FRG: Collaborative Research: Geometric and Topological Methods for Analyzing Shapes

当我们与我们的环境互动时，我们不断地评估、测量和比较环境中的对象。我们对这些物体的感知，无论是用我们的眼睛，还是通过扫描仪的结果，都是由它们的形状决定的，而它们的形状本身又是由它们表面的几何形状来表征的。我们目前正在经历一场关于物体形状的离散几何数据的爆炸式增长，这些数据来自扫描仪、相机、成像系统、传感器、卫星，甚至手机。迫切需要对这些几何数据进行自动处理，以进行形状匹配、形状比较和形状识别。这种需求出现在面部识别、对骨骼化石进行识别和分类、区分骨骼骨折、诊断器官中的肿瘤和异常以及测量大脑图像随时间的变化等领域。因此，创建一种数学理论并开发算法来识别和对齐这种几何形状是具有深远影响的主要研究挑战。过去几个世纪发展起来的几何和分析方面的深奥数学理论现在正在形状匹配领域找到应用。这个项目探索了这一令人兴奋的领域的基本问题，这一领域即将取得重大进展。它通过使用共形、调和和等距映射理论对齐曲面来实现这一点。虽然这些理论已经在数学背景下进行了广泛的研究，但它们对计算算法的适应性仍在开发中。该项目为这一新兴学科开发了一个连贯和全面的理论框架，以及与科学应用的具体联系。它计划实施并向公众提供一系列软件，这些软件将提供新的工具，并将为生物学、医学、人类学和其他领域的科学家开辟新的探究路线，在这些领域，形状分析发挥着核心作用。我们三维世界中的表面可以在数学上描述为二维黎曼流形。研究这类曲面上的几何结构是拓扑学和微分几何等数学领域的中心课题。它引出了共形几何、模空间、调和和共形映射以及黎曼曲面的经典理论。这些领域现在被应用于研究骨骼、大脑皮层、蛋白质和其他生物分子的表面。当用激光、雷达或CAT扫描观察物体时，我们得到这样一个表面的离散表示。经典理论不足以处理这些真实世界的数据。这个项目将发展曲面的共形映射和调和映射的离散对应，探索它们的存在、唯一性和微分同态性质，并建立离散理论到经典光滑理论的收敛。它还将创建和实现纳入这一理论的算法，为科学家和其他从业者创建可用的软件。通过这种方式，这个项目将在几何和拓扑学的数学理论和将这些思想应用于形状数据的算法分析之间架起一座桥梁。这个奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

As we interact with our environment we constantly assess, measure, and compare the objects within it. Our perception of such objects, either with our eyes, or through the results of scanners, is determined by their shapes, which are themselves characterized by the geometry of their surfaces. We are currently experiencing an explosion of discrete geometric data on shapes of objects obtained from scanners, cameras, imaging systems, sensors, satellites, and even cell phones. There is an urgent need for this geometric data to be processed automatically, for shape matching, shape comparison, and shape recognition. This need arises in areas such as facial recognition, identifying and classifying fossilized bones, distinguishing fractures in bones, diagnosing tumors and anomalies in organs, and measuring changes over time in brain images. Creating a mathematical theory and developing algorithms to recognize and to align such geometric shapes are therefore major research challenges that have far-reaching implications. Deep mathematical theories in geometry and analysis that were developed over the past centuries are now finding applications in this field of shape matching. This project explores fundamental issues in this exciting area, which is on the cusp of seeing major advances. It does so by using the theory of conformal, harmonic, and isometric mappings to align surfaces. While these theories have been extensively studied in a mathematical context, their adaptation to computational algorithms is still under development. This project develops a cohesive and comprehensive theoretical framework for this emerging discipline along with concrete connections to scientific applications. It plans to implement and make publicly available a collection of software that will offer new tools and will open new lines of inquiry to scientists in biology, medicine, anthropology and other fields where the analysis of shape plays a central role.The surfaces in our three-dimensional world can be described mathematically as two-dimensional Riemannian manifolds. Study of the geometric structures on such surfaces is a central topic in mathematical areas such as topology and differential geometry. It leads to classical theories of conformal geometry, moduli spaces, harmonic and conformal maps, and Riemann surfaces. These fields are now being applied to study surfaces of bones, brain cortices, proteins and other bio-molecules. When viewing an object with a laser, or radar, or CAT scan, we obtain a discrete representation of such a surface. Classical theories are inadequate for processing this real world data. This project will develop discrete counterparts of conformal and harmonic maps of surfaces, explore their existence, uniqueness, and diffeomorphism properties, and establish the convergence of the discrete theory to the classical smooth theory. It will also create and implement algorithms that incorporate this theory to create usable software for scientists and other practitioners. In this way, this project will bridge the gap between the mathematical theories of geometry and topology and the application of such ideas to algorithmic analysis of shape data.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.