Invariant higher-order variational problems

Abstract

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-Poincaré theory in higher-order tangent spaces on Lie groups. In particular, we develop the Euler-Poincaré 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-Poincaré 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-Poincaré theory for applications on the Hamiltonian side.

46 pages, 8 figures, First version -- comments welcome!

References in corpus (1)

• Momentum Maps and Measure-valued Solutions (Peakons, Filaments and Sheets) for the EPDiff Equation

Cited by in corpus (23)

• Invariant higher-order variational problems II
• Higher order mechanics on graded bundles
• Retraction maps: a seed of geometric integrators
• On the geometry of higher-order variational problems on Lie groups
• Higher-order discrete variational problems with constraints
• Inexact trajectory planning and inverse problems in the Hamilton--Pontryagin framework
• Geometric integrators for higher-order mechanics on Lie groups
• Quantum splines
• About simple variational splines from the Hamiltonian viewpoint
• Bundle-theoretic methods for higher-order variational calculus
• Beyond Euclid: An Illustrated Guide to Modern Machine Learning with Geometric, Topological, and Algebraic Structures
• Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions
• Optimal control of affine connection control systems from the point of view of Lie algebroids
• Un-reduction
• Regularity properties of fiber derivatives associated with higher-order mechanical systems
• Unified formalism for higher-order variational problems and its applications in optimal control
• A general framework for quantum splines
• Tulczyjew's Triplet for Lie Groups III : Higher Order Dynamics and Reductions for Iterated Bundles
• Reductions of Dynamics on Second Iterated Bundles of Lie Groups
• Variational integrators for underactuated mechanical control systems with symmetries
• Continuous matter creation and the acceleration of the universe: a replay
• Classification results for polyharmonic helices in space forms
• Dynamic interpolation for obstacle avoidance on Riemannian manifolds