Densities of mixed volumes for Boolean models

In generalization of the well-known formulae for quermass densities of stationary and isotropic Boolean models, we prove corresponding results for densities of mixed volumes in the stationary situation and show how they can be used to determine the intensity of non-isotropic Boolean models Z in d -dimensional space for d = 2, 3, 4. We then consider non-stationary Boolean models and extend results of Fallert on quermass densities to densities of mixed volumes. In particular, we present explicit formulae for a planar inhomogeneous Boolean model with circular grains.

[1]  R. Schneider.  Convex Bodies: The Brunn–Minkowski Theory: Minkowski addition , 1993 .

[2]  W. Weil.  On the mean shape of particle processes , 1997, Advances in Applied Probability.

[3]  Wolfgang Weil,et al.  Expectation formulas and isoperimetric properties for non‐isotropic Boolean models , 1988 .

[4]  Mixed Measures and Functionals of Translative Integral Geometry , 2001 .

[5]  D. Stoyan,et al.  Stochastic Geometry and Its Applications , 1989 .

[6]  Salvatore Torquato,et al.  Microstructure functions for a model of statistically inhomogeneous random media , 1997 .

[7]  H. Kellerer.  Minkowski functionals of Poisson processes , 1984 .

[8]  W. Weil.  Iterations of translative integral formulae and non-isotropic Poisson processes of particles , 1990 .

[9]  P. Davy.  Stereology - a statistical viewpoint , 1978, Bulletin of the Australian Mathematical Society.

[10]  J. Quintanilla,et al.  Clustering in a Continuum Percolation Model , 1997, Advances in Applied Probability.

[11]  I. Molchanov.  Statistics of the Boolean Model for Practitioners and Mathematicians , 1997 .

[12]  Stoyan,et al.  Stereological analysis and modelling of gradient structures , 1999, Journal of microscopy.

[13]  Wolfgang Weil,et al.  Densities for stationary random sets and point processes , 1984, Advances in Applied Probability.

[14]  R. E. Miles.  Estimating aggregate and overall characteristics from thick sections by transmission microscopy , 1976 .

[15]  Klaus Mecke,et al.  Integral Geometry in Statistical Physics , 1998 .

[16]  Wolfgang Weil,et al.  Intensity analysis of Boolean models , 1999, Pattern Recognit..

[17]  W. Weil.  The Estimation of mean shape and mean particle number in overlapping particle systems in the plane , 1995, Advances in Applied Probability.

[18]  G. Matheron.  Random Sets and Integral Geometry , 1976 .