不变高阶变分问题
Invariant higher-order variational problems
摘要：
本文研究了李群上的高阶几何$k$ -样条模板匹配。这是由于需要将同构模板匹配应用于一系列图像，例如，计算解剖学的纵向研究我们的方法在李群的高阶切空间中建立了欧拉-庞加莱理论。特别是，我们开发的高阶变分问题，李群变换下不变的欧拉-庞加莱形式主义。然后将该理论应用于高阶模板匹配，并在李群变换上显示出相应的曲线满足高阶Euler-Poincaré方程。给出了球面上模板匹配的SO（3）的例子。各种余切束动量映射自然出现，有助于组织公式。我们还提出了哈密顿和哈密顿Ostrogradsky李泊松公式的高阶欧拉庞加莱理论的应用程序的哈密顿侧。
英文摘要：
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.
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