A Survey of the Statistical Theory of Shape

Kendall, David G.

doi:10.1214/ss/1177012582

Cited by 508 publications

(338 citation statements)

References 19 publications

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“…what remains invariant under the action of a given group of transformation (usually rigid body ones or similarities). The statistics on these shape spaces [20,21,22,23] raised the need for intrinsic tools. However, the link between the tools developed in these works, the metric used and the space structure was not always very clear.…”

Section: Introductionmentioning

confidence: 99%

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

2006

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In medical image analysis and high level computer vision, there is an intensive use of geometric features like orientations, lines, and geometric transformations ranging from simple ones (orientations, lines, rigid body or affine transformations, etc.) to very complex ones like curves, surfaces, or general diffeomorphic transformations. The measurement of such geometric primitives is generally noisy in real applications and we need to use statistics either to reduce the uncertainty (estimation), to compare observations, or to test hypotheses. Unfortunately, even simple geometric primitives often belong to manifolds that are not vector spaces. In previous works [1, 2], we investigated invariance requirements to build some statistical tools on transformation groups and homogeneous manifolds that avoids paradoxes. In this paper, we consider finite dimensional manifolds with a Riemannian metric as the basic structure. Based on this metric, we develop the notions of mean value and covariance matrix of a random element, normal law, Mahalanobis distance and χ 2 law. We provide a new proof of the characterization of Riemannian centers of mass and an original gradient descent algorithm to efficiently compute them. The notion of Normal law we propose is based on the maximization of the entropy knowing the mean and covariance of the distribution. The resulting family of pdfs spans the whole range from uniform (on compact manifolds) to the point mass distribution. Moreover, we were able to provide tractable approximations (with their limits) for small variances which show that we can effectively implement and work with these definitions.

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Section: Introductionmentioning

confidence: 99%

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

2006

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“…Fréchet [19] is probably one of the first mathematicians to develop some interest in the analysis of random shapes, i.e., curves. 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].…”

Section: Introductionmentioning

confidence: 99%

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

Charpiat¹,

Faugeras

Keriven³

2004

Found Comput Math

130

116

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Abstract. This paper proposes a framework for dealing with several problems related to the analysis of shapes. Two related such problems are the definition of the relevant set of shapes and that of defining a metric on it. Following a recent research monograph by Delfour and Zolésio [11], we consider the characteristic functions of the subsets of R 2 and their distance functions. The L 2 norm of the difference of characteristic functions, the L ∞ and the W 1,2 norms of the difference of distance functions define interesting topologies, in particular the well-known Hausdorff distance. Because of practical considerations arising from the fact that we deal with image shapes defined on finite grids of pixels, we restrict our attention to subsets of R 2 of positive reach in the sense of Federer [16], with smooth boundaries of bounded curvature. For this particular set of shapes we show that the three previous topologies are equivalent. The next problem we consider is that of warping a shape onto another by infinitesimal gradient descent, minimizing the corresponding distance. Because the distance function involves an inf, it is not differentiable with respect to the shape. We propose a family of smooth approximations of the distance function which are continuous with respect to the Hausdorff topology, and hence with respect to the other two topologies. We compute the corresponding Gâteaux derivatives. They define deformation flows that can be used to warp a shape onto another by solving an initial value problem. We show several examples of this warping and prove properties of our approximations that relate to the existence of local minima. We then use this tool to produce computational definitions of the empirical mean and covariance of a set of shape examples. They yield an analog of the notion of principal modes of variation. We illustrate them on a variety of examples.

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“…rigid body) transformation, we obtain the celebrated shape (resp. size and shape) spaces of Kendall at al [12,4,13,31]. One can define similarly the shape spaces of curves by removing the effect of re-parameterizations and the global pose of the curve [27,38,29,28].…”

Section: Shapesmentioning

confidence: 99%

Statistical Shape Analysis of Surfaces in Medical Images Applied to the Tetralogy of Fallot Heart

McLeod

Mansi

Sermesant

et al. 2012

Modeling in Computational Biology and Biomedicine

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International audienceThere is an increasing need for shape statistics in medical imaging to provide quantitative measures to aid in diagnosis, prognosis and therapy planning. In view of this, we describe methods for computing such statistics by utilizing a well-posed framework for representing the shape of surfaces as currents. Given this representation we can compute an atlas as a mean representation of the population and the main modes of variation around this mean. The modes are computed using principal component analysis (PCA) and applying standard correlation analysis to these allows to correlate shape features with clinical indices. Beyond this, we can compute a generative model of growth using partial least squares regression (PLS) and canonical correlation analysis (CCA). In this chapter, we investigate a clinical application of these statistical techniques on the shape of the heart for patients with repaired Tetralogy of Fallot (rToF), a severe congenital heard defect that requires surgical repair early in infancy. We relate the shape to the severity of the pathology and we build a bi-ventricular growth model of the rToF heart from cross-sectional data which gives insights about the evolution of the disease

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A Survey of the Statistical Theory of Shape

Cited by 508 publications

References 19 publications

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

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

Statistical Shape Analysis of Surfaces in Medical Images Applied to the Tetralogy of Fallot Heart

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