Skip to main content
    • Aa
    • Aa
  • Get access
    Check if you have access via personal or institutional login
  • Cited by 9
  • Cited by
    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.


    Svane, Anne Marie 2014. Estimation of Intrinsic Volumes from Digital Grey-Scale Images. Journal of Mathematical Imaging and Vision, Vol. 49, Issue. 2, p. 352.

    Svane, Anne Marie 2014. On Multigrid Convergence of Local Algorithms for Intrinsic Volumes. Journal of Mathematical Imaging and Vision, Vol. 49, Issue. 1, p. 148.

    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.

    Ambrosio, Luigi Colesanti, Andrea and Villa, Elena 2008. Outer Minkowski content for some classes of closed sets. Mathematische Annalen, Vol. 342, Issue. 4, p. 727.


On Infinitesimal Increase of Volumes of Morphological Transforms

  • Markus Kiderlen (a1) and Jan Rataj (a2)
  • DOI:
  • Published online: 21 December 2009

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.

Linked references
Hide All

This list contains references from the content that can be linked to their source. For a full set of references and notes please see the PDF or HTML where available.

1H. Federer , Curvature measures. Trans. Amer. Math. Soc. 93 (1959), 418491.

3P. Gutkowski , E. B. V. Jensen and M. Kiderlen , Directional analysis of digitized 3D images by configuration counts. J. Microsc. 216 (2004), 175185.

4P. Hall and I. Molchanov , Corrections for systematic boundary effects in pixel-based area counts. Pattern Recog. 32 (1999), 15191528.

8D. Hug , G. Last and W. Weil , Generalized contact distributions of inhomogeneous Boolean models. Adv. Appl. Probab. (SGSA) 34 (2002), 2147.

9D. Hug , G. Last and W. Weil , A local Steiner-type formula for general closed sets and applications. Math. Z. 246 (2004), 237272.

10M. Kiderlen and E. B. V. Jensen , Estimation of the directional measure of planar random sets by digitization. Adv. in Appl. Probab. 35 (2003), 583602.

11G. Matheron , La formule de Steiner pour les érosions. J. Appl. Prob. 15 (1978), 126135.

14J. Rataj , Determination of spherical area measures by means of dilation volumes. Math. Nachr. 235 (2002), 143162.

16J. Rataj and M. Zähle , Curvatures and currents for unions of sets with positive reach, II. Ann. Global Anal. Geom. 20 (2001), 121.

17R. Schneider , Additive Transformationen konvexer Körper. Geom. Dedicata 3 (1974), 221228.

19R. Schneider , On the mean normal measures of a particle process. Adv. Appl. Probab. 33 (2001), 2538.

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

24M. Zähle , Integral and current representation of Federer's curvature measures. Arch. Math. 46 (1986), 557567.

25M. Zähle , Curvatures and currents for unions of sets with positive reach. Geom. Dedicata 23 (1987), 155171.

Recommend this journal

Email your librarian or administrator to recommend adding this journal to your organisation's collection.

  • ISSN: 0025-5793
  • EISSN: 2041-7942
  • URL: /core/journals/mathematika
Please enter your name
Please enter a valid email address
Who would you like to send this to? *