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Markus Kiderlen

On infinitesimal increase of volumes of morphological transforms

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  • Department of Mathematical Sciences

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.

Original languageEnglish
Pages (from-to)103-127
Number of pages25
Publication statusPublished - 2007

    Research areas

  • Surface area measure, dilation, erosion, hit-or-miss transform, volumethreshold,

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