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On Infinitesimal Increase of Volumes of Morphological Transforms

  • Markus Kiderlen (a1) and Jan Rataj (a2)

Let B (“black”) and W (“white”) be disjoint compact test sets in ℝd, and consider the volume of all its simultaneous shifts keeping B inside and W outside a compact set A ⊂ ℝd. If the union BW 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 BA and WAC 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.

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18 R. Schneider , Convex Bodies: the Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications 44, Cambridge Univ. Press (Cambridge, 1993).

20 R. Schneider and W. Weil , Stochastische Geometrie. Teubner (Stuttgart, 2000).

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  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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