  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.
