Geman condition remains a sufficient condition for $N_{X'} (T, v)$ to admit a second moment. No assumption of isotropy is required.

We consider an isotropic bi-phasic (pore and solid) medium, draw many lines through it, and see each line as a one-dimensional level-cut process with values 0 or 1 according to whether a regular stationary process $X$ is less or greater than a given level. The lengths of time intervals when $X$ is in a given phase are named chord functions. We are interested into obtaining information on chord functions in order to get most characterizations of the medium such as the volume fractions, the two-point correlation function or the specific surface area. Working with the Palm probability measure and using level crossings techniques, Rice methods in particular, allow to obtain the exact analytical formula of the chord-distribution function as well as the joint distribution function of two successive chords and the two-point correlation function. Also the ``Independent Interval Assumption'' is discussed.

International audience; We consider Poisson random balls, with the pair (center, radius) being given by a Poisson point process. According to the intensity measure of the Poisson process, we investigate the eventuality of covering the whole space with the union of the balls. We exhibit a disjunction phenomenon between the coverage with large balls (low frequency) and the coverage with small balls (high frequency). Concerning the second type of coverage, we prove the existence of a critical regime which separates the case where coverage occurs a.s. and the case where coverage does not occur a.s. We give an explicit value of the critical intensity and we prove that the Hausdorff measure of the set of points which are not covered by the union of balls is linked with this value. We also compare with other critical regimes appearing in continuum percolation.

The fractional Poisson field (fPf) can be interpreted in term of the number of balls falling down on each point of $\R^D$, when the centers and the radii of the balls are thrown at random following a Poisson point process in $\R^D\times \R^+$ with an appropriate intensity measure. It provides a simple description for a non Gaussian random field that has the same covariance function as the fractional Brownian field. In the present paper, we concentrate on the restrictions of the fPf to finite sets of points in $\R^D$. Actually, since it takes discrete values, it seems natural to adapt this field to a discrete context. We are particularly interested in its finite-dimensional distributions, in its representation on a finite grid, and in its discrete variations which yield an estimator for its Hurst index.

This paper considers random balls in a D-dimensional Euclidean space whose centers are prescribed by a homogeneous Poisson point process and whose radii are prescribed by a specific power law. A random field is constructed by counting the number of covering balls at each point. Even though it is not Gaussian, this field shares the same covariance function as the fractional Brownian field (fBf). By analogy it is called fractional Poisson field (fPf). In this paper, we are mainly interested in the simulation of fPfs with index H in (0,1/2) and in the estimation of the H index. Our method is based on the analysis of structure functions. The fPf exhibits a multifractal behavior, contrary to that of the fBf which is monofractal.

We study a random field obtained by counting the number of balls containing each point, when overlapping balls are thrown at random according to a Poisson random measure. We are particularly interested in the local asymptotical self-similarity (lass) properties of the field, as well as the action of X-ray transforms. We discover two different lass properties when considering the asymptotic either "in law" or "on the second order moment" and prove a relationship between the lass behavior of the field and the lass behavior of its X-ray transform. We also describe a microscopic process which leads to a multifractional behavior. These results can be exploited to model and analyze granular media, images or connections network., A multifractional case is investigated and reference [10] is added. Former title: "Poisson microballs: self-similarity and directional analysis"

Deux des auteurs ont récemment établi que la régularité Rn d'un fractal continu isotrope de dimension n (nD) est égale à celle sa projection Rn-1 dans un espace (n-1)D diminuée de 0.5, ou plus clairement, Rn = Rn-1 - 0.5. Dans un premier temps, l'objectif de cette étude est d'expliquer brièvement ce résultat et de l'illustrer sur des volumes fractals synthétiques, continus et binaires, et leurs projections. Pour les fractals continus, les résultats sont proches de ceux qu'il convient de trouver pour H faible. Pour H élevé, il apparaît des difficultés liées essentiellement aux défauts de la méthode de synthèse retenue. Pour les fractals binaires, l'écart de 0.5 qu'il est possible de trouver entre la régularité du volume et celle de sa projection n'est pas clairement établie. Dans un deuxième temps, nous avons expérimentalement étudié si cette relation est valide pour l'os trabéculaire et sa projection. Nous avons analysé 22 échantillons de tête de fémur imagés par micro-scanner. Nous trouvons une différence entre la régularité 2D et celle en 3D de 0.53 ± 0.046. Ceci indique que le la relation qui lie la régularité 3D d'un volume fractal à celle de sa projection l'applique pour cet objet binaire. Une simple mesure 2D permettrait alors de quantifier la structure trabéculaire 3D et pourrait se révéler d'une grande force pour le diagnostic précoce de l'ostéoporose.

In this communication, we present the Directional Averages Method (DAM) to analyse Anisotropic Fractional Brownian Fields (AFBF) defined in the spectral domain by a Hurst parameter H( θ) depending on the direction θ. For a given direction ϕ on the image, the average is formed in that direction and the regularity R( ϕ) is estimated. We show that for θ = ϕ + π/2, the function H( θ) is equal to R( ϕ) -1/2 and can be thus recovered. We have tested DAM on isotropic fractal images as well as on AFBF generated by the inverse Fourier method. Results are fair even if difficulties arise linked to the synthesis of true AFBF, to the analysis of signals which Hölder exponent is greater than one, and to the digitisation of images. Interesting perspectives can be developed concerning the analysis of fractal porous volumes only known from a 2D projection as it is the case for radiographs.

International audience; Let $X=\{X(t), t \in \R^N\}$ be a Gaussian random field with values in $\R^d$ defined by $$X(t) = \big(X_1(t), \ldots, X_d(t)\big), \qquad \forall \ t \in \R^N, $$ where $X_1, \ldots, X_d$ are independent copies of a centered real-valued Gaussian random field $X_0$. We consider the case when $X_0$ is anisotropic and study the packing dimension of the range $X(E)$, where $E\subseteq \R^N$ is a Borel set. For this purpose we extend the original notion of packing dimension profile due to Falconer and Howroyd (1997) to the anisotropic metric space $(\R^N, \rho)$, where $\rho(s, t) = \sum_{j=1}^N |s_j - t_j|^{H_j}$ and $(H_1, \ldots, H_N) \in (0, 1)^N$ is a given vector. The extended notion of packing dimension profile is of independent interest.