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    This article has been cited by the following publications. This list is generated based on data provided by CrossRef.

    Svane, Anne Marie 2015. Local Digital Algorithms for Estimating the Integrated Mean Curvature of r-Regular Sets. Discrete & Computational Geometry, Vol. 54, Issue. 2, p. 316.


    Svane, Anne Marie 2015. Asymptotic variance of grey-scale surface area estimators. Advances in Applied Mathematics, Vol. 62, p. 41.


    Kampf, Jürgen 2014. A LIMITATION OF THE ESTIMATION OF INTRINSIC VOLUMES VIA PIXEL CONFIGURATION COUNTS. Mathematika, Vol. 60, Issue. 02, p. 485.


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    Svane, Anne Marie 2014. Local Digital Estimators of Intrinsic Volumes for Boolean Models and in the Design-Based Setting. Advances in Applied Probability, Vol. 46, Issue. 01, p. 35.


    2013. Stochastic Geometry and its Applications.


    Ballani, Felix 2011. Multiple-point hit distribution functions and vague convergence of related measures. Mathematische Nachrichten, Vol. 284, Issue. 8-9, p. 938.


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On Infinitesimal Increase of Volumes of Morphological Transforms

  • Markus Kiderlen (a1) and Jan Rataj (a2)
  • DOI: http://dx.doi.org/10.1112/S002557930000005X
  • Published online: 21 December 2009
Abstract
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 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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Mathematika
  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
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