Miles formulae for boolean models observed on lattices

Image Analysis and Stereology

Volumn 28, Issue 2, 2009, Pages 77-92

  Ohser, Joachim ^a   Nagel, Werner ^b   Schladitz, Katja ^c  

^a DARMSTADT UNIVERSITY OF APPLIED SCIENCES   (Germany)

^b FRIEDRICH SCHILLER UNIVERSITY JENA   (Germany)

^c FRAUNHOFER INSTITUTE FOR INDUSTRIAL MATHEMATICS ITWM   (Germany)

Author keywords

Boolean models; Digital topology; Image analysis; Miles' formulae; Minkowski functionals; Quermass densities; Volume images

Indexed keywords

EID: 77950184797     PISSN: 15803139     EISSN: 18545165     Source Type: Journal    
DOI: 10.5566/ias.v28.p77-92     Document Type: Article

References (30)

1. Blasquez I, Poiraudeau JF. (2003). Efficient processing of Minkowski functionals on a 3d binary image using binary decision diagrams. In: Journal ofWSCG, vol. 11 (1). WSCG, Plzen, Czech Republic: UNION Agency-Science Press.
2. Guderlei R, Klenk S, Mayer J, Schmidt V, Spodarev E. (2007). Algorithms for the computation of Minkowski functionals of deterministic and random polyconvex sets. Image Vision Comput 25:464-74.
3. Hadwiger H. (1957). Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Berlin: Springer-Verlag.
4. Hall P, Molchanov IS. (1999). Corrections for systematic boundary effects in pixel-based area counts. Pattern Recogn 32:1519-28.
5. Jernot J, Jouannot-Chesney P, Lantuejoul C. (2004). Local contributions to the Euler-Poincaré characteristic of a set. J Microsc 215:40-9.
6. Kiderlen M, Rataj J. (2006). On infinitesimal increase of volumes of morphological transforms. Mathematica 53:103-27.
7. Klenk S, Schmidt V, Spodarev E. (2006). A new algorithmic approach to the computation of Minkiwski functionals of germ-grain models. Comp Geom Th Appl 34:127- 48.
8. Klette R, Rosenfeld A. (2004). Digital Geometry. Amsterdam: Morgan & Kaufman Publ.
9. Lachaud JO, Montanvert A. (2000). Continuous analogs of digital boundaries: A topological approach to isosurfaces. Graph Models 62:129-64.
10. Lang C, Ohser J, Hilfer R. (2001). On the analysis of spatial binary images. J Microsc 203:303-13.
11. Lindblad J. (2005). Surface area estimation of digitized 3D objects using weighted local computations. Image Vision Comput 23:111-22.
12. Lindblad J, Nyström I. (2002). Surface area estimation of digitized 3D objects using local computations. In: 10th International Conference on Discrete Geometry for Computer Imagery, vol. 2301 of LNCS. DGCI, Bordeaux, France, Berlin, Heidelberg, New York: Springer.
13. Matheron G. (1975). Random Sets and Integral Geometry. New York, London: John Wiley & Sons.
14. Miles RE. (1976). Estimating aggregate and overall characteristics from thick selections by transmission microscopy. J Microsc 107:227-33.
15. Molchanov IS. (1997). Statistics of the Boolean Model for Practitioners and Mathematicans. New York: J Wiley & Sons.
16. Mrkvička T, Rataj J. (2008). On the estimation of intrinsic volume densities of stationary random closed sets. Stoch Proc Appl 118:213-31.
17. Mullikin J, Verbeek P. (1993). Surface estimation of digital planes. Bioimaging 1:6-16.
18. Nagel W, Ohser J, Pischang K. (2000). An integralgeometric approach for the Euler-Poincarč characteristic of spatial images. J Microsc 198:54-62.
19. Ohser J, Mücklich F. (2000). Statistical Analysis of Microstructures in Materials Science. Chichester, New York: J Wiley & Sons.
20. Ohser J, Nagel W, Schladitz K. (2002). The Euler number of discretized sets - on the choice of adjacency in homogeneous lattices. In: Mecke KR, Stoyan D, eds., Morphology of Condensed Matter, Lecture Notes in Physics. Berlin: Springer.
21. Ohser J, Nagel W, Schladitz K. (2003). The Euler number of discretised sets - surprising results in three dimensions. Image Anal Stereol 22:11-9.
22. Rataj J, Zähle M. (2003). Normal cycles of Lipschitz manifolds by approximation with parallel sets. Differ Geom Appl 19:113-26.
23. Schladitz K, Ohser J, Nagel W. (2006). Measurement of intrinsic volumes of sets observed on lattices. In: Kuba A, Nyul LG, Palagyi K, eds., 13th Int Conf Discrete Geom Computer Imagery, LNCS. DGCI, Szeged, Hungary, Berlin, Heidelberg, New York: Springer.
24. Schmidt V, Spodarev E. (2005). Joint estimators for the specific intrinsic volumes of stationary random sets. Stoch Proc Appl 115:959-81.
25. Schneider R. (1993). Convex Bodies: The Brunn-Minkowski Theory. Cambridge: Encyclopedia of Mathematics and Its Application Vol. 44, Cambridge University Press.
26. Schneider R, Weil W. (2008). Stochastic and Integral Geometry. Probability and Its Application. Heidelberg: Springer.
27. Serra J. (1982). Image Analysis and Mathematical Morphology, Vol. 1. London: Academic Press.
28. Wills JM. (1970). Zum Verhältnis von Volumen zu Oberfläche bei konvexen Körpern. Arch Math 21:557-60.
29. Windreich G, Kiryati N, Lohmann G. (2003). Surface area estimation in practice. In: Nyström I, di Baja GS, Svensson S, eds., 11th Int Conf Discrete Geom Computer Imagery, vol. 2886 of LNCS. DGCI, Naples, Italy, Berlin, Heidelberg, New York: Springer.
30. Ziegel J, Kiderlen M. (2009). Estimation of the surface area and surface areameasure of three-dimensional sets from digitizations. Image Vision Comput, in press.