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.