Hostname: page-component-78c5997874-mlc7c Total loading time: 0 Render date: 2024-11-05T12:07:40.008Z Has data issue: false hasContentIssue false
  • English
  • Français

Objects arranged randomly in space: an accessible theory

Published online by Cambridge University Press:  01 July 2016

Richard Cowan
Richard Cowan*
Affiliation:
CSIRO Division of Mathematics and Statistics, Lindfield
*
∗Present address: Department of Statistics, University of Hong Kong, Pokfulam Road, Hong Kong.
Get access Rights & Permissions [Opens in a new window]

Abstract

This expository paper deals with many problems concerning bounded objects arranged randomly in space. The objects are of rather general shapes and sizes, whilst the random mechanisms for positioning and orienting them are also fairly general. There are no restrictions on the dependence between shapes, sizes, orientations and positions of objects. The only substantive assumption is that the objects are distributed in a ‘statistically uniform' way throughout the whole of the space. We focus on the statistical properties of features seen in an observation window, itself of general size and shape.

Keywords

RANDOM GEOMETRYINTEGRAL GEOMETRYGEOMETRIC STATISTICSRANDOM SETS
Type
Research Article
Information
Advances in Applied Probability , Volume 21 , Issue 3 , September 1989 , pp. 543 - 569
Copyright
Copyright © Applied Probability Trust 1989 

