Poisson random balls: self-similarity and X-ray images

Hermine Biermé (1) , Anne Estrade (1)

1 MAP5 - UMR 8145 - Mathématiques Appliquées Paris 5

We study a random field obtained by counting the number of balls containing a given point, when overlapping balls are thrown at random according to a Poisson random measure. We describe a microscopic process which exhibits a multifractional behavior. We are particularly interested in the local asymptotical self-similarity (lass) properties of the field, as well as in its X-ray transforms. We obtain two different lass properties when considering the asymptotics either "in law" or "in the sense of second-order moments" and prove a relationship between the lass behavior of the field and the lass behavior of its X-ray transform. These results can be used to model and analyze porous media, images or connection networks.

Mots clés: Random field; random set; shot-noise; overlapping spheres; Poisson point process; X-ray transform; asymptotic self-similarity; fractional Brownian motion

Domaines: Probabilités [math.PR]

Hermine Biermé, Anne Estrade. Poisson random balls: self-similarity and X-ray images. Advances in Applied Probability, 2006, 38, pp.853-872. ⟨hal-00170277⟩