Abstract
Random closed sets (RACS) in the d—dimensional Euclidean space are considered, whose realizations belong to the extended convex ring. A family of nonparametric estimators is investigated for the simultaneous estimation of the vector of all specific Minkowski functionals (or, equivalently, the specific intrinsic volumes) of stationary RACS. The construction of these estimators is based on a representation formula for the expected local connectivity number of stationary RACS intersected with spheres, whose radii are small in comparison with the size of the whole sampling window. Asymptotic properties of the estimators are given for unboundedly increasing sampling windows. Numerical results are provided as well.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Preview
Unable to display preview. Download preview PDF.
References
J.-P. Chilès and P. Delfiner. Geostatistics: Modelling Spatial Uncertainty. J. Wiley & Sons, New York, 1999.
N. A. C. Cressie. Statistics for Spatial Data. J. Wiley & Sons, New York, 2nd edition, 1993.
A. V. Ivanov and N. N. Leonenko. Statistical Analysis of Random Fields. Kluwer, Dordrecht, 1989.
S. Klenk, J. Mayer, V. Schmidt, and E. Spodarev. Algorithms for the computation of Minkowski functionals of deterministic and random polyconvex sets. Preprint, 2005.
S. Klenk, V. Schmidt, and E. Spodarev. A new algorithmic approach for the computation of Minkowski functionals of polyconvex sets. Preprint, 2004. Submitted.
G. Matheron. Random Sets and Integral Geometry. J. Wiley & Sons, New York, 1975.
I. S. Molchanov. Statistics of the Boolean Model for Practitioners and Mathematicians. J. Wiley & Sons, Chichester, 1997.
I. S. Molchanov and S. A. Zuyev. Variational analysis of functionals of a Poisson process. Mathematics of Operations Research, 25:485–508, 2000.
W. G. Müller. Collecting Spatial Data. Physica-Verlag, Heidelberg, 2001.
J. Ohser and F. Mticklich. Statistical Analysis of Microstructures in Materials Science. J. Wiley & Sons, Chichester, 2000.
V. Schmidt and E. Spodarev. Joint estimators for the specific intrinsic volumes of stationary random sets. Preprint, 2004. To appear in Stochastic Processes and their Applications.
R. Schneider. Convex Bodies. The Brunn-Minkowski Theory. Cambridge University Press, Cambridge, 1993.
R. Schneider and W. Weil. Stochastische Geometrie. Teubner Skripten zur Mathematischen Stochastik. Teubner, Stuttgart, 2000.
J. Serra. The Boolean model and random sets. Computer Graphics and Image Processing, 12:99–126, 1980.
J. Serra. Image Analysis and Mathematical Morphology. Academic Press, London, 1982.
J. Serra. Image Analysis and Mathematical Morphology: Theoretical Advances, volume 2. Academic Press, London, 1988.
E. Spodarev. Isoperimetric problems and roses of neighborhood for stationary flat processes. Mathematische Nachrichten, 251(4):88–100, 2003.
D. Stoyan, W. S. Kendall, and J. Mecke. Stochastic Geometry and its Applications. J. Wiley & Sons, Chichester, 2nd edition, 1995.
H. Wackernagel. Multivariate Geostatistics. Springer, Berlin, 2nd edition, 1998.
Author information
Authors and Affiliations
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2005 Springer
About this paper
Cite this paper
Spodarev, E., Schmidt, V. (2005). On the Local Connectivity Number of Stationary Random Closed Sets. In: Ronse, C., Najman, L., Decencière, E. (eds) Mathematical Morphology: 40 Years On. Computational Imaging and Vision, vol 30. Springer, Dordrecht. https://doi.org/10.1007/1-4020-3443-1_31
Download citation
DOI: https://doi.org/10.1007/1-4020-3443-1_31
Publisher Name: Springer, Dordrecht
Print ISBN: 978-1-4020-3442-8
Online ISBN: 978-1-4020-3443-5
eBook Packages: Computer ScienceComputer Science (R0)