174830
20180128034110.0
10.1007/s00220-011-1313-y
doi
000298802200005
ISI
ARTICLE
Invariant Higher-Order Variational Problems
2012
2012
Journal Articles
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.
Geodesic-Flows
Lie Quadratics
Reduction
Geometry
Splines
Metrics
Cubics
Gay-Balmaz, Francois
128168
240497
Holm, Darryl D.
Meier, David M.
Ratiu, Tudor S.
118378
243113
Vialard, Francois-Xavier
413-458
Communications In Mathematical Physics
309
CAG2
252609
oai:infoscience.tind.io:174830
article
118378
EPFL-ARTICLE-174830
EPFL
PUBLISHED
REVIEWED
ARTICLE