Intensity analysis of Boolean models

We consider the problem to estimate the intensity of a stationary, but not necessarily isotropic Boolean model Z in d-dimensional space from the observation of global and local functionals. After a short description of the recent solution in the planar case, we show that the spatial case can be solved in a similar way by using an iterated translative integral formula for support functions.

[1]  Translative and kinematic integral formulae for support functions , 1995 .

[2]  K. Ball.  CONVEX BODIES: THE BRUNN–MINKOWSKI THEORY , 1994 .

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

[4]  Albrecht M. Kellerer,et al.  On the number of clumps resulting from the overlap of randomly placed figures in a plane , 1983, Journal of Applied Probability.

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

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

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

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

[9]  W. Nagel,et al.  The estimation of the Euler‐Poincare characteristic from observations on parallel sections , 1996 .

[10]  Estimation of oriented direction distribution of a planar body , 1996, Advances in Applied Probability.

[11]  Ilya S. Molchanov,et al.  Statistics of the Boolean model: from the estimation of means to the estimation of distributions , 1995, Advances in Applied Probability.

[12]  S. Campi,et al.  Approximation of zonoids by zonotopes in fixed directions , 1994, Discret. Comput. Geom..

[13]  Crofton's formula generalized to projected thick sections , 1981 .

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

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

[16]  Paul Goodey,et al.  Spherical Projections and Centrally Symmetric Sets , 1997 .

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

[18]  P. Davy.  Projected thick sections through multi-dimensional particle aggregates , 1976, Journal of Applied Probability.

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

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

[21]  Michel Schmitt.  Estimation of the density in a stationary Boolean model , 1991 .

[22]  Section and projection means of convex bodies , 1998 .

[23]  H. Groemer.  On translative integral geometry , 1977 .