Research Article

On Infinitesimal Increase of Volumes of Morphological Transforms

Markus Kiderlena1 and Jan Rataja2

a1 Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark. E-mail:

a2 Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 186 75 Praha 8, Czech Republic. E-mail:


Let B (“black”) and W (“white”) be disjoint compact test sets in d, and consider the volume of all its simultaneous shifts keeping B inside and W outside a compact set A 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 AC are replaced by prescribed threshold volumes of B in A and W in AC. 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.

(Received November 08 2005)

MSC 2000

  • Primary, 28A75;
  • Secondary, 52C07, 60D05


Dedicated to Rolf Schneider on the occasion of his 65th birthday