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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.

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