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

当我们与环境互动时，我们会不断地评估、测量和比较环境中的物体。我们对这些物体的感知，无论是通过我们的眼睛，还是通过扫描仪的结果，都是由它们的形状决定的，而形状本身又由它们表面的几何形状来表征。我们目前正经历着从扫描仪、照相机、成像系统、传感器、卫星甚至手机中获得的关于物体形状的离散几何数据的爆炸式增长。迫切需要对这些几何数据进行自动处理，以便进行形状匹配、形状比较和形状识别。这种需求出现在面部识别、鉴定和分类化石骨骼、区分骨骼骨折、诊断肿瘤和器官异常以及测量大脑图像随时间的变化等领域。因此，创建数学理论和开发算法来识别和对齐这些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学和分析学的深奥数学理论现在正在形状匹配领域中得到应用。该项目探索了这一令人兴奋的领域的基本问题，该领域正处于取得重大进展的风口浪尖。它通过使用共形、调和和等距映射的理论来对齐曲面。虽然这些理论已经在数学背景下得到了广泛的研究，但它们对计算算法的适应仍在发展中。该项目为这一新兴学科建立了一个有凝聚力和全面的理论框架，并与科学应用建立了具体联系。它计划实施并公开一系列软件，这些软件将提供新的工具，并为生物学、医学、人类学和其他以形状分析为核心的领域的科学家开辟新的研究方向。三维世界中的曲面可以用二维黎曼流形在数学上描述。研究此类曲面上的几何结构是拓扑和微分几何等数学领域的中心课题。它导致了共形几何，模空间，调和和共形映射，和黎曼曲面的经典理论。这些领域现在被应用于研究骨骼表面、大脑皮层、蛋白质和其他生物分子。当用激光、雷达或CAT扫描观察一个物体时，我们获得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。本项目将发展曲面的共形映射和调和映射的离散对应物，探索它们的存在性、唯一性和微分同胚性，并建立离散理论对经典光滑理论的收敛性。它还将创建和实现算法，将这一理论结合起来，为科学家和其他实践者创建可用的软件。通过这种方式，该项目将弥合几何和拓扑的数学理论与这些思想在形状数据算法分析中的应用之间的差距。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值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.