The diffusion of Radon shape

Abstract

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 Σ₁^k 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 Σ_n^k.