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On Infinitesimal Increase of Volumes of Morphological Transforms

Part of: Classical measure theory Discrete geometry

Published online by Cambridge University Press:  21 December 2009

Markus Kiderlen  and
Jan Rataj
Markus Kiderlen
Affiliation:
Department of Mathematical Sciences, University of Aarhus, Ny Munkegade, DK-8000 Aarhus C, Denmark. E-mail: kiderlen@imf.au.dk
Jan Rataj
Affiliation:
Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 186 75 Praha 8, Czech Republic. E-mail: rataj@karlin.mff.cuni.cz
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Abstract

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 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.

MSC classification

Secondary: 28A75: Length, area, volume, other geometric measure theory 52C07: Lattices and convex bodies in $n$ dimensions
Type
Research Article
Information
Mathematika , Volume 53 , Issue 1 , June 2006 , pp. 103 - 127
Copyright
Copyright © University College London 2006

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