On the empty space function of some germ-grain models

We derive and discuss formulas for the density and the hazard rate of the e mpty space function of a germ-grain model Xi in R-d generated by a stationa ry point process Phi and i.i.d. convex primary grains Xi(n),, n is an eleme nt of N, that are independent of Phi. Our formulas are based on the Palm pr obability of the germ process and the mean generalized curvature measure of the grain. Particular attention is paid to cluster models, where the grain s form a Poisson cluster process. Our discussion of specific Gauss-Poisson models with spherical grains provides some motivation for the use of the fa ilure rate of F to detect clustering effects. In the general case we propos e a family of functions comparing the behaviour in the neighbourhood of a t ypical germ with the neighbourhood of an arbitrary point in space. These ch aracteristics can be used to measure effects of clustering and spatial inte ractions between the locations of the individual grains. (C) 1999 Pattern R ecognition Society. Published by Elsevier Science Ltd. All rights reserved.