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research article

Invariant Higher-Order Variational Problems

Gay-Balmaz, Francois  
•
Holm, Darryl D.
•
Meier, David M.
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2012
Communications In Mathematical Physics

We investigate higher-order geometric k-splines for template matching on Lie groups. This is motivated by the need to apply diffeomorphic template matching to a series of images, e.g., in longitudinal studies of Computational Anatomy. Our approach formulates Euler-Poincar, theory in higher-order tangent spaces on Lie groups. In particular, we develop the Euler-Poincar, formalism for higher-order variational problems that are invariant under Lie group transformations. The theory is then applied to higher-order template matching and the corresponding curves on the Lie group of transformations are shown to satisfy higher-order Euler-Poincar, equations. The example of SO(3) for template matching on the sphere is presented explicitly. Various cotangent bundle momentum maps emerge naturally that help organize the formulas. We also present Hamiltonian and Hamilton-Ostrogradsky Lie-Poisson formulations of the higher-order Euler-Poincar, theory for applications on the Hamiltonian side.

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Type
research article
DOI
10.1007/s00220-011-1313-y
Web of Science ID

WOS:000298802200005

Author(s)
Gay-Balmaz, Francois  
Holm, Darryl D.
Meier, David M.
Ratiu, Tudor S.  
Vialard, Francois-Xavier
Date Issued

2012

Published in
Communications In Mathematical Physics
Volume

309

Start page

413

End page

458

Subjects

Geodesic-Flows

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Lie Quadratics

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Reduction

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Geometry

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Splines

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Metrics

•

Cubics

Editorial or Peer reviewed

REVIEWED

Written at

EPFL

EPFL units
CAG2  
Available on Infoscience
February 9, 2012
Use this identifier to reference this record
https://infoscience.epfl.ch/handle/20.500.14299/77644
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