Geodesic Regression for Image Time-Series

Niethammer, Marc; Huang, Yang; Vialard, François-Xavier

doi:10.1007/978-3-642-23629-7_80

Cited by 119 publications

(113 citation statements)

References 13 publications

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“…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. Geodesic models are useful, but are limited by their lack of flexibility when modelling complex trends.…”

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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“…4). This is not quite the same as (10), not just because of the additional regularization, but also since here the point-to-point correspondence…”

Section: The Optimization Algorithmmentioning

confidence: 85%

“…A variational formulation of geodesic regression is given in [9], where for given input shapes S i at times t i the (in a least squares sense) best approximating geodesic is computed as the minimizer of the energy i dist 2 (exp p (t i v), S i ) over the initial shape S of the geodesic path and its initial velocity or momentum v. Here, dist(·, ·) is the Riemannian distance and exp the exponential map. A computationally efficient method in the group of diffeomorphisms shape space is based on duality calculus in constrained optimization and presented in [10]. In [11], a generalization allowing for image metamorphosis-simultaneous diffeomorphic image deformation and image intensity modulation-is proposed.…”

Section: Introductionmentioning

confidence: 99%

Discrete Geodesic Regression in Shape Space

Berkels

Fletcher

Heeren

et al. 2013

Lecture Notes in Computer Science

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Abstract.A new approach for the effective computation of geodesic regression curves in shape spaces is presented. Here, one asks for a geodesic curve on the shape manifold that minimizes a sum of dissimilarity measures between given two-or three-dimensional input shapes and corresponding shapes along the regression curve. The proposed method is based on a variational time discretization of geodesics. Curves in shape space are represented as deformations of suitable reference shapes, which renders the computation of a discrete geodesic as a PDE constrained optimization for a family of deformations. The PDE constraint is deduced from the discretization of the covariant derivative of the velocity in the tangential direction along a geodesic. Finite elements are used for the spatial discretization, and a hierarchical minimization strategy together with a Lagrangian multiplier type gradient descent scheme is implemented. The method is applied to the analysis of root growth in botany and the morphological changes of brain structures due to aging.

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“…Rentmeesters [24], Fletcher [11] and Hinkle et al [15] address the problem of geodesic fitting on Riemannian manifolds, mostly focusing on symmetric spaces. Niethammer et al [22] generalized linear regression to the manifold of diffeomorphisms to model image time-series data, followed by works extending this concept [16,25,26].…”

Section: Related Workmentioning

confidence: 99%

“…In principle, we can distinguish between two groups of approaches: first, geodesic shooting based strategies which address the problem using adjoint methods from an optimal-control point of view [22,16,25,26]; the second group comprises strategies which are based on optimization techniques that leverage Jacobi fields to compute the required gradients [11,24]. Unlike Jacobi field approaches, solutions using adjoint methods do not require computation of the curvature explicitly and easily extend to higher-order models, e.g., polynomials [15], splines [26], or piecewise regression models.…”

Section: Related Workmentioning

confidence: 99%

Geodesic Regression on the Grassmannian

Hong

Kwitt

Singh

et al. 2014

Computer Vision – ECCV 2014

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Abstract. This paper considers the problem of regressing data points on the Grassmann manifold over a scalar-valued variable. The Grassmannian has recently gained considerable attention in the vision community with applications in domain adaptation, face recognition, shape analysis, or the classification of linear dynamical systems. Motivated by the success of these approaches, we introduce a principled formulation for regression tasks on that manifold. We propose an intrinsic geodesic regression model generalizing classical linear least-squares regression. Since geodesics are parametrized by a starting point and a velocity vector, the model enables the synthesis of new observations on the manifold. To exemplify our approach, we demonstrate its applicability on three vision problems where data objects can be represented as points on the Grassmannian: the prediction of traffic speed and crowd counts from dynamical system models of surveillance videos and the modeling of aging trends in human brain structures using an affine-invariant shape representation.

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Geodesic Regression for Image Time-Series

Cited by 119 publications

References 13 publications

Intrinsic Polynomials for Regression on Riemannian Manifolds

Intrinsic Polynomials for Regression on Riemannian Manifolds

Discrete Geodesic Regression in Shape Space

Geodesic Regression on the Grassmannian

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