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Point processes of cylinders, particles and flats

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Abstract

Point processesX of cylinders, compact sets (particles), or flats inR d are mathematical models for fields of sets as they occur, e.g., in practical problems of image analysis and stereology. For the estimation of geometric quantities of such fields, mean value formulas forX are important. By a systematic approach, integral geometric formulas for curvature measures are transformed into density formulas for geometric point processes. In particular, a number of results which are known for stationary and isotropic Poisson processes of convex sets are generalized to nonisotropic processes, to non-Poissonian processes, and to processes of nonconvex sets. The integral geometric background (including recent results from translative integral geometry), the fundamentals of geometric point processes, and the resulting density formulas are presented in detail. Generalizations of the theory and applications in image analysis and stereology are mentioned shortly.

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References

  • Ambartzumian R. V. (1974) ‘Convex Polygons and Random Tessellations’, in E. F. Harding and D. G. Kendall (eds.),Stochastic Geometry, Wiley, London, pp. 176–191.

    Google Scholar 

  • Ambartzumian R. V. (1977) ‘Stochastic Geometry from the Standpoint of Integral Geometry’,Adv. Appl. Prob. 9, 792–823.

    Google Scholar 

  • Ambartzumian R. V. (1982)Combinatorial Integral Geometry, Wiley, Chichester.

    Google Scholar 

  • Cowan R. (1980) ‘Properties of Ergodic Random Mosaic Processes’,Math. Nachr. 97, 89–102.

    Google Scholar 

  • Davy P. (1976) ‘Projected Thick Sections through Multidimensional Particle Aggregates’,J. Appl. Prob. 13, 714–722. Correction:J. Appl. Prob. 15, 456 (1978).

    Google Scholar 

  • Davy, P. (1978) ‘Stereology—a Statistical Viewpoint’, Thesis, Australian National Univ., Canberra.

  • De Hoff R. T. (1978), ‘Stereological Uses of the Area Tangent Count’, in R. E. Miles and J. Serra (eds),Geometrical Probability and Biological Structures, Lect. Notes Biomath.23, Springer, Berlin, pp. 99–113.

    Google Scholar 

  • Fava N. A. and Santaló L. A. (1978) ‘Plate and Line Segment Processes’,J. Appl. Prob. 15, 494–501.

    Google Scholar 

  • Fava N. A. and Santaló L. A. (1979) ‘Random Processes of Manifolds inR n ’,Z. Wahrscheinlich-keitstheorie verw. Gebiete 50, 85–96.

    Google Scholar 

  • Goodey, P. and Weil, W. (1986) ‘Translative Integral Formulas for Convex Bodies’,Aequationes Mathematicae (to appear).

  • Gundersen H. J. (1978) ‘Estimators of the Number of Objects per Area Unbiased by Edge Effects’,Microseapia Acta 81, 107–117.

    Google Scholar 

  • Jensen E. B. and Sundberg R. (1985) ‘On Edge Effect in Planar Sampling’,Acta Stereologica 4, 89.

    Google Scholar 

  • Jensen E. B. and Sundberg R. (1986) ‘Generalized Associated Point Methods for Sampling Planar Objects’,J. Microscopy 144, 55–70.

    Google Scholar 

  • Kellerer A. M. (1985) ‘Counting Figures in Planar Random Configurations’,J. Appl. Prob. 22, 68–81.

    Google Scholar 

  • Kellerer H. G. (1984) ‘Minkowski Functionals of Boisson Processes’,Z. Wahrscheinlichkeitsthearie verw. Gebiete 67, 63–64.

    Google Scholar 

  • Matheron G. (1975)Random Sets and Integral Geometry, Wiley, New York.

    Google Scholar 

  • Matheron G. (1976) ‘La formule de Crofton pour les sections épaisses’,J. Appl. Prob. 13, 707–713.

    Google Scholar 

  • McMullen P. and Schneider R. (1983) ‘Valuations on Convex Bodies’, in P. Gruber and J. M. Wills (eds),Convexity and Its Applications, Birkhäuser, Basle, pp. 170–247.

    Google Scholar 

  • Mecke J. (1980) ‘Palm Methods for Stationary Random Mosaics’, in R. V. Ambartzumian (ed.),Combinatorial Principles in Stochastic Geometry, Armenian Academy of Sciences, Yerevan, pp. 124–132.

