Determination of the mean normal measure from isotropic means of flat sections

Advances in Applied Probability

Volumn 34, Issue 3, 2002, Pages 505-519

  Kiderlen, Markus ^a  

^a UNIVERSITY OF KARLSRUHE   (Germany)

Author keywords

Local Crofton formula; Oriented mean normal measure; Rose of normal directions; Spherical harmonics; Spherical lifting

Indexed keywords

COMPUTATIONAL METHODS; HARMONIC ANALYSIS; INTEGRATION; MATHEMATICAL TRANSFORMATIONS; PROBABILITY;

SPHERICAL HARMONICS;

SET THEORY;

EID: 0036756316     PISSN: 00018678     EISSN: None     Source Type: Journal    
DOI: 10.1239/aap/1033662163     Document Type: Article

References (27)

1. Fallert, H., Goodey, P. and Weil, W. (1997). Spherical projections and centrally symmetric sets. Adv. Math. 129, 301-322.
2. Goodey, P. (1998). Minkowski sums of projections of convex bodies. Mathematika 45, 253-268.
3. Goodey, P. and Jiang, W. (2000). Minkowski sums of three dimensional projections of convex bodies. Rend. Circ. Mat. Palermo II Suppl. 65, 105-119.
4. Goodey, P. and Weil, W. (1998). The determination of convex bodies from the mean of random sections. Math. Proc. Camb. Phil. Soc. 112, 419-430.
5. Goodey, P., Kiderlen, M. and Weil, W. (1998). Section and projection means of convex bodies. Monatshefte Math. 126, 37-54.
6. Groemer, H. (1996). Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge University Press.
7. Hilliard, J.E. (1962). Specification and measurement of microstructural anisotropy. Trans. Metall. Soc. AIME 224, 1201-1211.
8. Hug, D. (2000). Contact distributions of Boolean models. Rend. Circ. Mat. Palermo II Suppl. 65, 137-181.
9. Hug, D. and Last, G. (2000). On support measures in Minkowski spaces and contact distributions in stochastic geometry. Ann. Prob. 28, 796-850.
10. Kiderlen, M. (1999). Schnittmittelungen und äquivariante Endomorphismen konvexer Körper. Doctoral Thesis, University of Karlsruhe.
11. Kiderlen, M. (2001). Non-parametric estimation of the directional distribution of stationary line and fibre processes. Adv. Appl. Prob. 33, 6-24.
12. Kiderlen, M. (2001). Blaschke and Minkowski endomorphisms of convex bodies. In preparation.
13. Mecke, J. and Stoyan, D. (1980). Formulas for stationary planar fibre processes I-general theory. Math Operationsforsch. Statist. Ser. Statist. 11, 267-279.
14. Miles, R.E. (1971). Poisson flats in Euclidean spaces. Part II: Homogeneous Poisson flats and the complementary theorem. Adv. Appl. Prob. 3, 1-43.
15. Molchanov, I.S. (1995) Statistics of the Boolean model: from the estimation of means to the estimation of distributions. Adv. Appl. Prob. 27, 63-86.
16. Pohlmann, S., Mecke, J. and Stoyan, D. (1981). Stereological formulas for stationary surface processes. Math. Operationsforsch. Statist. Ser. Statist. 12, 429-440.
17. Rataj, J. (1996). Estimation of oriented direction distribution of a planar body. Adv. Appl. Prob. 28, 394-404.
18. Schneider, R. (2001). On the mean normal measure of a particle process. Adv. Appl. Prob. 33, 25-38.
19. Schneider, R. and Weil, W. (1992). Integralgeometrie. Teubner, Stuttgart.
20. Schneider, R. and Weil, W. (2000). Stochastische Geometrie. Teubner, Stuttgart.
21. Stoyan, D., Kendall, W.S. and Mecke, J. (1995). Stochastic Geometry and Its Applications, 2nd edn. John Wiley, New York.
22. Weil, W. (1982). Zonoide und verwandte Klassen konvexer Körper. Monatshefte Math. 94, 73-84.
23. 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.
24. Weil, W. (1997). The mean normal distribution of stationary random sets and particle processes. In Advances in the Theory and Applications of Random Sets (Proc. Internat. Symp., Fontainebleau, 1996), ed. D. Jeulin, World Scientific, Singapore, pp. 21-33.
25. Weil, W. (1997). Mean bodies associated with random closed sets. Rend. Circ. Mat. Palermo II Suppl. 50, 387-412.
26. Weil, W. (1999). Intensity analysis of Boolean models. Pattern Recognition 32, 1675-1684.
27. Zähle, M. (1982). Random processes of Hansdorff rectifiable closed sets. Math. Nachr. 108, 49-72.