Mixed measures of convex cylinders and quermass densities of Boolean models
A Boolean model of convex bodies is the best known example of a random closed set. It has been studied extensively in recent years and is among the most popular models from stochastic geometry. For the latter, quermass densities (also called densities of Minkowski functionals) are very interesting characteristics because a lot of algorithms for their estimation from real data are available. Whereas quermass densities can be defined for stationary random closed sets in the extended convex ring, for a stationary and isotropic Boolean model of convex bodies, iterations of the principal kinematic formula for quermass integrals can be used to express quermass densities in terms of the underlying Poisson particle process. To extend these results to stationary but not necessarily isotropic processes of convex particles and their Boolean models, translative integral formulas for curvature measures by R. Schneider and W. Weil were applied. For non-stationary Boolean models of convex bodies, quermass densities were introduced by \textit{H. Fallert} [Math. Nachr. 181, 165--184 (1996; Zbl 0858.60015)] using an iterated version of these translative integral formulas where the latter was obtained by W. Weil. Yet in some applications [\textit{K. Schladitz, S. Peters, D. Reinel-Bitzer, A. Wiegmann} and \textit{J. Ohser}, Comput. Mat. Sci. 38, 56--66 (2006)] it is necessary to work with Boolean models of convex cylinders, i.e. Minkowski sums of convex bodies and linear subspaces of \({\mathbb R}^d\) (= \(d\)-dimensional Euclidean space). In this case formulas for all quermass densities are only available for stationary and isotropic Boolean models for which they were proved by \textit{P. Davy} [Stereology -- a stationary viewpoint. Dissertation, Australian National University, Camberra (1978)]. Recently, \textit{M. Spiess} and \textit{E. Spodarev} obtained results for the specific volume and specific surface area of stationary Boolean models [``Anisotropic dilated Poisson \(k\)-flat processes'' (to appear)]. In the present paper the author uses a special form of the iterated translative integral formulas by W. Weil for convex cylinders (Section 3) to express quermass densities of non-stationary Boolean models of convex cylinders in terms of the underlying Poisson process (Sections 4 and 5). This generalizes the respective results by P. Davy, H. Fallert and M. Spiess and E. Spodarev [loc. cit.].
scientific article; zbMATH DE number 5636289