Younes Laurent, Arrate Felipe, Miller Michael I
Center for Imaging Science, The Johns Hopkins University, 3400N Charles St., Baltimore, MD 21218, USA.
Neuroimage. 2009 Mar;45(1 Suppl):S40-50. doi: 10.1016/j.neuroimage.2008.10.050. Epub 2008 Nov 12.
One of the main purposes in computational anatomy is the measurement and statistical study of anatomical variations in organs, notably in the brain or the heart. Over the last decade, our group has progressively developed several approaches for this problem, all related to the Riemannian geometry of groups of diffeomorphisms and the shape spaces on which these groups act. Several important shape evolution equations that are now used routinely in applications have emerged over time. Our goal in this paper is to provide an overview of these equations, placing them in their theoretical context, and giving examples of applications in which they can be used. We introduce the required theoretical background before discussing several classes of equations of increasingly complexity. These equations include energy minimizing evolutions deriving from Riemannian gradient descent, geodesics, parallel transport and Jacobi fields.
计算解剖学的主要目的之一是对器官的解剖变异进行测量和统计研究，尤其是大脑或心脏的解剖变异。在过去十年中，我们团队针对这个问题逐步开发了几种方法，所有这些方法都与微分同胚群的黎曼几何以及这些群作用的形状空间有关。随着时间的推移，出现了几个现在在应用中经常使用的重要形状演化方程。本文的目标是对这些方程进行概述，将它们置于理论背景中，并给出可以使用它们的应用示例。在讨论几类复杂度不断增加的方程之前，我们先介绍所需的理论背景。这些方程包括源自黎曼梯度下降、测地线、平行移动和雅可比场的能量最小化演化。
