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

AF：小型：协作研究：Reeb 图流：指标、绘图和分析

• 批准号: 1907612
• 负责人: Erin Chambers
• 金额: $ 15.26万
• 依托单位: Saint Louis University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-10-01 至 2023-09-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1907612&HistoricalAwards=false
• 关键词: AF; Small; Collaborative; Research; Reeb

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.

形状数据（例如图像和3D扫描）来自多种来源，包括医学成像，动植物的生物学分析，人工制品的数字保存，计算机图形和动画等。为了将这些数据提炼到本质上，该项目使用了计算拓扑的工具，该工具结合了数学和计算机科学的工具，以找到有效且实用的方法来简化，存储和分析这些图像。其中一种工具，即REEB图，提供了结构的骨架，可用于可视化。该项目创建了以时间变化的方式量化，可视化和分析这些骨架的工具，即何时输入图像数据实际上是一部电影。 鉴于从运动中或随着时间的推移对象收集多个图像或扫描的频率，因此该项目的结果在许多领域都有广泛的应用。这些结果将通过出版物，演示文稿，开源软件以及参与各种研讨会和活动的参与来传播，所有这些结果都将在更大的形状分析和计算研究人员网络中继续进行协作和跨学科工作。 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. 但是，如果给出时间变化的形状，很少有策略利用先前的扫描来快速更新和维持具有理想特性的可视化家族。 在该项目中，PIS将合作开发改进的可视化工具包，用于分析此类数据的主要方法之一，REEB图（在计算拓扑，形状分析和可视化中都大量使用）。 这些骨骼提供了压缩，准确的方法来存储形状和计算形状不变性和统计数据，但是仍然很大且难以可视化。特别是，该项目的重点是分析时变的Reeb图，称为Reeb图流。在项目过程中，PI将研究来自各种自然输入域的此类Reeb流的特性，并确定哪些设置允许更快，更准确的可视化和分析。该项目的结果将包括一系列理论和应用出版物，开源软件开发，并将被传播到更大的对形状分析和使用拓扑用于算法和分析的研究人员网络。这项奖项反映了NSF的法定任务，并通过使用该基金会的知识优点和广泛的影响来评估NSF的法定任务，并被认为是值得的。

项目成果

On Complexity of Computing Bottleneck and Lexicographic Optimal Cycles in a Homology Class

论同调类中计算瓶颈的复杂性和字典序最优循环

• DOI: 10.4230/lipics.socg.2022.25
• 发表时间: 2022
• 期刊: International Symposium on Computational Geometry
• 作者: Chambers, Erin Wolf;Parsa, Salman;Schreiber, Hannah
• 通讯作者: Schreiber, Hannah

Perceptually grounded quantification of 2D shape complexity

• DOI: 10.1007/s00371-022-02634-8
• 发表时间: 2022-08
• 期刊: The Visual Computer
• 作者: Dena Bazazian;Bonnie Magland;C. Grimm;E. Chambers;Kathryn Leonard

Subtrajectory Clustering: Finding Set Covers for Set Systems of Subcurves

• 发表时间: 2021-03
• 期刊: Comput. Geom. Topol.
• 作者: Frederik Brüning;H. Akitaya;E. Chambers;Anne Driemel

A Cautionary Tale: Burning the Medial Axis Is Unstable (Media Exposition)

警示故事：烧伤内侧轴不稳定（媒体博览会）

• DOI: 10.4230/lipics.socg.2022.66
• 发表时间: 2022
• 期刊: Leibniz international proceedings in informatics
• 作者: Chambers, Erin;Fillmore, Christopher;Stephenson, Elizabeth;Wintraecken, Mathijs
• 通讯作者: Wintraecken, Mathijs

Minimum Cuts in Surface Graphs

• DOI: 10.1137/19m1291820
• 发表时间: 2019-10
• 期刊: SIAM J. Comput.
• 作者: E. Chambers;Jeff Erickson;K. Fox;A. Nayyeri