Hostname: page-component-586b7cd67f-t7czq Total loading time: 0 Render date: 2024-11-27T22:16:09.677Z Has data issue: false hasContentIssue false
  • English
  • Français

On the rose of intersections of stationary flat processes

Part of: Integral transforms, operational calculus Geometric probability and stochastic geometry Stochastic processes

Published online by Cambridge University Press:  01 July 2016

Eugene Spodarev
Eugene Spodarev*
Affiliation:
Friedrich-Schiller-Universität Jena
*
∗ Postal address: Friedrich-Schiller-Universität Jena, Institut für Stochastik, Ernst-Abbe Platz 1-4, 07743 Jena, Germany. Email address: [email protected]
Get access Rights & Permissions [Opens in a new window]

Abstract

The paper yields retrieval formulae of the directional distribution of a stationary k-flat process in ℝd if its rose of intersections with all r-flats is known. Cases k = d −1, 1 ≤ r ≤ d - 1 for arbitrary d and d = 4, k = 2, r = 2 are considered. Some generalizations to manifold processes in ℝd are made. The proofs use the methods of harmonic analysis on higher Grassmannians (spherical harmonics, integral transforms).

Keywords

Grassmann manifoldsstationary flat (Poisson) processesmanifold processesspherical harmonics(spherical) Radon transformsine transformcosine transformrose of intersectionsstochastic geometry

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry
Secondary: 60G55: Point processes 44A12: Radon transform 44A99: None of the above, but in this section
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 33 , Issue 3 , September 2001 , pp. 584 - 599
Copyright
Copyright © Applied Probability Trust 2001 

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

Ambartzumian, R. V. (1990). Factorizational Calculus and Geometric Probability (Encyclopaedia Math. Appl. 33). Cambridge University Press.Google Scholar
Erdélyi, A., et al. (1953). Higher Transcendental Functions, Vol. II. McGraw-Hill, New York.Google Scholar
Gardner, R. J. (1995). Geometric Tomography (Encyclopaedia Math. Appl. 58). Cambridge University Press.Google Scholar
Goodey, P. and Howard, R. (1990). Processes of flats induced by higher dimensional processes I. Adv. Math. 80, 92–109.Google Scholar
Goodey, P. and Howard, R. (1990). Processes of flats induced by higher dimensional processes II. Contemp. Math. 113, 111–119.CrossRefGoogle Scholar
Goodey, P. and Weil, W. (1992). Centrally symmetric convex bodies and the spherical Radon transform. J. Differential Geom. 35, 675–688.Google Scholar
Goodey, P. and Weil, W. (1993). Zonoids and generalisations. In Handbook of Convex Geometry, Vol. B, eds Gruber, P. M. and Wills, J. M.. Elsevier, Amsterdam, pp. 1297–1326.CrossRefGoogle Scholar
Goodey, P., Howard, R. and Reeder, M. (1996). Processes of flats induced by higher dimensional processes III. Geometriae Dedicata 61, 257–269.Google Scholar
Groemer, H. (1993). Fourier series and spherical harmonics in convexity. In Handbook of Convex Geometry, Vol. B, eds Gruber, P. M. and Wills, J. M.. Elsevier, Amsterdam, pp. 1259–1295.Google Scholar
Groemer, H. (1996). Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge University Press.Google Scholar
Helgason, S. (1959). Differential operators on homogeneous spaces. Acta Math. 102, 239–299.CrossRefGoogle Scholar
Helgason, S. (1984). Groups and Geometric Analysis. Academic Press, Orlando, FL.Google Scholar
Helgason, S. (1990). The totally-geodesic Radon transform on constant curvature spaces. Contemp. Math. 113, 141–149.Google Scholar
Helgason, S. (1994). Geometric Analysis on Symmetric Spaces (Math. Surveys Monogr. 39). American Mathematical Society, Providence, RI.Google Scholar
Klingenberg, W. (1995). Riemannian Geometry. De Gruyter, Berlin.Google Scholar
Klingenberg, W. (1996). Grassmannian manifolds in geometry. Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar. Kluwer, Dordrecht, pp. 281–284.CrossRefGoogle Scholar
Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.Google Scholar
Mecke, J. (1981). Formulas for stationary planar fibre processes III—intersections with fibre systems. Math. Operationsforsch. Statist. Ser. Statist. 12, 201–210.Google Scholar
Mecke, J. (1981). Stereological formulas for manifold processes. Prob. Math. Statist. 2, 31–35.Google Scholar
Mecke, J. (1988). An extremal property of random flats. J. Microscopy 151, 205–209.CrossRefGoogle Scholar
Mecke, J. and Nagel, W. (1980). Stationäre räumliche Faserprozesse und ihre Schnittzahlrosen. Elektron. Informationsverarb. Kyb. 16, 475–483.Google Scholar
Mecke, J. and Thomas, C. (1986). On an extreme value problem for flat processes. Commun. Statist. Stoch. Models 2, 273–280.Google Scholar
Mecke, J., Schneider, R., Stoyan, D. and Weil, W. (1990). Stochastische Geometrie. Birkhäuser, Basel.Google Scholar
Molchanov, I. and Stoyan, D. (1994). Directional analysis of fibre processes related to Boolean models. Metrica 41, 183–199.Google Scholar
Müller, C., (1966). Spherical Harmonics (Lecture Notes Math. 17). Springer, Berlin.Google Scholar
Pogorelov, A. V. (1979). Hilbert's Fourth Problem. Winston and Sons, Washington.Google Scholar
Rubin, B. (2000). Spherical Radon transforms and intertwining fractional integrals. Preprint, The Hebrew University of Jerusalem.Google Scholar
Rubin, B. (2000). Inversion formulas for the spherical Radon transform, the generalized cosine transform and intertwining fractional integrals. Preprint, The Hebrew University of Jerusalem.Google Scholar
Rubin, B. and Ryabogin, D. (2000). The k-dimensional Radon transform on the n-sphere and related wavelet transforms. Preprint, The Hebrew University of Jerusalem.Google Scholar
Schneider, R. (1970). Über eine Integralgleichung in der Theorie der konvexen Körper. Math. Nachr. 44, 55–75.Google Scholar
Spodarev, E. (2000). On the roses of intersections of stationary flat processes. Preprint Math/Inf/00/11, Friedrich-Schiller-Universität Jena.Google Scholar
Stoyan, D., Kendall, W. and Mecke, J. (1995). Stochastic Geometry and its Applications. John Wiley, New York.Google Scholar
Weil, W. (1976). Centrally symmetric convex bodies and distributions. Israel J. Math. 24, 352–367.CrossRefGoogle Scholar