On the second derivative of the spherical contact distribution function of smooth grain models

Probability Theory and Related Fields

Volumn 121, Issue 1, 2001, Pages 49-72

  Last, G ^a   Schassberger, R ^b  

^a UNIVERSITY OF KARLSRUHE   (Germany)

^b TECHNICAL UNIVERSITY OF BRAUNSCHWEIG   (Germany)

Author keywords

Boolean model; Curvature measures; Grain model; Palm probability; Random measure; Random set; Spherical contact distribution; Stochastic geometry; Surface measure

Indexed keywords

EID: 0035636115     PISSN: 01788051     EISSN: None     Source Type: Journal    
DOI: 10.1007/PL00008797     Document Type: Article

References (22)

1. Chiu, S.N., Stoyan, D.: Estimators of distance distributions for spatial patterns, Stat. Neerl., 52, 239-246 (1998)
2. Eder, G., Janesitch-Kriegl, H.: Crystallization. In: Material Science and Technology., Vol. 18 (R.W. Cahn, P. Haasen, and E.J. Kramer, eds), Verlag-Chemie, Weinheim, 269-342 (1997)
3. Federer, H.: Geometric Measure Theory. Springer, Berlin (1969)
4. Hansen, M.B., Gill, R.D., Baddeley, A.J.: First contact distributions for spatial patterns: regularity and estimation, Adv. Appl. Probab., 31, 15-33 (1999)
5. Heinrich, L.: Asymptotic properties of minimum contrast estimators for parameters of Boolean models, Metrika 40, 67-94 (1993)
6. Hewitt, E., Stromberg, K.: Real and Abstract Analysis, Springer, Berlin (1969)
7. Hug, D.: Generalized curvature measures and singularities of sets with positive reach, Forum Math., 10, 699-728 (1998)
8. Hug, D.: Contact distributions of Boolean models, Preprint, Universität Freiburg (1999)
9. Hug, D., Last, G.: On support measures in Minkowski spaces and contact distributions in stochastic geometry, Annals of Probability 28, 796-850 (2000a)
10. Hug, D., Last, G.: On generalized curvatures of sets in the extended convex ring with applications to stochastic geometry, in preparation (2000b)
11. Last, G., Schassberger, R.: On the distribution of the spherical contact vector of stationary grain models, Adv. Appl. Probab., 30, 36-52 (1998)
12. Last, G., Holtmann, M.: On the empty space function of some germ-grain models, Pattern Recognition 32, 1587-1600 (1999)
13. Matheron, G.: Random Sets and Integral Geometry, Wiley, New York (1975)
14. Mecke, K.: Integralgeometrie in der Statistischen Physik, Verlag Harri Deutsch, Frankfurt am Main (1994)
15. Parthasarathy, K.R.: Probability Measures on Metric Spaces, Academic Press, New York (1967)
16. Rataj, J., Zähle, M.: Curvatures and currents for unions of sets with positive reach II, Preprint (1999)
17. Schneider, R.: Parallelmengen mit Vielfachheit und Steiner-Formeln, Geom. Dedicata., 9, 111-127 (1980)
18. Stoyan, D., Kendall, W.S., Mecke, J.: Stochastic Geometry and Its Applications, Second Edition, Wiley, Chichester (1995)
19. Weil, W.: Mean bodies associated with random closed sets, Rend. Circ. Mat. Palermo., II. Ser., Suppl. 50, 387-412 (1997)
20. Weil, W.: Intensity analysis of Boolean models, Pattern Recognition 32, 1675-1684 (1999)
21. Weil, W., Wieacker, J.A.: Densities for stationary random sets and point processes, Adv. Appl. Probab., 16, 324-346 (1984)
22. Zähle, M.: Integral and current representation of Federer's curvature measures, Arch. Math., 46, 557-567 (1986)