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Adler, R. J. (1981) The Geometry of Random Fields. Wiley, Chichester.Google Scholar
Aleksandrov, A. D. (1963) Curves and surfaces. In Mathematics, its Content, Methods and Meaning, ed. Aleksandrov, A. D., Kolmogorov, A. N. and Lavrent'ev, V. V.. American Mathematical Society Translation, MIT Press, Cambridge, Mass. Google Scholar
Ambartzumian, R. V. (1970) Random fields of segments and random mosaics on a plane. Proc. 6th Berkeley Symp. Math. Statist. Prob. 369–381.Google Scholar
Ambartzumian, R. V. (1974a) Convex polygons and random tessellations. In Stochastic Geometry, ed. Harding, E. F. and Kendall, D. G.. Wiley, London.Google Scholar
Ambartzumian, R. V. (1974b) On random fields of threads in Rn. Soviet Math. Dokl. 15, 83–86.Google Scholar
Ambartzumian, R. V. (1977) Stochastic geometry from the standpoint of integral geometry. Adv. Appl. Prob. 9, 792–823.CrossRefGoogle Scholar
Baddeley, A. J., Gundersen, H. J. G. and Cruz-Orive, L. M. (1986) Estimation of surface area from vertical sections. J. Microsc. 142, 259–276.CrossRefGoogle ScholarPubMed
Berman, M. (1977) Distance distributions associated with Poisson processes of geometric figures. J. Appl. Prob. 14, 195–199.CrossRefGoogle Scholar
Blaschke, W. (1936) Vorlesungen über Integralgeometric. Deutsch Verlag Wissenschaft, Berlin.Google Scholar
Chern, S. S. (1952) On the kinematic formula in the euclidean space of n dimensions. Amer. J. Math. 74, 227–236.CrossRefGoogle Scholar
Coleman, R. (1972) Sampling procedures for the lengths of random straight lines. Biometrika 59, 415–426.CrossRefGoogle Scholar
Cowan, R. (1978) The use of the ergodic theorems in random geometry. Suppl. Adv. Appl. Prob. 10, 47–57.Google Scholar
Cowan, R. (1979) Homogeneous line-segment processes. Math. Proc. Camb. Phil. Soc. 86, 481–489.CrossRefGoogle Scholar
Cowan, R. (1980) Properties of ergodic random mosaic processes. Math. Nachr. 97, 89–102.Google Scholar
Cruz-Orive, L. M. (1980) Best linear unbiased estimators for stereology. Biometrics 36, 595–605.Google Scholar
Daley, D. J. and Vere-Jones, D. (1972) A summary of the theory of point processes. In Stochastic Point Processes ed. Lewis, P. A. W.. Wiley, New York.Google Scholar
Davidson, R. (1974) Stochastic processes of flats and exchangeability. In Stochastic Geometry, ed. Harding, E. F. and Kendall, D. G., Wiley, London.Google Scholar
Davy, P. and Miles, R. E. (1977) Sampling theory for opaque spatial specimens. J. R. Statist. Soc. B39, 56–65.Google Scholar
Davy, P. (1978) Stereology—A Statistical Viewpoint. , Australian National University.CrossRefGoogle Scholar
Diggle, P. J. (1983) Statistical Analysis of Spatial Point Patterns. Academic Press, London.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 in Rn. Z. Wahrscheinlichkeitsth. 50, 85–96.CrossRefGoogle Scholar
Federer, H. (1959) Curvature measures. Trans. Amer. Math. Soc. 93, 418–491.Google Scholar
Gilbert, E. N. (1967) Random plane networks and needle-shaped crystals. In Applications of Undergraduate Mathematics in Engineering, by Noble, B.. Macmillan, New York.Google Scholar
Gundersen, H. J. G. (1986) Stereology of arbitary particles. J. Microsc. 143, 3–45.Google Scholar
Hadwiger, H. (1957) Vorlesungen über Inhalt, Oberfläche und Isoperimetrie. Springer-Verlag, Berlin.Google Scholar
Hadwiger, H. and Giger, H. (1968) Über Treffzahlwahrscheinlichkeiten im Eikörperfeld. Z. Wahrscheinlichkeitsth. 10, 329–334.Google Scholar
Jensen, E. B., Baddeley, A. J., Gundersen, H. J. G. and Sundberg, R. (1985) Recent trends in stereology. Internat. Statist. Rev. 53, 99–108.Google Scholar
Hall, P. (1988) Introduction to the Theory of Coverage Processes. Wiley, New York.Google Scholar
Kellerer, A. M. (1983) On the number of clumps resulting from the overlap of randomly placed figures in a plane. J. Appl. Prob. 20, 126–135.Google Scholar
Kellerer, A. M. (1986) The variance of a Poisson process of domains. J. Appl. Prob. 23, 307–321.CrossRefGoogle Scholar
Kellerer, H. G. (1984) Minkowski functionals of Poisson processes. Z. Wahrscheinlichkeitsth. 67, 63–84.Google Scholar
Kendall, D. G. (1974) Foundations of a theory of random sets. In Stochastic Geometry, ed. Harding, E. F. and Kendall, D. G., Wiley, London.Google Scholar
Kendall, M. G. and Moran, P. A. P. (1963) Geometrical Probability. Griffin, London.Google Scholar
Maillardet, R. J. (1982) Generalised Voronoi Tesselations. , Australian National University.Google Scholar
Matheron, G. (1975) Random Sets and Integral Geometry. Wiley, New York.Google Scholar
