Let B (“black”) and W (“white”) be disjoint compact test sets in the d-dimensional Euclidean space and consider the volume of all its simultaneous shifts keeping B inside and W outside a compact set A. If the union of B and W is rescaled by a factor tending to zero, then the rescaled volume converges to a value determined by the surface area measure of A and the support functions of B and W, provided that A is regular enough (e.g. polyconvex). An analogous formula is obtained for the case when the conditions "B in A" and "W in complement(A)" are replaced with prescribed threshold volumes of B in A and W in the complement of A. Applications in stochastic geometry are discussed. Firstly, the hit distribution function of a random set with an arbitrary compact structuring element B is considered. Its derivative at 0 is expressed in terms of the rose of directions and B. An analogue result holds for the hit-or-miss function. Secondly, in a design based setting, different random digitizations of a deterministic set A are treated. It is shown how the number of configurations in such a digitization is related to the surface area measure of A as the lattice distance converges to zero.
Mathematika, 2007, Vol 53, p. 103-127
Surface area measure, dilation, erosion, hit-or-miss transform, volumethreshold,