具有高阶动量分布的LDDMM的对称性
Symmetries in LDDMM with higher order momentum distributions
摘要：
在用于图像配准的大变形超纯度量映射公式的一些实现中，我们考虑局部平移图像数据的粒子的运动。然后，我们提升粒子的运动，以获得整个图像上的运动。然而，当然也可以考虑不只是平移的粒子，这就是本文将要描述的。由于与EPDiff相关的未约拉格朗日量具有$\Diff(M)$对称性，因此对于任何李群，它也必须具有$G \subset \Diff(M)$对称性。在本文中，我们将描述一个塔李群$G^{(0)} \subseteq G^{(1)} \subseteq G^{(2)} \subseteq...$，它对应于保持$k$阶喷流数据。简化的组态空间$Q^{(k)} := \Diff(M) / G^{(k)}$将是有限维的（特别地，$Q^{(0)}$是$M$中$N$粒子的组态流形）。我们将观察到$G^{(k)}$是$G^{(0)}$的正规子群，因此商$G^{(0)} / G^{(k)}$本身是作用于$Q^{(k)}$的（有限维）李群。这使得$Q^{(k)}$成为$Q^{(0)}$上的一个主丛，并且在$Q^{(k)}$上的约化测地线方程将具有$G^{(0)} / G^{(k)}$ -对称性。Noether定理揭示了$T^{\ast}Q^{(k)}$上约化系统守恒动量的存在性。
英文摘要：
In some implementations of the Large Deformation Diffeomorphic Metric Mapping formulation for image registration we consider the motion of particles which locally translate image data. We then lift the motion of the particles to obtain a motion on the entire image. However, it is certainly possible to consider particles which do more than translate, and this is what will be described in this paper. As the unreduced Lagrangian associated to EPDiff possesses $\Diff(M)$ symmetry, it must also exhibit $G \subset \Diff(M)$ symmetry, for any Lie subgroup. In this paper we will describe a tower of Lie groups $G^{(0)} \subseteq G^{(1)} \subseteq G^{(2)} \subseteq...$ which correspond to preserving $k$-th order jet-data. The reduced configuration spaces $Q^{(k)} := \Diff(M) / G^{(k)}$ will be finite-dimensional (in particular, $Q^{(0)}$ is the configuration manifold for $N$ particles in $M$). We will observe that $G^{(k)}$ is a normal subgroup of $G^{(0)}$ and so the quotient $G^{(0)} / G^{(k)}$ is itself a (finite dimensional) Lie group which acts on $Q^{(k)}$. This makes $Q^{(k)}$ a principle bundle over $Q^{(0)}$ and the reduced geodesic equations on $Q^{(k)}$ will possess $G^{(0)} / G^{(k)}$-symmetry. Noether's theorem implies the existence of conserved momenta for the reduced system on $T^{\ast}Q^{(k)}$.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
