Intensity analysis of Boolean models

Pattern Recognition

Volumn 32, Issue 9, 1999, Pages 1675-1684

  Weil, Wolfgang ^a  

^a UNIVERSITY OF KARLSRUHE   (Germany)

Author keywords

Boolean model; Convex ring; Euler characteristic; Intensity; Poisson process; Stationarity; Support function; Surface area measure

Indexed keywords

BOOLEAN FUNCTIONS; MATHEMATICAL MODELS; POISSON DISTRIBUTION; PROBABILITY DISTRIBUTIONS; RANDOM PROCESSES; SET THEORY;

BOOLEAN MODEL; CONVEX RING; EULER CHARACTERISTICS; INTENSITY; POISSON PROCESS; STATIONARITY; SUPPORT FUNCTION; SURFACE AREA MEASURE;

PATTERN RECOGNITION;

EID: 0032676930     PISSN: 00313203     EISSN: None     Source Type: Journal    
DOI: 10.1016/S0031-3203(99)00030-8     Document Type: Article

References (32)

1. [1] G. Matheron, Random Sets and Integral Geometry, Wiley, New York, 1975.
2. [2] D. Stoyan, W.S. Kendall, J. Mecke, Stochastic Geometry and Its Applications, Second ed., Wiley, New York, 1995.
3. [3] I.S. Molchanov, Statistics of the Boolean model for Practitioners and Mathematicians, Wiley, New York, 1997.
4. [4] R. Schneider, Convex Bodies: the Brunn-Minkowski Theory, Cambridge Univ. Press, Cambridge, 1993.
5. [5] R. Schneider, W. Weil, Integralgeometrie, Teubner, Stuttgart, 1992.
6. [6] I.S. Molchanov, Statistics of the Boolean model: from the estimation of means to the estimation of distributions, Adv. Appl. Probab. 27 (1995) 63-86.
7. [7] M. Schmitt, Estimation of the density in a stationary Boolean model, J. Appl. Probab. 28 (1991) 702-708.
8. [8] W. Weil, The estimation of mean shape and mean particle number in overlapping particle systems in the plane, Adv. Appl. Probab. 27 (1995) 102-119.
9. [9] W. Weil, J.A. Wieacker, Densities for stationary random sets and point processes, Adv. Appl. Probab. 16 (1984) 324-346.
10. [10] W. Weil, Densities of quermassintegrals for stationary random sets, in: R.V. Ambartzumian, W. Weil (Eds.), Stochastic Geometry, Geometric Statistics, Stereology, Teubner, Leipzig, 1984, pp. 233-247.
11. [11] R.E. Miles, Estimating aggregate and overall characteristics from thick sections by transmission microscopy, J. Microsc. 107 (1976) 227-233.
12. [12] P. Davy, Projected thick sections through multidimensional particle aggregates, J. Appl. Probab. 13 (1976) 714-722. Correction: J. Appl. Probab. 15 (1978) 456.
13. [12] P. Davy, Projected thick sections through multidimensional particle aggregates, J. Appl. Probab. 13 (1976) 714-722. Correction: J. Appl. Probab. 15 (1978) 456.
14. [13] P. Davy, Stereology - a statistical viewpoint, Thesis, Australian National Univ., Canberra, 1978.
15. [14] A.M. Kellerer, On the number of clumps resulting from the overlap of randomly placed figures in the plane, J. Appl. Probab. 20 (1983) 126-135.
16. [15] H.G. Kellerer, Minkowski functionals of Poisson processes, Z. Wahrscheinlichkeitstheorie verw. Gebiete 67 (1984) 63-84.
17. [16] M. Zähle, Curvature mesures and random sets II, Z. Wahrscheinlichkeitstheorie verw. Gebiete 71 (1986) 37-58.
18. [17] W. Weil, Expectation formulas and isoperimetric properties for non-isotropic Boolean models, J. Microsc. 151 (1988) 235-245.
19. [18] W. Weil, Iterations of translative integral formulae and non-isotropic Poisson processes of particles, Math. Z. 205 (1990) 531-549.
20. [19] W. Weil, Translative integral geometry, in: A. Hübler, W. Nagel, B.D. Ripley, G. Werner (Eds.), Geobild '89, Akademie-Verlag, Berlin, 1989, pp. 75-86.
21. [20] W. Weil, Lectures on translative integral geometry and stochastic geometry of anisotropic random geometric structures, Atti del Primo Convegno Italiano di Geometria Integrale, Rend. Sem. Mat. Messina, II. Ser. 13 (1990) 79-97.
22. [21] W. Weil, Mean bodies associated with random closed sets, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 50 (1997) 387-412.
23. [22] W. Weil, Support functions in the plane and densities of random sets and point processes, Suppl. Rend. Circ. Mat. Palermo, II. Ser. 35 (1994) 323-344.
24. [23] W. Weil, The mean normal distribution of stationary random sets and particle processes, in: D. Jeulin (Ed.), Advances in Theory and Applications of Random Sets, Proceedings of the International Symposium, October 9-11, 1996. Fontainebleau, France, World Scientific, Singapore, 1997. pp. 21-33.
25. [24] W. Weil, Translative and kinematic integral formulae for support functions, Geometriae Dedicata 57 (1995) 91-103.
26. [25] W. Weil, Translative and kinematic integral formulae for support functions II, in preparation.
27. [26] J. Rataj, Estimation of oriented direction distribution of a planar body, Adv. Appl. Probab. (SGSA) 28 (1996) 394-404.
28. [27] P. Goodey, M. Kiderlen, W. Weil, Section and projection means of convex bodies, Monatsh. Math. 126 (1998) 37-54.
29. [28] H. Fallert, P. Goodey, W. Weil, Spherical projections and centrally symmetric sets, Adv. Math. 129 (1997) 301-322.
30. [29] S. Campi, D. Haas, W. Weil, Approximation of zonoids by zonotopes in fixed directions, Discrete Comput. Geom. 11 (1994) 419-431.
31. [30] R. Schneider, Crofton's formula generalized to projected thick sections, Rend. Circ. Mat. Palermo, II. Ser. 30 (1981) 157-160.
32. [31] J. Ohser, W. Nagel, The estimation of the Euler-Poincaré characteristic from observations on parallel sections, J. Microsc. 184 (1996) 117-126.