Cubic Splines on Curved Spaces

Noakes, Lyle; Heinzinger, G.; Paden, B.

doi:10.1093/imamci/6.4.465

Cited by 228 publications

(202 citation statements)

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“…use, available at https://www.cambridge.org/core/terms. https://doi.org/10.1017/S1446788700036417 [11] Elastica in 50 (3) We now show that y satisfies a differential equation from the theory of elliptic functions. First, we have an additional integral for elastica in 50(3).…”

Section: Lemma 41 For Any a E 50(3) And Any T 0 E R (I) T \-Y A(v(mentioning

confidence: 75%

“…https://doi.org/10.1017/S1446788700036417 [7] Elastica in 50 (3) 111 By analogy with the Lie quadratics of [7][8][9] and [11] (unconstrained solutions of V = [V, V] + C), a curve V : I -+9 satisfying (2.3) and (2.4) for some C e 9 and all t e I will be called an elastic Lie quadratic with constant C. If x : / -> G is an elastic curve, the curve V defined by (2.2) is an elastic Lie quadratic; it will be called the elastic Lie quadratic associated with x. For later reference, we re-state the observation (2.9) from the preceding proof.…”

Section: Elastica In Lie Groupsmentioning

confidence: 99%

“…The critical points of <t>, Riemannian cubics, are studied in [7][8][9]11] and references therein. When vo and v\ are unit vectors, the elastic problem is to minimise 4> over [3] Elastica in 50 (3) 107…”

Section: Introductionmentioning

confidence: 99%

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Elastica in SO(3)

Popiel¹,

Noakes²

2007

J. Aust. Math. Soc.

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In a Riemannian manifold M, elastica are solutions of the Euler-Lagrange equation of the following second order constrained variational problem: find a unit-speed curve in M, interpolating two given points with given initial and final (unit) velocities, of minimal average squared geodesic curvature. We study elastica in Lie groups G equipped with bi-invariant Riemannian metrics, focusing, with a view to applications in engineering and computer graphics, on the group 50(3) of rotations of Euclidean 3-space. For compact G, we show that elastica extend to the whole real line. For G = 50(3), we solve the Euler-Lagrange equation by quadratures.2000 Mathematics subject classification: primary 49K99,49Q99.

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Section: Lemma 41 For Any a E 50(3) And Any T 0 E R (I) T \-Y A(v(mentioning

confidence: 75%

Section: Elastica In Lie Groupsmentioning

confidence: 99%

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Elastica in SO(3)

Popiel¹,

Noakes²

2007

J. Aust. Math. Soc.

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“…While differential geometric concepts, such as geodesics and intrinsic higherorder curves, have been well studied [23,5], their use for regression has only recently gained interest. A variety of methods extending concepts of regression in Euclidean spaces to nonflat manifolds have been proposed.…”

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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“…As we will see in examples, a natural choice of Lagrangian leads to Riemannian cubics and their higher-order generalizations. This class of curves was introduced in Noakes et al [8] and has since been studied in a series of papers including [9][10][11][12][13][14]. Riemannian cubics appear in a variety of applications, for example, in the quantum control problem mentioned above, but also in computer graphics, robotics and spacecraft control [15][16][17][18].…”

Section: Introductionmentioning

confidence: 99%

Inexact trajectory planning and inverse problems in the Hamilton–Pontryagin framework

2013

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We study a trajectory-planning problem whose solution path evolves by means of a Lie group action and passes near a designated set of target positions at particular times. This is a higher-order variational problem in optimal control, motivated by potential applications in computational anatomy and quantum control. Reduction by symmetry in such problems naturally summons methods from Lie group theory and Riemannian geometry. A geometrically illuminating form of the Euler-Lagrange equations is obtained from a higher-order Hamilton-Pontryagin variational formulation. In this context, the previously known node equations are recovered with a new interpretation as Legendre-Ostrogradsky momenta possessing certain conservation properties. Three example applications are discussed as well as a numerical integration scheme that follows naturally from the Hamilton-Pontryagin principle and preserves the geometric properties of the continuous-time solution.

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Cubic Splines on Curved Spaces

Cited by 228 publications

References 0 publications

Elastica in SO(3)

Elastica in SO(3)

Geodesic Regression on the Grassmannian

Inexact trajectory planning and inverse problems in the Hamilton–Pontryagin framework

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