Matthes, K., Kerstan, J. and Mecke, J. (1978) Infinitely Divisible Point Processes. Wiley, Chichester.Google Scholar
Mecke, J. (1967) Stationäre zufällige Maße auf lokalkompakten Abelschen Gruppen. Z. Wahrscheinlichkeitsth. 9, 36–58.CrossRefGoogle Scholar
Mecke, J. (1980) Palm methods for stationary random mosaics. In Combinational Principles in Stochastic Geometry, ed. Ambartzumian, R. V.. Armenian Academy of Sciences, Erevan.Google Scholar
Mecke, J. (1981a) Formulas for stationary planar fibre processes III—Intersections with fibre systems. Math. Operationsf. Statist. Ser. Statist. 12, 201–210.Google Scholar
Mecke, J. (1981b) Stereological formulas for manifold processes. Prob. Math. Statist. 2, 31–35.Google Scholar
Mecke, J. (1983) Random tesselations generated by hyperplanes. In Stochastic Geometry, Geometric Statistics, Stereology, ed. Weil, W. and Ambartzumian, R. V.. Teubner, Leipzig.Google Scholar
Mecke, J. and Stoyan, D. (1980) Formulas for stationary planar fibre processes I—General theory. Math. Operationsf. Statist. Ser. Statist. 11, 267–279.Google Scholar
Miles, R. E. (1961) Random Polytopes: The Generalization to n Dimensions of the Intervals of a Poisson Process. , Cambridge University.Google Scholar
Miles, R. E. (1964) Random polygons determined by random lines in the plane. I. II. Proc. Nat. Acad. Sci. USA 52, 901–907; 1157–1160.CrossRefGoogle Scholar
Miles, R. E. (1970) On the homogeneous planar Poisson process. Math. Biosci. 6, 85–127.Google Scholar
Miles, R. E. (1973) The various aggregates of random polygons determined by random lines in a plane. Adv. in Math. 10, 256–290.Google Scholar
Miles, R. E. (1977) The importance of proper model specification in stereoogy. In Geometrical Probability and Biological Structures. Buffon's 200th Anniversary ed. Miles, R. E. and Serra, J., Springer-Verlag, Berlin.Google Scholar
Miles, R. E. (1985) A comprehensive set of stereological formulae for embedded aggregates of not-necessarily-convex particles. J. Microsc. 138, 115–125.Google Scholar
Parker, P. and Cowan, R. (1976) Some properties of line-segment processes. J. Appl. Prob. 13, 96–107.CrossRefGoogle Scholar
Pohlmann, S., Mecke, J. and Stoyan, D. (1981) Stereological formulas for stationary surface processes. Math. Operationsf. Statist. Ser. Statist. 12, 429–440.Google Scholar
Ripley, B. D. (1981) Spatial Statistics. Wiley, New York.CrossRefGoogle Scholar
Santaló, L. A. (1953) Introduction to Integral Geometry. Hermann, Paris.Google Scholar
Santaló, L. A. (1976) Integral Geometry and Geometric Probability. Addison-Wesley, Reading, Mass.Google Scholar
Santaló, L. A. (1977) Random processes of linear segments and graphs. In Geometrical Probability and Biological Structures. Buffon's 200th Anniversary, ed. Miles, R. E. and Serra, J., Springer-Verlag, Berlin.Google Scholar
Serra, J. (1982) Image Analysis and Mathematical Morphology. Academic Press, London.Google Scholar
Solomon, H. (1978) Geometric Probability. SIAM, Philadelphia.CrossRefGoogle Scholar
Stoyan, D. (1979a) On some qualitative properties of the Boolean model of stochastic geometry. Z. angew. Math. Mech. 59, 447–454.CrossRefGoogle Scholar
Stoyan, D. (1979b) Proof of some fundamental formulas of stereology for non-Poisson grain models. Math. Operationsf. Statist. Ser. Optimization 10, 573–581.Google Scholar
Stoyan, D. (1981) On the second-order analysis of stationary planar fibre processes. Math. Nachr. 102, 189–199.CrossRefGoogle Scholar
Stoyan, D. (1982) Stereological formulae for size distributions via marked point processes. Prob. Math. Statist. 2, 161–166.Google Scholar
Stoyan, D. (1986) On generalized planar random tessellations. Math. Nachr. 128, 215–219.Google Scholar
Stoyan, D., Kendall, W. S. and Mecke, J. (1987) Stochastic Geometry and its Applications. Wiley, Chichester.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. Operationsf. Statist. Ser. Statist. 11, 281–286.Google Scholar
Streit, F. (1970) On multiple integral-geometric integrals and their application to probability theory. Canad. J. Math. 22, 151–163.CrossRefGoogle Scholar
Voss, K. (1982) Frequencies of n-polygons in planar sections of polyhedra. J. Microsc. 128, 111–120.CrossRefGoogle Scholar
Weil, W. (1983) Stereology: A survey for geometers. In Convexity and Its Applications, ed. Gruber, J. and Wills, J. M., Birkhäuser, Basel.Google Scholar
Weil, W. (1987) Point processes of cylinders, particles and flats. Acta Appl. Math. 9, 103–136.CrossRefGoogle Scholar
Zähle, M. (1982) Random processes of Hausdorff rectifiable sets. Math. Nachr. 108, 49–72.CrossRefGoogle Scholar