  A stationary Boolean model is the union set of random compact particles which
are attached to the points of a stationary Poisson point process. For a
stationary Boolean model with convex grains we consider a recently developed
collection of shape descriptors, the so called Minkowski tensors. By combining
spatial and probabilistic averaging we define Minkowski tensor densities of a
Boolean model. These densities are global characteristics of the union set
which can be estimated from observations. In contrast local characteristics
like the mean Minkowski tensor of a single random particle cannot be observed
directly, since the particles overlap. We relate the global to the local
properties by density formulas for the Minkowski tensors. These density
formulas generalize the well known formulas for intrinsic volume densities and
are obtained by applying results from translative integral geometry. For an
isotropic Boolean model we observe that the Minkowski tensor densities are
proportional to the intrinsic volume densities, whereas for a non-isotropic
Boolean model this is usually not the case. Our results support the idea that
the degree of anisotropy of a Boolean model may be expressed in terms of the
Minkowski tensor densities. Furthermore we observe that for smooth grains the
mean curvature radius function of a particle can be reconstructed from the
Minkowski tensor densities. In a simulation study we determine numerically
Minkowski tensor densities for non-isotropic Boolean models based on ellipses
and on rectangles in two dimensions and find excellent agreement with the
derived analytic density formulas. The tensor densities can be used to
characterize the orientational distribution of the grains and to estimate model
parameters for non-isotropic distributions.
