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Estimation of oriented direction distribution of a planar body

Part of: General convexity Geometric probability and stochastic geometry

Published online by Cambridge University Press:  01 July 2016

Jan Rataj
Jan Rataj*
Affiliation:
Charles University
*
∗ Postal address: Mathematical Institute, Charles University, Sokolovská 83, 186 00 Praha 8, Czech Republic.
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Abstract

Methods of estimation of the oriented direction distribution (i.e. the distribution of unit outer normals over the boundary) of a planar set from the convex ring are proposed. The methods are based on an estimation of the area dilation of the investigated set by chosen test sets.

Keywords

CONVEX RINGDIRECTION DISTRIBUTIONMIXED AREADILATIONEXOSKELETON

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 52A39: Mixed volumes and related topics
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 28 , Issue 2 , June 1996 , pp. 394 - 404
Copyright
Copyright © Applied Probability Trust 1996 

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Footnotes

Supported partially by the Grant Agency of Czech Republic, project No. 201/93/2172.

References

Coleman, R. (1989) Inverse problems. J. Microsc. 153, 233–248.Google Scholar
Federer, H. (1969) Geometric Measure Theory. Springer. Berlin.Google Scholar
Lešanovský, A. and Rataj, J. (1990) Determination of compact sets in Euclidean spaces by the volume of their dilation. In DIANA III. pp. 165–177. MÚCSAV, Praha.Google Scholar
Matern, B. (1989) Precision of area estimation. J. Microsc. 153, 269–284.Google Scholar
Molchanov, I. S. and Stoyan, D. (1994) Directional analysis of fibre processes related to Boolean models. Metrika 41, 183–199.Google Scholar
Nagel, W. (1991) Das geometrische Kovariogramm und verwandte Größen zweiter Ordnung. Habilitationsschrift, Friedrich-Schiller-Universität Jena.Google Scholar
Rataj, J. and Saxl, I. (1989) Analysis of planar anisotropy by means of the Steiner compact set. J. Appl. Prob. 26, 490–502.Google Scholar
Rataj, J. and Zähle, M. (1995) Mixed curvature measures and a translative integral formula for sets of positive reach. Geom. Dedicata 57, 259–283.Google Scholar
Schneider, R. (1974) Additive Transformationen konvexer Körper. Geom. Dedicata 3, 221–228.Google Scholar
Schneider, R. (1993) Convex Bodies: The Brunn-Minkowski Theory. Cambridge University Press, Cambridge.Google Scholar
Schneider, R. and Weil, W. (1986) Translative and kinematic integral formulae for curvature measures. Math. Nachricht. 129, 67–80.CrossRefGoogle Scholar
Serra, J. (1982) Mathematical Morphology and Image Analysis. Academic Press, London.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and Its Applications. Akademie, Berlin.Google Scholar
Weil, W. (1994) Support functions on the convex ring in the plane and support densities for random sets and point processes. Suppl. Rend. Circ. Mat. Palermo (Series II) 35, 323–344.Google Scholar
Weil, W. (1995). The estimation of mean shape and mean particle number in overlapping particle systems in the plane. Adv. Appl. Prob. 27, 102–119.Google Scholar