Hostname: page-component-78c5997874-g7gxr Total loading time: 0 Render date: 2024-11-10T17:01:05.115Z Has data issue: false hasContentIssue false
  • English
  • Français

Local Digital Estimators of Intrinsic Volumes for Boolean Models and in the Design-Based Setting

Part of: Communication, information Geometric probability and stochastic geometry

Published online by Cambridge University Press:  22 February 2016

Anne Marie Svane
Anne Marie Svane*
Affiliation:
Aarhus University
*
∗ Postal address: Department of Mathematics, Aarhus University, 8000 Aarhus C, Denmark. Email address: anne.marie.svane@gmail.com
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In order to estimate the specific intrinsic volumes of a planar Boolean model from a binary image, we consider local digital algorithms based on weighted sums of 2×2 configuration counts. For Boolean models with balls as grains, explicit formulas for the bias of such algorithms are derived, resulting in a set of linear equations that the weights must satisfy in order to minimize the bias in high resolution. These results generalize to larger classes of random sets, as well as to the design-based situation, where a fixed set is observed on a stationary isotropic lattice. Finally, the formulas for the bias obtained for Boolean models are applied to existing algorithms in order to compare their accuracy.

Keywords

Digitization in 2Dintrinsic volumelocal estimatorconfigurationBoolean modeldesign-based digitization

MSC classification

Primary: 94A08: Image processing (compression, reconstruction, etc.)
Secondary: 60D05: Geometric probability and stochastic geometry
Type
Research Article
Information
Advances in Applied Probability , Volume 46 , Issue 1 , March 2014 , pp. 35 - 58
Copyright
© Applied Probability Trust 

References

Bieri, H. and Nef, W. (1984). Algorithms for the Euler characteristic and related additive functionals of digital objects. Comput. Vision Graphics Image Process. 28, 166–175.CrossRefGoogle Scholar
Dorst, L. and Smeulders, A. W. M. (1987). Length estimators for digitized contours. Comput. Vision Graphics Image Process. 40, 311–333.CrossRefGoogle Scholar
Freeman, H. (1970). Boundary encoding and processing. In Picture Processing and Psychopictorics, eds Lipkin, B. S. and Rosenfeld, A., Academic Press, New York, pp. 241–266.Google Scholar
Hug, D., Last, G. and Weil, W. (2004). A local Steiner-type formula for general closed sets and applications. Math. Z. 246, 237–272.CrossRefGoogle Scholar
Kampf, J. (2012) A limitation of the estimation of intrinsic volumes via pixel configuration counts. WiMa Rep. 144, University of Kaiserslautern.Google Scholar
Kampf, J. and Kiderlen, M. (2013). Large parallel volumes of finite and compact sets in d-dimensional Euclidean space. Documenta Math. 18, 275–295.CrossRefGoogle Scholar
Kiderlen, M. and Rataj, J. (2006). On infinitesimal increase of volumes of morphological transforms. Mathematika 53, 103–127.CrossRefGoogle Scholar
Kiderlen, M. and Vedel Jensen, E. B. (2003). Estimation of the directional measure of planar random sets by digitization. Adv. Appl. Prob. 35, 583–602.CrossRefGoogle Scholar
Klette, R. and Rosenfeld, A. (2004). Digital Geometry. Morgan Kaufmann, San Fransisco, CA.Google Scholar
Ohser, J. and Mücklich, F. (2000). Statistical Analysis of Microstructures in Materials Science. John Wiley, New York.Google Scholar
Ohser, J., Nagel, W. and Schladitz, K. (2003). The Euler number of discretised sets—surprising results in three dimensions. Image Anal. Stereol. 22, 11–19.CrossRefGoogle Scholar
Ohser, J., Nagel, W. and Schladitz, K. (2009). Miles formulae for Boolean models observed on lattices. Image Anal. Stereol. 28, 77–92.CrossRefGoogle Scholar
Schneider, R. (1993). Convex Bodies: The Brunn–Minkowski Theory (Encyclopedia Math. Appl. 44). Cambridge University Press.CrossRefGoogle Scholar
Schneider, R. and Weil, W. (2008). Stochastic and Integral Geometry. Springer, Berlin.CrossRefGoogle Scholar
Stelldinger, P. and Köthe, U. (2005). Shape preserving digitization of binary images after blurring. In Discrete Geometry for Computer Iimagery (Lecture Notes Comput. Sci. 3429), Springer, Berlin, pp. 383–391.CrossRefGoogle Scholar