STEREOLOGICAL FORMULAE FOR SIZE DISTRIBUTIONS VIAMARKED POINT PROCESSES
Dietrich Stoyan
Abstract: This paper deals with the intersection of a non-random -dimensional flat with a
subset of which is the union of infinitely many particles. It is described by a stationary
marked point process , where the are the positions and the determine the
forms of particles. Then the intersection set can be described similarly by a stationary
marked point process . The intensity and the Palm mark distribution of can be
expressed in terms of the corresponding characteristics of . The formulae generalize
well-known formulae for the Boolean model (i.e. independently marked Poisson process)
to a quite general case. Furthermore, they are similar or equivalent to those for
cross-sections of compact sets by Poisson fiat processes. Since the formulae connect
-dimensional characteristics with -dimensional ones, they are of stereological interest.
-dimensional flat with a
subset of 
which is the union of infinitely many particles. It is described by a stationary
marked point process 
, where the 
are the positions and the 
determine the
forms of particles. Then the intersection set can be described similarly by a stationary
marked point process 
. The intensity and the Palm mark distribution of 
can be
expressed in terms of the corresponding characteristics of 
. The formulae generalize
well-known formulae for the Boolean model (i.e. independently marked Poisson process)
to a quite general case. Furthermore, they are similar or equivalent to those for
cross-sections of compact sets by Poisson fiat processes. Since the formulae connect
-dimensional characteristics with 
-dimensional ones, they are of stereological interest.