Invariant higher-order variational problems

Autor: François Gay-Balmaz, Tudor S. Ratiu, Darryl D. Holm, David M. Meier, François-Xavier Vialard
Přispěvatelé: CEntre de REcherches en MAthématiques de la DEcision (CEREMADE), Centre National de la Recherche Scientifique (CNRS)-Université Paris Dauphine-PSL, Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL), Laboratoire de Météorologie Dynamique (UMR 8539) (LMD), Département des Géosciences - ENS Paris, École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École normale supérieure - Paris (ENS Paris), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-Centre National de la Recherche Scientifique (CNRS)-École des Ponts ParisTech (ENPC)-École polytechnique (X)-Institut national des sciences de l'Univers (INSU - CNRS)-Université Pierre et Marie Curie - Paris 6 (UPMC), Department of Mathematics [Imperial College London], Imperial College London, Section de Mathématiques and Bernoulli Center, École Polytechnique Fédérale de Lausanne, CH-1015 Lausanne, Switzerland, Université Pierre et Marie Curie - Paris 6 (UPMC)-Institut national des sciences de l'Univers (INSU - CNRS)-École polytechnique (X)-École des Ponts ParisTech (ENPC)-Centre National de la Recherche Scientifique (CNRS)-Département des Géosciences - ENS Paris, École normale supérieure - Paris (ENS-PSL), Université Paris sciences et lettres (PSL)-Université Paris sciences et lettres (PSL)-École normale supérieure - Paris (ENS-PSL)
Rok vydání: 2012
Předmět:
Splines
Pure mathematics
[PHYS.MPHY]Physics [physics]/Mathematical Physics [math-ph]
Complex system
Geometry
FOS: Physical sciences
01 natural sciences
Lie Quadratics
Mathematics - Analysis of PDEs
[MATH.MATH-MP]Mathematics [math]/Mathematical Physics [math-ph]
0502 economics and business
Tangent space
FOS: Mathematics
Cubics
[MATH.MATH-AP]Mathematics [math]/Analysis of PDEs [math.AP]
0101 mathematics
Invariant (mathematics)
Mathematical Physics
Reduction
Mathematics
Template matching
010102 general mathematics
05 social sciences
Lie group
Statistical and Nonlinear Physics
Geodesic-Flows
Mathematical Physics (math-ph)
Nonlinear Sciences - Chaotic Dynamics
16. Peace & justice
Computational anatomy
[NLIN.NLIN-CD]Nonlinear Sciences [physics]/Chaotic Dynamics [nlin.CD]
Cotangent bundle
Metrics
Diffeomorphism
Chaotic Dynamics (nlin.CD)
050203 business & management
[PHYS.PHYS.PHYS-DATA-AN]Physics [physics]/Physics [physics]/Data Analysis
Statistics and Probability [physics.data-an]
Analysis of PDEs (math.AP)
Zdroj: Communications in Mathematical Physics
Communications in Mathematical Physics, Springer Verlag, 2012, 309 (2), pp.413-458. ⟨10.1007/s00220-011-1313-y⟩
Communications in Mathematical Physics, 2012, 309 (2), pp.413-458. ⟨10.1007/s00220-011-1313-y⟩
ISSN: 0010-3616
1432-0916
DOI: 10.1007/s00220-011-1313-y
Popis: 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\'e theory in higher-order tangent spaces on Lie groups. In particular, we develop the Euler-Poincar\'e 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\'{e} 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\'e theory for applications on the Hamiltonian side.
Comment: 46 pages, 8 figures, First version -- comments welcome!
Databáze: OpenAIRE
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