    Google Scholar 

  • Mecke J. (1981a) ‘Stereological Formulas for Manifold Processes’,Probab, Math, Statist. 2, 31–35.

    Google Scholar 

  • Mecke J. (1981b) ‘Formulas for Stationary Planar Fibre Processes III—Intersections with Fibre Systems’,Math, Operationsforsch. Statist., Ser. Statist. 12, 201–210.

    Google Scholar 

  • Mecke J. (1984) ‘Parametric Representation of Mean Values for Stationary Random Mosaics’,Math. Operationsforsch. Statist., Ser. Statist. 15, 437–442.

    Google Scholar 

  • Mecke J. and Nagel W. (1980) ‘Statlonäre räumliche Faserprozesse und ihre Schnittzahlrosen’,Elektron. Informationsverarb. Kybernet. 16, 475–483.

    Google Scholar 

  • Mecke J. and Stoyan D. (1980a) ‘Formulas for Stationary Planar Fibre Processes I—General Theory’,Math. Operationsforsch. Statist., Ser. Statist. 11, 267–279.

    Google Scholar 

  • Mecke J. and Stoyan D. (1980b) ‘Stereological Problems for Spherical Particles’,Math. Nachr. 96, 311–317.

    Google Scholar 

  • Miles R. E. (1974) ‘On the Elimination of Edge Effects in Planar Sampling’ in E. F. Harding and D. G. Kendall (eds.),Stochastic Geometry, Wiley, London, pp. 228–247.

    Google Scholar 

  • Miles R. E. (1978) ‘The Importance of Proper Model Specification in Stereology’, in R. E. Miles and J. Serra (eds.),Geometrical Probability and Biological Structures, Lect. Notes Biomath.23, Springer, Berlin, pp. 115–136.

    Google Scholar 

  • Nagel W. (1983) ‘Dünne Schnitte von stationären räumlichen Faserprozessen’,Math. Operationsforsch. Statist., Ser. Statist. 14, 569–576.

    Google Scholar 

  • Neveu J. (1977)Processus Ponctuels, Lect. Notes Math.598, Springer, Berlin.

    Google Scholar 

  • Nguyen X. X. and Zessin H. (1979) ‘Ergodic Theorems for Spatial Processes’,Z. Wahrscheinlichkeitstheorie verw. Gebiete 48, 133–158.

    Google Scholar 

  • Ohser J. (1981) ‘A Remark on the Estimation of the Rose of Directions of Fibre Processes’,Math. Operationsforsch. Statist., Ser. Statist. 12, 581–585.

    Google Scholar 

  • Pohlmann S., Mecke J. and Stoyan D. (1981) ‘Stereological Formulas for Stationary Surface Processes’,Math. Operationsforsch. Statist., Ser. Statist. 12, 429–440.

    Google Scholar 

  • Radecke W. (1980) ‘Some Mean Value Relations on Stationary Random Mosaics in the Space’,Math. Nachr. 97, 203–210.

    Google Scholar 

  • Santaló L. A. (1976)Integral Geometry and Geometric Probability, Addison-Wesley, Reading, Mass.

    Google Scholar 

  • Schneider R. (1979) ‘Boundary Structure and Curvature of Convex Bodies’, in J. Tölke and J. M. Wills (eds.),Contributions to Geometry, Birkhäuser, Basle, pp. 13–59.

    Google Scholar 

  • Schneider R. (1980a) ‘Parallelmengen mit Vielfachheit und Steiner-Formeln’,Geometriae Dedicata 9, 111–127.

    Google Scholar 

  • Schneider R. (1980b) ‘Curvature Measures and Integral Geometry of Convex Bodies’,Rend. Sem. Mat. Univers. Politecn. Torino 38, 79–98.

    Google Scholar 

  • Schneider R. (1981) ‘Crofton's Formula Generalized to Projected Thick Sections’,Rend. Circ. Math. Palermo (2)30, 157–160.

    Google Scholar 

  • Schneider R. and Weil W. (1983) ‘Zonoids and Related Toples’, in P. Gruber and J. M. Wills (eds),Convexity and Its Applications, Birkhäuser, Basle, pp. 296–317.

    Google Scholar 

  • Schneider R. and Weil W. (1986) ‘Translative and Kinematic Integral Formulae for Curvature Measures’,Math. Nachr. 129, 67–80.

