AF: Small: Collaborative Research: Reeb graph flows: Metrics, Drawings, and Analysis

图像和3D扫描等形状数据来自各种来源，包括医学成像、动植物生物分析、人工制品的数字保存、计算机图形和动画等许多其他方面。为了提取这些数据的本质，这个项目使用了计算拓扑学中的工具，该领域结合了数学和计算机科学的工具，以找到有效和实用的方法来简化、存储和分析这些图像。其中一个工具，Reeb图，提供了结构的骨架，这对可视化很有用。该项目创建工具来以时变的方式量化、可视化和分析这些骨架，即当输入的图像数据实际上是电影时。考虑到从运动或随时间推移的物体收集多幅图像或扫描的频率，该项目的结果在许多领域都有广泛的应用。这些成果将通过出版物、演示文稿、开放源码软件以及参加各种讲习班和活动来传播，所有这些讲习班和活动都将在一个更大的形状分析和计算拓扑学研究人员网络中继续开展协作和跨学科工作。除了新工具的核心理论开发，这两个PI都致力于扩大对数学和计算机科学的参与，作为项目的一部分，将广泛关注教授和指导学生和初级研究人员，包括在主要关注代表性不足的群体的社会中积极的领导和指导。在许多应用领域，越来越需要分析通过扫描某些3D结构生成的形状的拓扑和几何信息。然而，当给定随时间变化的形状时，很少有策略利用先前的扫描来快速更新和维护具有所需属性的连续的可视化家族。在这个项目中，PI将合作开发改进的可视化工具包，用于分析这种类型的数据的主要方法之一-Reeb图，它在计算拓扑、形状分析和可视化中被广泛使用。这些骨架提供了压缩和准确的方法来存储形状并计算形状不变量和统计数据，但仍然可能很大，很难可视化。特别是，该项目的重点是分析随时间变化的Reeb图，即众所周知的Reeb图流。在项目过程中，PI将调查来自各种自然输入域的此类Reeb流的属性，并确定哪些设置允许更快、更准确的可视化和分析。该项目的成果将包括一系列理论和应用出版物、开源软件开发，并将传播给对形状分析和拓扑学用于算法和分析感兴趣的更大的研究人员网络。该奖项反映了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。

Shape data such as images and 3d scans come from a wide variety of sources, including medical imaging, biological analysis of plants and animals, digital preservation of artifacts, computer graphics and animation, and many others. To distill this data to its essence, this project uses tools from computational topology, a field which combines tools from mathematics and computer science to find efficient and practical ways to simplify, store, and analyze these images. One of these tools, the Reeb graph, provides a skeleton of the structure which is useful for visualization. This project creates tools to quantify, visualize, and analyze these skeletons in a time-varying fashion, i.e. when the input image data is actually a movie. Given how often multiple images or scans are collected from an object in motion or over time, the results of this project have broad applications in many fields. These results will be disseminated through publications, presentations, open-source software, and participation in a variety of workshops and activities, all of which continue collaborative and interdisciplinary work in a larger network of shape-analysis and computational-topology researchers. In addition to core theoretical development of new tools, both PIs are dedicated to broadening participation in mathematics and computer science, and as part of the project will focus extensively on teaching and mentoring students and junior researchers, including active leadership and mentoring in societies that focus primarily on underrepresented groups.In numerous application fields, there is an increasing need to analyze topological and geometric information about shapes that are generated by scanning some 3d structure. However, when given a time-varying shape, there are few strategies that utilize prior scans to quickly update and maintain a continuous family of visualizations with desirable properties. In this project, the PIs will collaborate to develop improved visualization toolkits for one of the major ways to analyze this type of data, the Reeb graph, which is heavily used in computational topology, shape analysis, and visualization. These skeletons give compressed and accurate ways to store the shape and compute shape invariants and statistics, but can nonetheless be large and difficult to visualize. In particular, the project is focused on the analysis of a time-varying Reeb graph, known as a Reeb graph flow. Over the course of the project, the PIs will investigate properties of such Reeb flows coming from a variety of natural input domains, and determine which settings allow faster and more accurate visualization and analysis. Results from this project will include a range of theoretical and applied publications, open-source software development, and will be disseminated to a larger network of researchers interested in shape analysis and the use of topology for algorithms and analysis.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.