Skip to main content
    • Aa
    • Aa

On Infinitesimal Increase of Volumes of Morphological Transforms

  • Markus Kiderlen (a1) and Jan Rataj (a2)

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.

Hide All
1Federer H., Curvature measures. Trans. Amer. Math. Soc. 93 (1959), 418491.
2Federer H., Geometric Measure Theory. Springer (Heidelberg, 1969).
3Gutkowski P., Jensen E. B. V. and Kiderlen M., Directional analysis of digitized 3D images by configuration counts. J. Microsc. 216 (2004), 175185.
4Hall P. and Molchanov I., Corrections for systematic boundary effects in pixel-based area counts. Pattern Recog. 32 (1999), 15191528.
5Hug D., Contact distributions of Boolean models. Suppl. Rend. Circ. Mat. Palermo II 65 (2000), 137181.
6Hug D. and Last G., On support measures in Minkowski spaces and contact distributions in stochastic geometry. Ann. Probab. 28 (2000), 796850.
7Hug D., Last G. and Weil W., A survey on contact distributions. Morphology of Condensed Matter. Physics and Geometry of Spatially Complex Systems. Lecture Notes in Physics (Mecke K. and Stoyan D., eds.), 600 Springer (Berlin, 2002), 317357.
8Hug D., Last G. and Weil W., Generalized contact distributions of inhomogeneous Boolean models. Adv. Appl. Probab. (SGSA) 34 (2002), 2147.
9Hug D., Last G. and Weil W., A local Steiner-type formula for general closed sets and applications. Math. Z. 246 (2004), 237272.
10Kiderlen M. and Jensen E. B. V., Estimation of the directional measure of planar random sets by digitization. Adv. in Appl. Probab. 35 (2003), 583602.
11Matheron G., La formule de Steiner pour les érosions. J. Appl. Prob. 15 (1978), 126135.
12Molchanov I., Statistics of the Boolean model for practitioners and mathematicians. Wiley Series in Probab. and Statist., Wiley (New York, 1997).
13Molchanov I., Grey-scale images and random sets. In Mathematical Morphology and its Applications to Image and Signal Processing (Heijmans H. and Roerdink J., eds.), Kluwer Acad. Publ. (Amsterdam, 1998), 247257.
14Rataj J., Determination of spherical area measures by means of dilation volumes. Math. Nachr. 235 (2002), 143162.
15Rataj J., On boundaries of unions of sets with positive reach. Beiträge Alg. Geom. 46 (2005), 397404.
16Rataj J. and Zähle M., Curvatures and currents for unions of sets with positive reach, II. Ann. Global Anal. Geom. 20 (2001), 121.
17Schneider R., Additive Transformationen konvexer Körper. Geom. Dedicata 3 (1974), 221228.
18Schneider R., Convex Bodies: the Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications 44, Cambridge Univ. Press (Cambridge, 1993).
19Schneider R., On the mean normal measures of a particle process. Adv. Appl. Probab. 33 (2001), 2538.
20Schneider R. and Weil W., Stochastische Geometrie. Teubner (Stuttgart, 2000).
21Stoyan D., Kendall W. S. and Mecke J., Stochastic Geometry and its Applications (2nd edition). John Wiley (New York, 1995).
22Serra J., Image Analysis and Mathematical Morphology. Academic Press (New York, 1982).
23Schütt C. and Werner E., The convex floating body. Math. Scand. 66 (1990), 275290.
24Zähle M., Integral and current representation of Federer's curvature measures. Arch. Math. 46 (1986), 557567.
25Zähle M., 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? *



Full text views

Total number of HTML views: 0
Total number of PDF views: 5 *
Loading metrics...

Abstract views

Total abstract views: 61 *
Loading metrics...

* Views captured on Cambridge Core between September 2016 - 24th October 2017. This data will be updated every 24 hours.