    Google Scholar 

  • Schwandtke, A., Ohser, J. and Stoyan, D. (1986) ‘Improved Estimation in Planar Sampling’,Acta Stereol.6 (to appear).

  • Stoyan D. (1979) ‘Proofs of Some Fundamental Formulas of Stereology for Non-Poisson Grain Models’,Math. Operationsforsch. Statist., Ser. Optimization 10, 575–583.

    Google Scholar 

  • Stoyan D. (1982) ‘Stereological Formulae for Size Distributions via Marked Point Processes’,Prob. Math. Statist. 2, 161–166.

    Google Scholar 

  • Stoyan D. (1984) ‘Further Stereological Formulae for Spatial Fibre Processes’,Math. Operationsforsch. Statist., Ser. Statist. 15, 421–428.

    Google Scholar 

  • Stoyan D. (1985a) ‘Estimating the Volume Density from Thin Sections’,Biom. J. 27, 427–430.

    Google Scholar 

  • Stoyan D. (1985b) ‘Stereological Determination of Orientations, Second-Order Quantities and Correlations for Random Spatial Fibre Systems’,Biom. J. 27, 411–425.

    Google Scholar 

  • Stoyan D., and Mecke J. (1983)Stochastische Geometrie, Akademie-Verlag, Berlin.

    Google Scholar 

  • Stoyan D. Mecke J. and Pohlmann S. (1980) ‘Formulas for Stationary Planar Fibre Processes II—Partially Oriented Fibre Systems’,Math. Operationsforsch. Statist., Ser. Statist. 11, 281–286.

    Google Scholar 

  • Voss K. and Stoyan D. (1985) ‘On the Stereological Estimation of Numerical Density of Particle Systems by an Object Counting Method’,Biom. J. 27, 919–924.

    Google Scholar 

  • Weil W. (1982) ‘Inner Contact Probabilities for Convex Bodies’,Adv. Appl. Prob. 14, 582–599.

    Google Scholar 

  • Weil W. (1983a) ‘Stereology—a Survey for Geometers’, in P. Gruber and J. M. Wills (eds.),Convexity and Its Applications, Birkhäuser, Basle, pp. 360–412.

    Google Scholar 

  • Weil W. (1983b) ‘Stereological Results for Curvature Measures’,Bull. Int. Statist. Inst. 50, 872–883.

    Google Scholar 

  • Weil W. (1984) ‘Densities of Quermassintegrals for Stationary Random Sets’, in R. V. Ambartzumian and W. Weil (eds.),Stochastic Geometry, Geometric Statistics, Stereology, Teubner, Leipzig, pp. 233–247.

    Google Scholar 

  • Weil W. and Wieacker J. A. (1984) ‘Densities for Stationary Random Sets and Point Processes’,Adv. Appl. Prob. 16, 324–346.

    Google Scholar 

  • Weil, W. and Wieacker, J. A. (1986) ‘A Representation Theorem for Random Sets’,Prob. Math. Statist. 9 (to appear).

  • Wieacker, J. A. (1982) ‘Translative stochastische Geometric der konvexen Körper’, Thesis, Albert-Ludwigs-Universität, Freiburg.

  • Weiss V. and Zähle M. (1986) ‘Geometric Measures for Random Curved Mosaics ofR d, Preprint, Friedrich-Schiller-Universitát, Jena.

    Google Scholar 

  • Zähle M. (1982) ‘Random Processes of Hausdorf Rectifiable Closed Sets’,Math. Nachr. 108, 49–72.

    Google Scholar 

  • Zähle M. (1984) ‘Thick Section Stereology for Random Fibres’,Math. Operationsforsch. Statist., Ser. Statist. 15, 429–435.

    Google Scholar 

  • Zähle M. (1986) ‘Curvature Measures and Random Sets II’,Z. Wahrscheinlichkeitstheorie verw. Gebiete 71, 37–58.

    Google Scholar 

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  1. Mathematisches Institut II, Universität Karlsruhe, Englerstrasse 2, 7500, Karlsruhe 1, F.R.G.

    Wolfgang Well

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Well, W. Point processes of cylinders, particles and flats. Acta Applicandae Mathematicae 9, 103–136 (1987). https://doi.org/10.1007/BF00580825

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