Collaborative Research: Data-Driven Elastic Shape Analysis with Topological Inconsistencies and Partial Matching Constraints

这个研究项目的重点是表示为曲线或曲面的数据结构。这种结构出现在从大脑解剖学、计算机视觉和分子生物学到气象和金融数据的应用中。数据可以通过激光扫描等设备直接获取，也可以通过显微镜或磁共振成像间接重建。这种图像和分析出现在例如人体解剖学和运动的研究中或计算机图形和运动的应用中。该项目在于统计形状分析的广泛领域，其中一个试图量化几何和/或拓扑变异内和跨人群。这项工作的目的是开发实用的方法来描述数据对象的形状，以确定它们在更大的系统中的作用。该项目将涉及研究生和生产开放源码软件。该项目依赖于弹性形状分析的范例，传统上，弹性形状分析关注的是分析所考虑的对象的几何形状的变化。它的核心是两个形状之间的距离的概念，它源于黎曼设置，使用一个度量，该度量对某些形状保持变换的作用是不变的，并且嵌入了空间的全局非线性及其局部线性。第一个目标是开发一个全面的理论和数值框架的弹性形状分析的曲线和曲面，允许拓扑不一致和部分匹配的约束。该框架将弹性形状分析与几何测量理论和拓扑数据分析方法相结合。第二个目标是将该框架应用于各种各样的合成和真实的数据，特别是高分辨率骨科表面复合物的形态分析。该奖项反映了NSF的法定使命，并被认为值得通过使用基金会的智力价值和更广泛的影响审查标准进行评估来支持。

This research project focuses on data structures that are represented as curves or surfaces. Such structures occur in applications ranging from brain anatomy, computer vision, and molecular biology to meteorological and financial data. Data can be either directly acquired by devices such as laser scans or indirectly reconstructed from microscopy or magnetic resonance imaging. Such images and analyses appear for example in the study of human anatomy and motion or in applications to computer graphics and motion. The project lies in the broad area of statistical shape analysis, in which one tries to quantify geometric and/or topological variability within and across populations. The work aims to develop practical methods to characterize the shape of data objects in order to ascertain their roles in larger systems. The project will involve graduate students and produce open source software. The project relies on the paradigms of elastic shape analysis, which is traditionally concerned with analyzing the variability in the geometries of the objects under consideration. At its core is the notion of a distance between two shapes, which stems from a Riemannian setting using a metric that is invariant to the action of certain shape-preserving transformations and embeds both the global nonlinearity of the space as well as its local linearity. The first goal is to develop a comprehensive theoretical and numerical framework for elastic shape analysis of curves and surfaces that allows for topological inconsistencies and partial matching constraints. This framework combines elastic shape analysis with methods from geometric measure theory and topological data analysis. The second goal is to apply the framework to a wide variety of synthetic and real data, in particular to morphological analysis of high-resolution orthopedic surface complexes.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.