Numerical Computation of Geodesics in the Framework of Metamorphosis

所提出的研究侧重于用于形状分析的变形，它依赖于形状变换模型，在该模型中，形状变化与数据的其他变换相耦合，允许拓扑变化或附加到变形对象的属性的部分平流。这一结果形成了一个通用的框架，其中可以基于任何数学结构来设计许多不同的模型，这些数学结构既可以通过微分同胚平流，也可以嵌入到希尔伯特或黎曼空间中。这种构造为可变形感兴趣对象的空间配备了新的黎曼度量，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的表示。这项研究将涉及变形模型，在该模型中，可变形的结构通过图像、密度或测量来表示，在二维或三维。这里的主要问题之一是测地线的计算，或者作为变分问题(流形上两点之间的最短路径)，或者作为初值问题(求解测地线演化的欧拉-拉格朗日方程)。这两个问题的数值分析都是具有挑战性的，特别是当人们增加了两个解在数值上一致的要求时，即第一个问题的离散解与第二个问题的离散解重合，这对应用非常重要。这项研究将通过开发初值问题的变分积分器和边值问题的打靶方法来解决这些问题，在涉及光滑和奇异分量的解决方案的情况下。PI和合作者还将部署和扩展一个综合软件，该软件提供与微分同胚匹配相关的算法集合。形状分析的目标是理解和表示可变形对象(如地标、图像、曲线或曲面的集合)数据集中的形状变化。这个问题对于描述医学图像中的解剖变异及其与病理学的关系尤其重要。在这方面的主要应用领域之一被称为计算解剖学，来自数学形状分析的方法已经被用于几个成功的应用。这一领域的发展例子包括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在分析大脑疾病和心力衰竭方面的合作。这项研究中将开发的理论和工具将能够分析以前方法无法处理的情况，这些方法的工作假设是，解剖变化基本上可以通过形状的平滑变化来描述。被提议的方法，称为变形法，将能够解决这些假设不满足的情况，并使例如，包括受试者之间的戏剧性变化的图像分析成为可能。这包括对衡量肿瘤进化的数据集的分析，或描述重大中风后的大脑恢复。这项研究将有助于在这种情况下出现新的解决方案，并向科学界提供相关软件。

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.