Minkowski tensor density formulas for Boolean models - Hoerrmann, Julia; Hug, Daniel; Klatt, Michael; Mecke, Klaus (Advances in Applied Mathematics, 2014)
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. 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 numerically determine Minkowski tensor densities for nonisotropic 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. In particular, the numerically determined values for the density of the Euler characteristic allow the estimation of certain mixed functionals of the grains. (C) 2014 Elsevier Inc. All rights reserved.
Keywords: Stochastic geometry; Boolean model; Stationarity; Anisotropy; Minkowski tensors; Poisson process; Translative integral geometry