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On Infinitesimal Increase of Volumes of Morphological Transforms
Author(s) -
Kiderlen Markus,
Rataj Jan
Publication year - 2006
Publication title -
mathematika
Language(s) - English
Resource type - Journals
SCImago Journal Rank - 0.955
H-Index - 29
eISSN - 2041-7942
pISSN - 0025-5793
DOI - 10.1112/s002557930000005x
Subject(s) - mathematics , disjoint sets , mathematical analysis , geometry , measure (data warehouse) , combinatorics , discrete mathematics , computer science , database
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.