Representation of Radon shape diffusions via hyperspherical Brownian motion

Abstract

A framework is introduced for the study of general Radon shape diffusions, that is, shape diffusions induced by projections of randomly rotating shapes. This is done via a convenient representation of unoriented Radon shape diffusions in (unoriented) D.G. Kendall shape space k n through a Brownian motion on the hypersphere. This representation leads to a coordinate system for the generalized version of Radon diffusions since it is shown that shape can be essentially identified with unoriented shape in the projected case. A bijective correspondence between Brownian motion on real projective space and Radon shape diffusions is established. Furthermore, equations are derived for the general (unoriented) Radon diffusion of shape-and-size, and stationary measures are discussed. Lewis (1932Lewis ( -2004 Dedicated to the memory of John Trevor

On the induced distribution of the shape of the projection of a randomly rotated configuration

“…which recovers the result of [5] in the sense that u X and u X/ u X are precisely the objects used there in studying the shape of the projection of a randomly rotated configuration for the case m = 2. It is worth noting that the approach we take in this paper is not quite the same as that of [6]. We concentrate on the evolution of the shape of the projection of a given configuration onto a fixed hyperplane after being randomly rotated, whereas Panaretos [6] studied the shape of the projection of a given configuration onto a randomly rotated hyperplane.…”

“…It is worth noting that the approach we take in this paper is not quite the same as that of [6]. We concentrate on the evolution of the shape of the projection of a given configuration onto a fixed hyperplane after being randomly rotated, whereas Panaretos [6] studied the shape of the projection of a given configuration onto a randomly rotated hyperplane. In terms of our current notation, (v R )X is the object considered in [6].…”

“…Since Kendall's seminal paper [1] on shape spaces over twenty years ago, many advances have been made in both theory and applications. A recent example is the study of the shape of the projection onto a fixed hyperplane of a given configuration after it is randomly rotated in [5] and [6] (see also [3]). As pointed out in the above papers, one motivation for this study is its importance in the field of biophysics, where, using electron microscopes, one views images of single particles in an aqueous environment in order to study the structure of biological molecules.…”

“…Panaretos also obtained, in the same paper, the equilibrium distribution of such a resulting shape diffusion. In [6], he studied the diffusion of the unoriented size-andshape of the projected configuration under the assumption that the rotation evolves as a Brownian motion in SO(n) when the given configuration lies in an n-dimensional Euclidean space.…”

On random tomography with unobservable projection angles

“…The sequence {2S n } is i.i.d. with mean Gram([ρ]) and covariance Σ (see (4.19) in Panaretos [42], with n = 2). Therefore, by the multidimensional central limit theorem, one has 2N −1/2 N n=1 S n N →∞ =⇒ H.…”