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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: [email protected]
Jan Rataj
Affiliation:
Faculty of Mathematics and Physics, Charles University, Sokolovska 83, 186 75 Praha 8, Czech Republic. E-mail: [email protected]
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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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References

1Federer, H., Curvature measures. Trans. Amer. Math. Soc. 93 (1959), 418–491.CrossRefGoogle Scholar
2Federer, H., Geometric Measure Theory. Springer (Heidelberg, 1969).Google Scholar
3Gutkowski, P., Jensen, E. B. V. and Kiderlen, M., Directional analysis of digitized 3D images by configuration counts. J. Microsc. 216 (2004), 175–185.CrossRefGoogle ScholarPubMed
4Hall, P. and Molchanov, I., Corrections for systematic boundary effects in pixel-based area counts. Pattern Recog. 32 (1999), 1519–1528.CrossRefGoogle Scholar
5Hug, D., Contact distributions of Boolean models. Suppl. Rend. Circ. Mat. Palermo II 65 (2000), 137–181.Google Scholar
6Hug, D. and Last, G., On support measures in Minkowski spaces and contact distributions in stochastic geometry. Ann. Probab. 28 (2000), 796–850.CrossRefGoogle Scholar
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), 317–357.CrossRefGoogle Scholar
8Hug, D., Last, G. and Weil, W., Generalized contact distributions of inhomogeneous Boolean models. Adv. Appl. Probab. (SGSA) 34 (2002), 21–47.CrossRefGoogle Scholar
9Hug, D., Last, G. and Weil, W., A local Steiner-type formula for general closed sets and applications. Math. Z. 246 (2004), 237–272.CrossRefGoogle Scholar
10Kiderlen, M. and Jensen, E. B. V., Estimation of the directional measure of planar random sets by digitization. Adv. in Appl. Probab. 35 (2003), 583–602.Google Scholar
11Matheron, G., La formule de Steiner pour les érosions. J. Appl. Prob. 15 (1978), 126–135.CrossRefGoogle Scholar
12Molchanov, I., Statistics of the Boolean model for practitioners and mathematicians. Wiley Series in Probab. and Statist., Wiley (New York, 1997).Google Scholar
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), 247–257.Google Scholar
14Rataj, J., Determination of spherical area measures by means of dilation volumes. Math. Nachr. 235 (2002), 143–162.3.0.CO;2-7>CrossRefGoogle Scholar
15Rataj, J., On boundaries of unions of sets with positive reach. Beiträge Alg. Geom. 46 (2005), 397–404.Google Scholar
16Rataj, J. and Zähle, M., Curvatures and currents for unions of sets with positive reach, II. Ann. Global Anal. Geom. 20 (2001), 1–21.CrossRefGoogle Scholar
17Schneider, R., Additive Transformationen konvexer Körper. Geom. Dedicata 3 (1974), 221–228.CrossRefGoogle Scholar
18Schneider, R., Convex Bodies: the Brunn-Minkowski Theory. Encyclopedia of Mathematics and its Applications 44, Cambridge Univ. Press (Cambridge, 1993).CrossRefGoogle Scholar
19Schneider, R., On the mean normal measures of a particle process. Adv. Appl. Probab. 33 (2001), 25–38.CrossRefGoogle Scholar
20Schneider, R. and Weil, W., Stochastische Geometrie. Teubner (Stuttgart, 2000).CrossRefGoogle Scholar
21Stoyan, D., Kendall, W. S. and Mecke, J., Stochastic Geometry and its Applications (2nd edition). John Wiley (New York, 1995).Google Scholar
22Serra, J., Image Analysis and Mathematical Morphology. Academic Press (New York, 1982).Google Scholar
23Schütt, C. and Werner, E., The convex floating body. Math. Scand. 66 (1990), 275–290.CrossRefGoogle Scholar
24Zähle, M., Integral and current representation of Federer's curvature measures. Arch. Math. 46 (1986), 557–567.CrossRefGoogle Scholar
25Zähle, M., Curvatures and currents for unions of sets with positive reach. Geom. Dedicata 23 (1987), 155–171. CrossRefGoogle Scholar