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Directional Statistics and Shape Analysis
, 1995
"... There have been various developments in shape analysis in the last decade. We describe here some relationships of shape analysis with directional statistics. For shape, rotations are to be integrated out or to be optimized over whilst they are the basis for directional statistics. However, various c ..."
Abstract

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There have been various developments in shape analysis in the last decade. We describe here some relationships of shape analysis with directional statistics. For shape, rotations are to be integrated out or to be optimized over whilst they are the basis for directional statistics. However, various concepts are connected. In particular, certain distributions of directional statistics have emerged in shape analysis, such a distribution is Complex Bingham Distribution. This paper first gives some background to shape analysis and then it goes on to directional distributions and their applications to shape analysis. Note that the idea of using tangent space for analysis is common to both manifold as well. 1 Introduction Consider shapes of configurations of points in Euclidean space. There are various contexts in which k labelled points (or "landmarks") x 1 ; :::; x k in IR m are given and interest is in the shape of (x 1 ; :::; x k ). Example 1 The microscopic fossil Globorotalia truncat...
Shapes of quantum states
 J. Phys. A: Math. Gen
"... Summary. The shape space of k labelled points on a plane can be identified with the space of pure quantum states of dimension k − 2. Hence, the machinery of quantum mechanics can be applied to the statistical analysis of planar configurations of points. Various correspondences between point configur ..."
Abstract

Cited by 1 (0 self)
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Summary. The shape space of k labelled points on a plane can be identified with the space of pure quantum states of dimension k − 2. Hence, the machinery of quantum mechanics can be applied to the statistical analysis of planar configurations of points. Various correspondences between point configurations and quantum states, such as linear superposition as well as unitary and stochastic evolution of shapes, are illustrated. In particular, a complete characterisation of shape eigenstates for an arbitrary number of points is given in terms of cyclotomic equations. 1. Statistical theory of shape The idea of a shapespace Σ k m, whose elements are the shapes of k labelled points in R m, at least two being distinct, was introduced in a statistical context by Kendall (1984). Here, it is natural to identify shapes differing only by translations, rotations, and dilations in R m (although there are situations of interest, not to be considered here, in which the scale is also relevant). However, this identification will not apply to reflections. Thus, the resulting shape space is the quotient Σ k m = S m(k−1)−1 /SO(m)