About simple variational splines from the Hamiltonian viewpoint

TL;DR

This paper explores simple variational splines on Riemannian manifolds using Hamiltonian methods, providing a unified approach to solving optimal control problems for smooth curves with applications in computational anatomy.

Contribution

It introduces a Hamiltonian framework for simple splines on manifolds, including explicit equations and analysis, especially on the 2-sphere, with potential applications in computational anatomy.

Findings

Derived intrinsic Hamiltonian equations for splines on manifolds.

Demonstrated Hamiltonian dynamics on the 2-sphere.

Discussed applications to landmark co-metrics in anatomy.

Abstract

In this paper, we study simple splines on a Riemannian manifold $Q$ from the point of view of the Pontryagin maximum principle (PMP) in optimal control theory. The control problem consists in finding smooth curves matching two given tangent vectors with the control being the curve's acceleration, while minimizing a given cost functional. We focus on cubic splines (quadratic cost function) and on time-minimal splines (constant cost function) under bounded acceleration. We present a general strategy to solve for the optimal hamiltonian within the PMP framework based on splitting the variables by means of a linear connection. We write down the corresponding hamiltonian equations in intrinsic form and study the corresponding hamiltonian dynamics in the case $Q$ is the $2$-sphere. We also elaborate on possible applications, including landmark cometrics in computational anatomy.