Invariant higher-order variational problems

TL;DR

This paper develops a higher-order Euler-Poincaré variational framework for invariant geometric problems on Lie groups, enabling advanced template matching techniques in computational anatomy.

Contribution

It introduces a higher-order Euler-Poincaré formalism on Lie groups and applies it to template matching, including explicit examples on SO(3).

Findings

Formulated higher-order Euler-Poincaré equations for Lie group variational problems.

Derived Hamiltonian and Lie-Poisson formulations for the higher-order theory.

Applied the framework to template matching on the sphere (SO(3)).

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-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.