Analysis of principal nested spheres

Jung, Sungkyu; Dryden, Ian L.; Marron, J. S.

doi:10.1093/biomet/ass022

Cited by 130 publications

(144 citation statements)

References 22 publications

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“…With the distance function ρ δ(c,r) (x, y) which measures the shortest arc-length between x, y ∈ δ(c, r) along the (small) circle via ρ δ(c,r) (x, y) = sin(r) arccos[(x y − cos 2 (r))/sin 2 (r)] (Jung et al, 2012) we have by definition…”

Section: Rigid Rotation Modelmentioning

confidence: 99%

“…The problem (7) is precisely the fitting of concentric (small) circles. Therefore, numerical algorithms for (7) are generalized algorithms of the well-studied fitting of small circles (Mardia and Gadsden, 1977;Rivest, 1999;Jung et al, 2011Jung et al, , 2012 and are discussed in the Appendix.…”

Section: Estimationmentioning

confidence: 99%

“…The problem (16) is a more general nonlinear least squares problem, which however can be solved in a similar manner to the former two problems. We propose a variant of the doubly iterative algorithm used in fitting small circles in S m (Jung et al, 2011(Jung et al, , 2012.…”

Section: Appendix Proof Of Lemmamentioning

confidence: 99%

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Analysis of Rotational Deformations From Directional Data

Schulz¹,

Jung²,

Huckemann³

et al. 2015

Journal of Computational and Graphical Statistics

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This paper discusses a novel framework to analyze rotational deformations of real 3D objects. The rotational deformations such as twisting or bending have been observed as the major variation in some medical applications, where the features of the deformed 3D objects are directional data. We propose modeling and estimation of the global deformations in terms of generalized rotations of directions. The proposed method can be cast as a generalized small circle fitting on the unit sphere. We also discuss the estimation of descriptors for more complex deformations composed of two simple deformations. The proposed method can be used for a number of different 3D object models. Two analyses of 3D object data are presented in detail: one using skeletal representations in medical image analysis as well as one from biomechanical gait analysis of the knee joint. Supplementary Materials are available online.

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Section: Rigid Rotation Modelmentioning

confidence: 99%

Section: Estimationmentioning

confidence: 99%

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Analysis of Rotational Deformations From Directional Data

Schulz¹,

Jung²,

Huckemann³

et al. 2015

Journal of Computational and Graphical Statistics

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“…First, it fits great circles, whereas our data live near small circles. Second, it is backward only at the first step, whereas the fully backward method of Jung [12] described next appears superior.…”

Section: Gpca: Geodesics With the Mean On The First Geodesicmentioning

confidence: 99%

Nested Sphere Statistics of Skeletal Models

Pizer

Jung

Goswami

et al. 2012

Mathematics and Visualization

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We seek a form of object model that exactly and completely captures the interior of most non-branching anatomic objects and simultaneously is well suited for probabilistic analysis on populations of such objects. We show that certain nearly medial, skeletal models satisfy these requirements. These models are first mathematically defined in continuous 3-space, and then discrete representations formed by a tuple of spoke vectors are derived. We describe means of fitting these skeletal models into manual or automatic segmentations of objects in a way stable enough to support statistical analysis, and we sketch means of modifying these fits to provide good correspondences of spoke vectors across a training population of objects. Understanding will be developed that these discrete skeletal models live in an abstract space made of a Cartesian product of a Euclidean space and a collection of spherical spaces. Based on this understanding and the way objects change under various rigid and nonrigid transformations, a method analogous to principal component analysis called composite principal nested spheres will be seen to apply to learning a more efficient collection of modes of object variation about a new and more representative mean object than those provided by other representations and other statistical analysis methods. The methods are illustrated by application to hippocampi.

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“…Generalizations of the Euclidean principal component analysis procedure to manifolds are particularly relevant for data exhibiting anisotropy. Approaches include principal geodesic analysis (PGA, [6]), geodesic PCA (GPCA, [11]), principal nested spheres (PNS, [12]), barycentric subspace analysis (BSA, [13]), and horizontal component analysis (HCA, [14]). Common to these constructions are explicit representations of approximating low-dimensional subspaces.…”

Section: Introductionmentioning

confidence: 99%

Anisotropically Weighted and Nonholonomically Constrained Evolutions on Manifolds

Sommer

2016

Entropy

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Abstract:We present evolution equations for a family of paths that results from anisotropically weighting curve energies in non-linear statistics of manifold valued data. This situation arises when performing inference on data that have non-trivial covariance and are anisotropic distributed. The family can be interpreted as most probable paths for a driving semi-martingale that through stochastic development is mapped to the manifold. We discuss how the paths are projections of geodesics for a sub-Riemannian metric on the frame bundle of the manifold, and how the curvature of the underlying connection appears in the sub-Riemannian Hamilton-Jacobi equations. Evolution equations for both metric and cometric formulations of the sub-Riemannian metric are derived. We furthermore show how rank-deficient metrics can be mixed with an underlying Riemannian metric, and we relate the paths to geodesics and polynomials in Riemannian geometry. Examples from the family of paths are visualized on embedded surfaces, and we explore computational representations on finite dimensional landmark manifolds with geometry induced from right-invariant metrics on diffeomorphism groups.

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Analysis of principal nested spheres

Cited by 130 publications

References 22 publications

Analysis of Rotational Deformations From Directional Data

Analysis of Rotational Deformations From Directional Data

Nested Sphere Statistics of Skeletal Models

Anisotropically Weighted and Nonholonomically Constrained Evolutions on Manifolds

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