III: Small: Multi-modal Neuroimaging Data Fusion and Analysis with Harmonic Maps Under Designed Riemannian Metric

获取多模式神经成像数据的快速发展为系统描述人脑结构、其与认知和行为的关系以及遗传和环境因素对大脑回路个体差异的贡献提供了令人兴奋的新机会。为了优化利用如此丰富的多模式数据，迫切需要强大的计算框架来整合和分析多源数据。当前的实践通常将来自不同数据源的可用数据特征组合在一起，而没有考虑数据源之间内在的几何和生物结构关系。基于形状分析的方法可以为多模型数据融合和分析提供一种通用的综合方法的桥梁。虽然许多研究致力于图像数据配准的研究，但在将具有某些物理性质和几何内在结构的不同形态数据集成方面取得的进展有限。该方案的重点是研究和发展具有规定黎曼度量的调和映射的计算算法，并为一般的多模式数据融合和分析问题提供理论上合理和实用有效的解决方案。这项提案中概述的工作将在许多研究领域中得到应用，包括(1)形状分析、神经成像和一般的医学成像。建议的研究统一和连接各种计算几何技术，并解决一些开放的问题，使其成为教授形状分析概念的理想框架，并为学生提供了一个更广泛的环境，不同的组成部分可以安装在一起。该项目开发的算法和工具将对神经成像研究产生直接影响。它可能使发现一些神经退行性疾病的多模式成像生物标记物成为可能，例如阿尔茨海默病。调和映射及其相关方法在许多领域都有应用，包括医学成像、计算机视觉、机器学习、计算机图形学和几何建模。PI将使社会能够接触到软件工具。该项目将促进神经成像研究的新课程和实验室基础设施的开发。它还为计算机科学专业的学生提供了一个独特的机会，让他们更有效地学习神经科学。这笔资金将使正在进行的努力得以继续，积极地从代表性不足的群体中招收学生并为他们提供建议。在这个项目中，概述了一个综合的研究和教育计划，以调查和开发计算定理和算法。第一个目标是发展一种方法来计算一般曲面之间在设计的黎曼度量下的调和映射。一个关键的创新之处在于，新方法将多源信息表示为黎曼度量，从而将多源融合问题转化为计算与目标表面上任何设计的黎曼度量相适应的表面调和映射。接下来，我们将建立一个通过调整黎曼度量来优化微分同胚调和映射的变分公式。调和映射保证了其具有全局最小变形。这将是一种实用的方法来优化曲面之间的微分同胚，并提供了引入由其他数据源定义的一般目标函数的灵活性。此外，还将实现在设计的黎曼度量下的体积调和映射的算法以及用于多源数据分析的一组新的多变量几何统计量。该框架探索了具有内在几何结构的多源数据融合，多变量统计可能为神经成像分析提供更敏感、更可靠和更容易获得的脑成像生物标志物。本研究项目的预期成果是：(1)调和映射的新计算算法，在医学成像、计算机视觉、机器学习、计算机图形学和几何建模等领域具有重要应用；(2)具有严格数学基础的实用软件包，用于分析多模式数据并生成敏感而全面的多变量成像统计数据。它将在一个大型公共神经成像数据集上进行测试，并通过各种分类任务进行评估。

The rapid development in acquiring multi-modal neuroimaging data provides exciting new opportunities to systematically characterize human brain structure, its relationship to cognition and behavior, and the contributions of genetic and environmental factors to individual differences in brain circuitry. To optimally use such rich multi-modal data, there is an urgent need for powerful computational frameworks to integrate and analyze multi-source data. The current practice usually combines available data features from different sources without considering the intrinsic geometry and biology structure relationship between data sources. Shape analysis based approach may serve as a bridge for a general and integrative approach to multi-model data fusion and analysis. Although numerous studies have been devoted to imaging data registration research, limited progress has been made to integrate different modality data with some physically nature and geometrically intrinsic structures. This proposal focuses on investigating and developing computational algorithms on harmonic map with prescribed Riemannian metric, and on producing theoretically sound and practically efficient solutions for general multi-modal data fusion and analysis problems. The work outlined in this proposal will have applications in a number of research fields, including (1) Shape Analysis, neuroimaging and medical imaging in general. The proposed research unifies and connects a variety of computational geometry techniques and tackles a few open problems making it an ideal framework for teaching concepts in shape analysis as well as providing students a broader context in which various components may fit together. The algorithms and tools developed in this project will have a direct impact on neuroimaging research. It may enable discovery of multi-modal imaging biomarkers for some neurodegenerative disease, such as Alzheimer's disease. Harmonic maps and their related methods have applications in many other fields, including medical imaging, computer vision, machine learning, computer graphics, and geometric modeling. The PI will make the software tools accessible to the society. This project will facilitate the development of new courses and laboratory infrastructure for neuroimaging research. It also provides a unique opportunity for students from computer science to learn neuroscience more efficiently. The funding will allow continuation of ongoing efforts to actively recruit and advise students from under-represented groups.An integrated research and education plan is outlined in this project to investigate and develop computational theorems and algorithms. The first goal is to develop a method to compute the harmonic map under a designed Riemannian metric between general surfaces. One key novelty is that the new method formulates multi-source information with a Riemannian metric and thus the multi-source fusion problem is converted to compute a surface harmonic map which is adapted to any designed Riemannian metric on the target surface. Next, a variational formulation that optimizes the diffeomorphic harmonic map via adjusting the Riemannian metric will be developed. The harmonic map guarantees that it has the global minimum deformation. It will be a practical way to optimize diffeomorphisms between surfaces and provide the flexibility to introduce general objective functions defined by other data sources. In addition, an algorithm for volumetric harmonic maps under a designed Riemannian metric and a set of novel multivariate geometry statistics for multi-source data analysis will be implemented. The framework explores multi-source data fusion with intrinsic geometry structures and the multivariate statistics may provide more sensitive, reliable and accessible brain imaging biomarkers for neuroimaging analysis. The anticipated outcomes of this research project are: (1) new computational algorithms on harmonic maps with significant applications in various fields, such as medical imaging, computer vision, machine learning, computer graphics and geometric modeling; (2) a practical software package with a rigorous mathematical foundation to analyze multi-modal data and produce sensitive and comprehensive multivariate imaging statistics. It will be tested on a large public neuroimaging dataset and evaluated by various classification tasks.