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The diffusion of Radon shape

Part of: Partial differential equations on manifolds; differential operators Integral transforms, operational calculus Physiological, cellular and medical topics Geometric probability and stochastic geometry Multivariate analysis

Published online by Cambridge University Press:  01 July 2016

Victor M. Panaretos
Victor M. Panaretos*
Affiliation:
University of California, Berkeley
*
∗ Postal address: Department of Statistics, University of California, 367 Evans Hall, Berkeley, CA 94720, USA. Email address: [email protected]
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Abstract

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In 1977 D. G. Kendall considered diffusions of shape induced by independent Brownian motions in Euclidean space. In this paper, we consider a different class of diffusions of shape, induced by the projections of a randomly rotating, labelled ensemble. In particular, we study diffusions of shapes induced by projections of planar triangular configurations of labelled points onto a fixed straight line. That is, we consider the process in Σ13 (the shape space of triads in ℝ) that results from extracting the ‘shape information’ from the projection of a given labelled planar triangle as this evolves under the action of Brownian motion in SO(2). We term the thus-defined diffusions Radon diffusions and derive explicit stochastic differential equations and stationary distributions. The latter belong to the family of angular central Gaussian distributions. In addition, we discuss how these Radon diffusions and their limiting distributions are related to the shape of the initial triangle, and explore whether the relationship is bijective. The triangular case is then used as a basis for the study of processes in Σ1k arising from projections of an arbitrary number, k, of labelled points on the plane. Finally, we discuss the problem of Radon diffusions in the general shape space Σnk.

Keywords

Single-particle biophysicscircular Brownian motionKendall's shape theoryangular central Gaussian distributionintegral geometrystochastic geometryrandom processes of geometrical objectsRadon diffusionRadon transformsingular value decomposition

MSC classification

Primary: 60D05: Geometric probability and stochastic geometry 44A12: Radon transform
Secondary: 62H11: Directional data; spatial statistics 58J65: Diffusion processes and stochastic analysis on manifolds 92C55: Biomedical imaging and signal processing
Type
Stochastic Geometry and Statistical Applications
Information
Advances in Applied Probability , Volume 38 , Issue 2 , June 2006 , pp. 320 - 335
Copyright
Copyright © Applied Probability Trust 2006 

Footnotes

Partially supported by an NSF Graduate Research Fellowship and an NSF VIGRE Fellowship.

References

Bookstein, F. L. (1986). Size and shape spaces for landmark data in two dimensions (with discussion). Statist. Sci. 1, 181–242.Google Scholar
Deans, S. R. (1993). The Radon Transform and some of Its Applications (Reprint). Krieger, Malabar, FL.Google Scholar
Glaeser, R. M. (1999). Review: Electron crystallography: present excitement, a nod to the past, anticipating the future. J. Struct. Biol. 128, 3–14.CrossRefGoogle Scholar
Glaeser, R. M. et al. (2006). Electron Crystallography of Biological Macromolecules. To appear from Oxford University Press.Google Scholar
Hartman, P. and Watson, G. S. (1974). ‘Normal’ distribution functions on spheres and the modified Bessel functions. Ann. Prob. 2, 593–607.Google Scholar
Helgason, S. (1980). The Radon Transform. Birkhäuser, Boston, MA.Google Scholar
Kallenberg, O. (2002). Foundations of Modern Probability, 2nd edn. Springer, New York.Google Scholar
Kendall, D. G. (1977). The diffusion of shape. Adv. Appl. Prob. 9, 428–430.CrossRefGoogle Scholar
Kendall, D. G., Barden, D., Carne, T. K. and Le, H. (1999). Shape and Shape Theory. John Wiley, Chichester.Google Scholar
Kendall, W. S. (1990). The diffusion of Euclidean shape. In Disorder in Physical Systems, eds Grimmett, G. and Welsh, D., Cambridge University Press, pp. 203–217.Google Scholar
Kendall, W. S. (1998). A diffusion model for Bookstein triangle shape. Adv. Appl. Prob. 30, 317–334.Google Scholar
Klotz, J. (1964). Small sample power of the bivariate sign tests of Blumen and Hodges. Ann. Math. Statist. 35, 1576–1582.Google Scholar
Le, H. (1991). On geodesics in Euclidean shape spaces. J. London Math. Soc. 44, 360–372.Google Scholar
Le, H. (1994). Brownian motions on shape and size-and-shape spaces. J. Appl. Prob. 31, 101–113.CrossRefGoogle Scholar
Le, H. and Kendall, D. G. (1993). The Riemannian structure of Euclidean shape spaces: a novel environment for statistics. Ann. Statist. 21, 1225–1271.Google Scholar
Mardia, K. V. (1972). Statistics of Directional Data. Academic Press, London.Google Scholar
McCullagh, P. (1996). Möbius transformation and Cauchy parameter estimation. Ann. Statist. 24, 787–808.Google Scholar
Øksendal, B. (2003). Stochastic Differential Equations. An Introduction with Applications, 6th edn. Springer, Berlin.Google Scholar
Panaretos, V. M. (2006). Representation of Radon shape diffusions via hyperspherical Brownian motion. Tech. Rep. 707, Department of Statistics, University of California, Berkeley.Google Scholar
Panaretos, V. M. (2006). Statistical inversion of stochastic Radon transforms. Unpublished manuscript.Google Scholar
Radon, J. (1917). Über die Bestimmung von Funktionen durch ihre Integralverte längs gewisser Mannigfaltigkeiten. Leipzig Ber. 69, 262–277.Google Scholar
Scheffé, H. (1947). A useful convergence theorem for probability distributions. Ann. Math. Statist. 18, 434–438.Google Scholar
Tyler, D. E. (1987). Statistical analysis for the central angular Gaussian distribution on the sphere. Biometrika 74, 579–589.Google Scholar
Watson, G. S. (1982). The estimation of paleomagnetic pole positions. In Statistics and Probability: Essays in Honour of C. R. Rao, eds Kallianpur, G. et al., North-Holland, Amsterdam, pp. 703–712.Google Scholar
Watson, G. S. (1983). Statistics on Spheres. John Wiley, New York.Google Scholar