Numerical Computation of Geodesics in the Framework of Metamorphosis

拟议的研究重点是变形的形状分析，它依赖于形状transformationmodel内的形状变化是与其他变换的数据相结合，cartingtopological变化，或部分平流属性附加到变形对象。这一结果是一个通用的框架，在其中可以设计出许多不同的模型，基于任何可以由同胚平流并嵌入希尔伯特或黎曼空间的结构。这种构造为感兴趣的可变形对象的空间配备了一个新的黎曼度量，允许对这些对象进行比较，并允许使用与黎曼流形中的数据分析相关的工具，如指数图中的数据集表示。该研究将涉及变形模型，其中可变形结构由二维或三维的图像、密度或测量来表示。在这方面的主要问题之一是测地线的计算，无论是作为一个变分问题（最短路径之间的两个点在流形）或作为一个初始值问题（解决欧拉-拉格朗日方程的发展测地线）。这两个问题的数值分析是具有挑战性的，特别是当一个增加的要求，两个解决方案是数值一致的，在这个意义上，第一个问题的离散解决方案与第二个的离散解决方案相吻合，这是重要的应用。本研究将针对这些问题，借由发展用于初值问题的变分积分器，以及用于边值问题的打靶法，在将涉及结合联合收割机平滑和奇异组件的解决方案的上下文中。PI和合作者还将部署和扩展一个综合软件，该软件提供了一系列与几何形态匹配相关的算法。形状分析的目标是理解和表示变化变形对象的数据集中的形状（如地标，图像，曲线或表面的集合）。这个问题是很重要的，特别是，在医学图像中的解剖变异的表征，和oftheir与病理的关系。在这方面的主要应用领域之一是被称为计算解剖学，从数学形状分析的方法已经被用于几个成功的应用。这一领域的发展包括PI与巴尔的摩肯尼迪克里格研究所或约翰霍普金斯大学计算医学研究所的研究人员合作，分析脑部疾病和心力衰竭。在这项研究中开发的理论和工具将能够分析以前的方法无法处理的情况，这些方法的工作假设是解剖学变化基本上可以通过形状的平滑变化来描述。所提出的方法，称为变形，将能够解决这些假设不满足的情况，并使之成为可能，例如，分析图像，包括主体之间的戏剧性变化。这包括对测量肿瘤进展或描述严重中风后大脑恢复的数据集进行分析。该研究将有助于在这种情况下出现新的解决方案，并使相关软件可供科学界使用。

The proposed research focuses on metamorphosis for shape analysis, which relies on a shape transformationmodel within which shape variation is coupled with other transformations of the data, permittingtopological changes, or partial advection of attributes attached to the deformed objects. This resultsin a versatile framework in which many different models can be devised, based on any mathematicalstructure that can both be advected by diffeomorphisms and embedded in a Hilbert orRiemannian space. This construction equips the space of deformable objects of interest with a newRiemannian metric, allowing for the comparison of these objects, and for the use of tools associatedto data analysis in Riemannian manifolds, like the representation of data sets in exponentialcharts. The research will involve models of metamorphosis in which the deformable structures are representedby images, densities, or measures, in two or three dimensions. One of the main issues inthis context is the computation of geodesics, either as a variational problem (shortest path betweentwo points in the manifold) or as an initial value problem (solving the Euler-Lagrange equation forthe evolution of geodesics). The numerical analysis of both problems is challenging, especiallywhen one adds the requirement for the two solutions to be numerically consistent, in the sense thatdiscrete solutions of the first problem coincide with discrete solutions of the second one, which isimportant for applications. This research will address these issues, by developing variationalintegrators for the initial value problems, and shooting methods for the boundary valueproblems, in contexts that will involve solutions that combine smooth and singular components.The PI and collaborators will also deploy and extend of a comprehensive softwarethat provides a collection of algorithms associated to diffeomorphic matching.The goal of shape analysis is to understand and represent variations of shapes in data sets ofdeformable objects (like collections of landmarks, images, curves or surfaces). This issue is important,in particular, for the characterization of anatomical variations in medical images, and oftheir relation with pathologies. One of the main areas of applications in this context is known asComputational Anatomy, and methods from mathematical shape analysis have already been used for several successful applications. Examples of developments in this domain include collaborations of the PI withresearchers at the Kennedy Krieger Institute in Baltimore, or at the Institute for Computational Medicine at Johns Hopkins University, on the analysis of brain disease and of cardiac failure. The theory and tools that will be developed in this research will enable the analysis of situations that cannot be handled by previous methods, which work under the assumption that anatomical variation can be essentially describedby smooth changes of shape. The proposed approach, called metamorphosis, will be able to address casesfor which these assumptions are not satisfied, and make possible, for example, the analysis of imagesthat include dramatic changes between subjects. This includes the analysis of datasets measuring the evolutionof tumors, or describing brain recovery after a major stroke. The research will contribute to the emergence of new solutionsin such contexts, and make the related software available to the scientific community.