On Infinitesimal Increase of Volumes of Morphological Transforms

Let B (“black”) and W (“white”) be disjoint compact test sets in ℝ d , and consider the volume of all its simultaneous shifts keeping B inside and W outside a compact set A ⊂ ℝ d . If the union B ∪ 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 ⊂ A and W ⊂ A C are replaced by prescribed threshold volumes of B in A and W in A C . Applications in stochastic geometry are discussed. First, 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 analogous result holds for the hit‐or‐miss function. Second, 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.