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On the rose of intersections of stationary flat processes

  • Eugene Spodarev (a1)

Abstract

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

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Postal address: Friedrich-Schiller-Universität Jena, Institut für Stochastik, Ernst-Abbe Platz 1-4, 07743 Jena, Germany. Email address: seu@minet.uni-jena.de

References

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[1] Ambartzumian, R. V. (1990). Factorizational Calculus and Geometric Probability (Encyclopaedia Math. Appl. 33). Cambridge University Press.
[2] Erdélyi, A., et al. (1953). Higher Transcendental Functions, Vol. II. McGraw-Hill, New York.
[3] Gardner, R. J. (1995). Geometric Tomography (Encyclopaedia Math. Appl. 58). Cambridge University Press.
[4] Goodey, P. and Howard, R. (1990). Processes of flats induced by higher dimensional processes I. Adv. Math. 80, 92109.
[5] Goodey, P. and Howard, R. (1990). Processes of flats induced by higher dimensional processes II. Contemp. Math. 113, 111119.
[6] Goodey, P. and Weil, W. (1992). Centrally symmetric convex bodies and the spherical Radon transform. J. Differential Geom. 35, 675688.
[7] 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. 12971326.
[8] Goodey, P., Howard, R. and Reeder, M. (1996). Processes of flats induced by higher dimensional processes III. Geometriae Dedicata 61, 257269.
[9] 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. 12591295.
[10] Groemer, H. (1996). Geometric Applications of Fourier Series and Spherical Harmonics. Cambridge University Press.
[11] Helgason, S. (1959). Differential operators on homogeneous spaces. Acta Math. 102, 239299.
[12] Helgason, S. (1984). Groups and Geometric Analysis. Academic Press, Orlando, FL.
[13] Helgason, S. (1990). The totally-geodesic Radon transform on constant curvature spaces. Contemp. Math. 113, 141149.
[14] Helgason, S. (1994). Geometric Analysis on Symmetric Spaces (Math. Surveys Monogr. 39). American Mathematical Society, Providence, RI.
[15] Klingenberg, W. (1995). Riemannian Geometry. De Gruyter, Berlin.
[16] Klingenberg, W. (1996). Grassmannian manifolds in geometry. Hermann Günther Graßmann (1809–1877): Visionary Mathematician, Scientist and Neohumanist Scholar. Kluwer, Dordrecht, pp. 281284.
[17] Matheron, G. (1975). Random Sets and Integral Geometry. John Wiley, New York.
[18] Mecke, J. (1981). Formulas for stationary planar fibre processes III—intersections with fibre systems. Math. Operationsforsch. Statist. Ser. Statist. 12, 201210.
[19] Mecke, J. (1981). Stereological formulas for manifold processes. Prob. Math. Statist. 2, 3135.
[20] Mecke, J. (1988). An extremal property of random flats. J. Microscopy 151, 205209.
[21] Mecke, J. and Nagel, W. (1980). Stationäre räumliche Faserprozesse und ihre Schnittzahlrosen. Elektron. Informationsverarb. Kyb. 16, 475483.
[22] Mecke, J. and Thomas, C. (1986). On an extreme value problem for flat processes. Commun. Statist. Stoch. Models 2, 273280.
[23] Mecke, J., Schneider, R., Stoyan, D. and Weil, W. (1990). Stochastische Geometrie. Birkhäuser, Basel.
[24] Molchanov, I. and Stoyan, D. (1994). Directional analysis of fibre processes related to Boolean models. Metrica 41, 183199.
[25] Müller, C., (1966). Spherical Harmonics (Lecture Notes Math. 17). Springer, Berlin.
[26] Pogorelov, A. V. (1979). Hilbert's Fourth Problem. Winston and Sons, Washington.
[27] Rubin, B. (2000). Spherical Radon transforms and intertwining fractional integrals. Preprint, The Hebrew University of Jerusalem.
[28] Rubin, B. (2000). Inversion formulas for the spherical Radon transform, the generalized cosine transform and intertwining fractional integrals. Preprint, The Hebrew University of Jerusalem.
[29] 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.
[30] Schneider, R. (1970). Über eine Integralgleichung in der Theorie der konvexen Körper. Math. Nachr. 44, 5575.
[31] Spodarev, E. (2000). On the roses of intersections of stationary flat processes. Preprint Math/Inf/00/11, Friedrich-Schiller-Universität Jena.
[32] Stoyan, D., Kendall, W. and Mecke, J. (1995). Stochastic Geometry and its Applications. John Wiley, New York.
[33] Weil, W. (1976). Centrally symmetric convex bodies and distributions. Israel J. Math. 24, 352367.

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On the rose of intersections of stationary flat processes

  • Eugene Spodarev (a1)

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