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The determination of convex bodies from the mean of random sections

  • Paul Goodey (a1) and Wolfgang Weil (a2)
Abstract

Random sectioning of particles (compact sets in ℝ3 with interior points) is a familiar procedure in stereology where it is used to estimate particle quantities like volume or surface area from planar or linear sections (see, for example, the survey [23] or the book [20]). In the following, we study the problem whether the whole shape of a convex particle K can be estimated from random sections. If E is an IUR (isotropic, uniform, random) line or plane intersecting K then the intersection Xk = KE is a (k-dimensional, k = 1 or 2) random set. It is clear that the distribution of Xk determines K uniquely and that if E1,…, En are such flats, the most natural estimator for K would be the convex hull

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[1]Adler R. J. and Pyke R.. Problem 91–3. IMS Bulletin 20 (1991), 406407.
[2]Artstein Z. and Vitale R. A.. A strong law of large numbers for random compact sets. Ann. Probab. 5 (1975), 879882.
[3]Bonnesen T. and Fenchel W.. Theorie der Konvexen Körper (Springer-Verlag, 1934).
[4]Erdélyi A., Magnus W., Oberhettinger F. and Tricomi E. G.. Higher Transcendental Functions, vol. 2 (McGraw-Hill, 1953).
[5]Goodey P. and Weil W.. Translative integral formulae for convex bodies. Aequationes Math. 34 (1987), 6477.
[6]Goodey P. and Weil W.. Integral geometric formulae for projection functions. Geom. Dedicata 41 (1992), 117126.
[7]Kingman J. F. C.. Random secants of a convex body. J. Appl. Probab. 6 (1969), 660672.
[8]Leichtweiβ K.. Konvexe Mengen (Springer-Verlag, 1980).
[9]Little J. J.. An iterative method for reconstructing convex polyhedra from extended Gaussian images. In Proceedings American Association for Artificial Intelligence (AAAI, 1983). pp. 247250.
[10]Mallows C. and Clark J.. Linear-intercept distributions do not characterize plane sets. J. Appl. Probab. 7 (1970), 240244.
[11]Matheron G.. Random Sets and Integral Geometry (Wiley, 1975).
[12]Müller C.. Spherical Harmonics (Springer-Verlag, 1966).
[13]Nagel W.. Orientation dependent chord length distributions characterize convex polygons, submitted.
[14]Schneider R.. Zu einem Problem von Shephard über die Projektionen konvexer Körper. Math. Z. 101 (1967), 7182.
[15]Schneider R.. Über eine Integralgleichung in der Theorie der konvexen Körper. Math. Nachr. 44 (1970), 5575.
[16]Schneider R.. Rekonstruktion eines konvexen Körpers aus seinen Projektionen. Math. Nachr. 79 (1977), 325329.
[17]Schneider R.. Boundary structure and curvature of convex bodies. In Contributions to Geometry, Proc. Geometry Sympos. Siegen 1978 (editors Tölke J. and Wills J. M.) (Birkhäuser, 1979). pp. 1359.
[18]Schneider R. and Weil W.. Zonoids and related topics. In Convexity and its Applications (editors Gruber P. and Wills J. M.) (Birkhäuser, 1983). pp. 296317.
[19]Schneider R. and Weil W.. Translative and kinematic integral formulae for curvature measures. Math. Nachr. 129 (1986), 6780.
[20]Stoyan D., Kendall W. S. and Mecke J.. Stochastic Geometry and its Applications (Akademie-Verlag, 1987).
[21]Waksman P.. Plane polygons and a conjecture of Blaschke's. Adv. in Appl. Probab. 17 (1985), 774793.
[22]Weil W.. Centrally symmetric convex bodies and distributions, II. Israel J. Math. 32 (1979), 173182.
[23]Weil W.. Stereology – a survey for geometers. In Convexity and its Applications (editors Gruber P. and Wills J. M.) (Birkhäuser, 1983). pp. 360412.
[24]Weil W.. Iterations of translative integral formulae and non-isotropic Poisson processes of particles. Math. Z. 205 (1990), 531551.
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Mathematical Proceedings of the Cambridge Philosophical Society
  • ISSN: 0305-0041
  • EISSN: 1469-8064
  • URL: /core/journals/mathematical-proceedings-of-the-cambridge-philosophical-society
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