CDS&E: Computational Riemannian Approaches for Statistical Analysis and Modeling of Complex Structures

在更大、更复杂的生物系统中，将生物部分的结构或形态与其功能联系起来是一个广泛应用的巨大挑战。解决这一挑战的方法需要有效的工具来量化感兴趣结构中的形状差异，提取正常的可变性模式，并在回归研究中使用形状作为预测因子。该项目旨在开发用于分析两类生物部分形状的计算技术：(1)人脑皮质下结构，表示为MRI成像的脑体积内的2D表面；(2)神经元，表示为使用高分辨率显微镜获得的树状结构。虽然最近的研究，包括人类连接组项目和NeuroMorpho数据库，都集中在基于成像的这类结构的大型数据库的生成上，但系统分析它们的技术远远落后。这项跨学科研究的目的是利用几个不同领域的工具，开发用于表征生物结构形状的基本解决方案。目前用于比较形态属性的技术或者是纯粹的拓扑学的(通常关注零件的计数而忽略它们的形状，例如树的编辑距离)，或者是纯粹的几何(关注曲线和曲面的几何)。该项目旨在通过比较对象的几何形状但允许某些拓扑可变性来弥合这两种方法之间的差距。它还将使用黎曼方法解决形状分析中最具挑战性的问题--跨对象的零件配准。这些工具还将自然而然地扩展到对具有时间演变结构的树木的分析。正在开发的框架的性质--大脑表面和神经元部分的自动注册和比较--使这种方法既新颖又具有挑战性，所需的工具是跨学科的。该项目将使用微分几何(特别是希尔伯特流形的几何)、代数、计算统计学和成像科学的元素来开发有效的解决方案。该项目的一个重要部分将涉及实时、可扩展算法的开发，以分析可用于研究皮质下解剖学和神经元形态的大数据集。

Associating structures or morphologies of biological parts to their functionalities in larger, complex biosystems is a grand challenge with widespread applications. Approaches to this challenge require efficient tools for quantifying shape differences in structures of interest, extracting normal modes of variability, and using shape as a predictor in regression studies. This project aims to develop computational techniques for analyzing shapes of two types of biological parts: (1) subcortical structures in human brain, represented as 2D surfaces inside MRI-imaged brain volumes, and (2) neurons, represented as tree-like structures obtained using high-resolution microscopy. While recent research, including the Human Connectome Project and the NeuroMorpho database, has focused on imaging-based generation of large databases of such structures, the techniques for systematically analyzing them lag far behind. This interdisciplinary research aims to develop fundamental solutions for characterizing shapes of biological structures using tools from several different areas. It is anticipated that the techniques under development will also be applicable in broader scientific contexts.Current techniques for comparing morphological properties are either purely topological (generally focusing on part counts while ignoring their shapes, e.g. the tree-edit distance) or purely geometrical (focusing on geometries of curves and surfaces). This project aims to bridge the gap between these two approaches by comparing geometries of objects but allowing certain topological variability. It will also address the most challenging issue of shape analysis -- registration of parts across objects -- using Riemannian methods. These tools will also naturally extend to analysis of trees with temporally-evolving structures. The nature of the framework under development -- automatic registration and comparison of parts across brain surfaces and neurons -- makes this approach both novel and challenging, and the tools needed are interdisciplinary. The project will use elements from differential geometry (especially the geometry of Hilbert manifolds), algebra, computational statistics, and imaging sciences to develop efficient solutions. A significant portion of this project will concern development of real-time, scalable algorithms for analyzing large datasets available for studying subcortical anatomy and neuronal morphology.