The Geometry of Shape Spaces

Abstract

In [2] D. G. Kendall introduced shape spaces which classsified TV-tuples of points in U K and explained their role for statistics. The aim of this paper is to study such shape spaces and, in particular, to explore the corresponding spaces for points on spheres. The ultimate purpose of the work is to be able to apply statistical techniques to observations where only the shape of the observations is significant. It is apparent that the geometry of the shape space is critical in determining the type of statistical technique which one should use. Therefore, the major part of this paper is devoted to an examination of the geometry, and the statistics is only introduced in a simple case when considering random processes on the shape space. We refer the reader to the surveys of the statistical background to this work in [2] and [5]. Further details of the background to this work, and related ideas are contained in the references [3] to [13] and [16] to [18].The general situation which we wish to study is as follows. Let G be a Lie group which acts smoothly on the differentiable manifold M. Then G acts smoothly on the space M N of TV-tuples from M. The shape space ~Z N (M, G) is defined to be the set of orbits for this action. So two TV-tuples (x n ) and (y n ) determine the same shape if there is a T e G with Tx n = y n for each n. There is a natural quotient map q: M N^1 . N (M, G) which sends an TV-tuple to its orbit. We wish to give the shape space more structure. It can certainly be given a quotient topology. We would like to give it a differentiable structure so that q were a submersion. This is not possible everywhere, for it can fail at points where the action of G is not free. However, for the cases which we will consider, there is a dense open set of regular points in H N (M, G) which can be made into a differentiable manifold so that q is a submersion above it. The remaining points are singularities of the shape space; they are often of importance in the related statistical problems. If M is a Riemannian manifold and G a Lie subgroup of the isometries, then we may find a Riemannian metric on the regular points of the shape space for which q is a Riemannian submersion. Furthermore, the Brownian motion on M N maps under q to a Markov process and we will wish to explore how this process compares with Brownian motion on the shape space. (By considering the densities of such a process at fixed times we may obtain information, in a particularly symmetric case, about the image of a probability density under q. See the references [3,7,8,10,11,12] for related results on the images of distributions.)We will only consider very special examples of shape spaces where M is either an Euclidean space IR* or a sphere S K . In the first section we examine in detail the case where M = U K and G is the orthogonal group O(K). This is simple because M N is flat. In the second section we consider the, largely superficial, changes that are necessary when we replace O(K) by the special orthogonal group SO(K). D. G. Kendall has studied...

Approximations of Shape Metrics and Application to Shape Warping and Empirical Shape Statistics

“…He was followed by Matheron [33] who founded, with Serra, the French School of Mathematical Morphology and by D. Kendall [24], [26], [27] and his colleagues, e.g., Small [42]. In addition, and independently, a rich body of theory and practice for the statistical analysis of shapes has been developed by Bookstein [4], Dryden and Mardia [13], Carne [5], and Cootes et al [8]. Except for the mostly theoretical work of Fréchet and Matheron, the tools developed by these authors are very much tied to the point-wise representation of the shapes they study: objects are represented by a finite number of salient points or landmarks.…”

Representation of Radon shape diffusions via hyperspherical Brownian motion

“…. , φ (2) n−1 ) by a rotation around the axis defined by φ (1) n , (ii) we have φ (1) n = −φ (2) n and (φ (1) 1 , . .…”

“…. , φ (2) n−1 ) by a rotoinversion around the axis defined by φ (1) n . Assume the first case is true.…”

“…Under this definition, the shapes of such ensembles can be thought of as points on a manifold, so as to allow the definition of statistical procedures that assess, compare and estimate shape characteristics. The geometry of shape spaces k n of k labelled points in n dimensions varies for different values of k and n and can be rather intricate (Kendall et al [7], Le and Kendall [12], Carne [1], Le [11]). …”

Edge Potential Functions (EPF) and Genetic Algorithms (GA) for Edge-Based Matching of Visual Objects

“…Bookestein [10], Dryden and Mardia [11], Cootes et al [13], Carne [14], and Small [15] developed the basic theory into practical statistical approaches for analyzing objects using probability distributions of shapes. From the viewpoint of applications, shape theory has been used to solve some problems in image analysis such as identifying landmarks of face images [12], or monitoring activities in a certain region from video data [16].…”