Shape-space smoothing splines for planar landmark data

Kume, Alfred; Dryden, Ian L.; Le, Huiling

doi:10.1093/biomet/asm047

Cited by 63 publications

(73 citation statements)

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“…In previous work, non-parametric kernel-based and spline-based methods have been extended to observations that lie on a Riemannian manifold with some success [8,18,22,26], but intrinsic parametric regression on Riemannian manifolds has received limited attention. Recently, Fletcher [12] and Niethammer et al [31] have each independently developed a form of parametric regression, geodesic regression, which generalizes the notion of linear regression to Riemannian manifolds.…”

Section: Regression Analysis and Curve-fittingmentioning

confidence: 99%

Intrinsic Polynomials for Regression on Riemannian Manifolds

2014

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We develop a framework for polynomial regression on Riemannian manifolds. Unlike recently developed spline models on Riemannian manifolds, Riemannian polynomials offer the ability to model parametric polynomials of all integer orders, odd and even. An intrinsic adjoint method is employed to compute variations of the matching functional, and polynomial regression is accomplished using a gradient-based optimization scheme. We apply our polynomial regression framework in the context of shape analysis in Kendall shape space as well as in diffeomorphic landmark space. Our algorithm is shown to be particularly convenient in Riemannian manifolds with additional symmetry, such as Lie groups and homogeneous spaces with right or left invariant metrics. As a particularly important example, we also apply polynomial regression to time-series imaging data using a right invariant Sobolev metric on the diffeomorphism group. The results show that Riemannian polynomials provide a practical model for parametric curve regression, while offering increased flexibility over geodesics.

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Section: Regression Analysis and Curve-fittingmentioning

confidence: 99%

Intrinsic Polynomials for Regression on Riemannian Manifolds

2014

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“…Human movement A human movement dataset introduced in Kume et al (2007) contains 50 samples of k = 4 landmarks on the lower back, shoulder, wrist, and index finger. The dataset consists of shape configurations of five different reaching movements projected in the plane of a table, each observed at ten different time-points.…”

Section: Real Data Analysismentioning

confidence: 99%

“…There has been a concern that when non-geodesic variation is major and apparent, geodesicbased methods do not give a fully effective decomposition of the space. As an example, a dataset of shapes representing human movements, discussed later in § 5·1 and introduced in Kume et al (2007), is plotted in the top panel of Fig. 1 using the first two principal component directions.…”

Section: Introductionmentioning

confidence: 99%

Analysis of principal nested spheres

Jung¹,

Dryden²,

Marron³

2012

Biometrika

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126

135

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SUMMARYA general framework for a novel non-geodesic decomposition of high-dimensional spheres or high-dimensional shape spaces for planar landmarks is discussed. The decomposition, principal nested spheres, leads to a sequence of submanifolds with decreasing intrinsic dimensions, which can be interpreted as an analogue of principal component analysis. In a number of real datasets, an apparent one-dimensional mode of variation curving through more than one geodesic component is captured in the one-dimensional component of principal nested spheres. While analysis of principal nested spheres provides an intuitive and flexible decomposition of the high-dimensional sphere, an interesting special case of the analysis results in finding principal geodesics, similar to those from previous approaches to manifold principal component analysis. An adaptation of our method to Kendall's shape space is discussed, and a computational algorithm for fitting principal nested spheres is proposed. The result provides a coordinate system to visualize the data structure and an intuitive summary of principal modes of variation, as exemplified by several datasets.

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“…Another approach to interpolation on manifolds consists of mapping the data points onto the affine tangent space at a particular point of M , then computing an interpolating curve in the tangent space, and finally mapping the resulting curve back to the manifold. The mapping can be defined, e.g., by a rolling procedure, see [HS07,KDL07].…”

Section: Previous Workmentioning

confidence: 99%

A Gradient-Descent Method for Curve Fitting on Riemannian Manifolds

et al. 2011

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Given data points p0, . . . , pN on a closed submanifold M of R n and time instants 0 = t0 < t1 < . . . < tN = 1, we consider the problem of finding a curve γ on M that best approximates the data points at the given instants while being as "regular" as possible. Specifically, γ is expressed as the curve that minimizes the weighted sum of a sum-of-squares term penalizing the lack of fitting to the data points and a regularity term defined, in the first case as the mean squared velocity of the curve, and in the second case as the mean squared acceleration of the curve. In both cases, the optimization task is carried out by means of a steepest-descent algorithm on a set of curves on M . The steepest-descent direction, defined in the sense of the first-order and second-order Palais metric, respectively, is shown to admit analytical expressions involving parallel transport and covariant integral along curves. Illustrations are given in R n and on the unit sphere.

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Shape-space smoothing splines for planar landmark data

Cited by 63 publications

References 10 publications

Intrinsic Polynomials for Regression on Riemannian Manifolds

Intrinsic Polynomials for Regression on Riemannian Manifolds

Analysis of principal nested spheres

A Gradient-Descent Method for Curve Fitting on Riemannian Manifolds

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