The Riemannian Structure of Euclidean Shape Spaces: A Novel Environment for Statistics

Le, Huiling; Kendall, David G.

doi:10.1214/aos/1176349259

Cited by 151 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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“…A first example is a set of K anatomical landmarks [7,10], which is a function from the space of labels to the real 3D space: φ : [1, ..., K] → R 3 . In this case, the manifold M of landmarks sets carries a right action of the group of permutations of the labels and a left action of the group of rotations and translations of the 3D space SE (3).…”

Section: Introductionmentioning

confidence: 99%

Biased Estimators on Quotient Spaces

Miolane

Pennec

2015

Lecture Notes in Computer Science

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Abstract. Usual statistics are defined, studied and implemented on Euclidean spaces. But what about statistics on other mathematical spaces, like manifolds with additional properties: Lie groups, Quotient spaces, Stratified spaces etc? How can we describe the interaction between statistics and geometry? The structure of Quotient space in particular is widely used to model data, for example every time one deals with shape data. These can be shapes of constellations in Astronomy, shapes of human organs in Computational Anatomy, shapes of skulls in Palaeontology, etc. Given this broad field of applications, statistics on shapes -and more generally on observations belonging to quotient spaces-have been studied since the 1980's. However, most theories model the variability in the shapes but do not take into account the noise on the observations themselves. In this paper, we show that statistics on quotient spaces are biased and even inconsistent when one takes into account the noise. In particular, some algorithms of template estimation in Computational Anatomy are biased and inconsistent. Our development thus gives a first theoretical geometric explanation of an experimentally observed phenomenon. A biased estimator is not necessarily a problem. In statistics, it is a general rule of thumb that a bias can be neglected for example when it represents less than 0.25 of the variance of the estimator. We can also think about neglecting the bias when it is low compared to the signal we estimate. In view of the applications, we thus characterize geometrically the situations when the bias can be neglected with respect to the situations when it must be corrected.

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“…The complexity of the geometry of general shape spaces Σ k n (see Kendall [9], Le & Kendall [15], Le [14]) makes such a study quite difficult, although some cases could be tractable. Nevertheless, using the so called inner product coordinates for shape and shape-and-size, certain basic results on the general case have been derived by the author (Panaretos [19]).…”

Section: Discussionmentioning

confidence: 99%

Optimal portfolio policies under fixed and proportional transaction costs

Irle¹,

Sass²

2006

Adv. in Appl. Probab.

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Almost thirty years ago, D.G. Kendall [8] 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 Σ 3 1 (the shape space of labelled triads in R 1 ) that results from extracting the "shape information" from the projection of a given labelled planar triangle, as this evolves under the action of a 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 pivot for the study of processes in Σ k 1 arising from projections of an arbitrary number k of labelled points on the plane. Finally, we discuss the problem of Radon diffusions in general shape-space Σ k n .

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The Riemannian Structure of Euclidean Shape Spaces: A Novel Environment for Statistics

Cited by 151 publications

References 0 publications

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

Intrinsic Statistics on Riemannian Manifolds: Basic Tools for Geometric Measurements

Biased Estimators on Quotient Spaces

Optimal portfolio policies under fixed and proportional transaction costs

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