Collaborative Research: AF: Medium: A Unified Framework for Geometric and Topological Signature-Based Shape Comparison

合作研究：AF：Medium：基于几何和拓扑签名的形状比较的统一框架

• 批准号: 2106672
• 负责人: Erin Chambers
• 金额: $ 36.36万
• 依托单位: Saint Louis University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-06-01 至 2025-05-31
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2106672&HistoricalAwards=false
• 关键词: Collaborative; Research; AF; Medium; Unified

A fundamental aspect for data analysis is the ability to compare data sets, in order to measure (dis)similarity and quantify patterns present in the data. However, data is often too large and complex to analyze in its entirety, and therefore different techniques are used to summarize the data in order to work with smaller, more manageable representations of it. This project studies the data-comparison problem through the lens of mathematics, using geometric and topological signatures to represent these shapes concisely. This project will consider a variety of different kinds of shape data which live in some larger geometric or topological space (e.g., GIS trajectories, point sets, meshes, 3d scans, or graphs), and consider classes of algebraic, geometric, and graphical signatures which can be used to represent these shapes concisely. The project draws primarily upon the nascent yet rapidly developing area of topological data analysis, where tools from topology like homology or homotopy are combined with geometric measures to create robust analysis tools for analyzing the shape of data. Graduate and undergraduate students will be tightly integrated into the project, and special efforts will be made to involve students from underrepresented groups. Additional efforts by the research team include planning a workshop focused on women in this field, as well as broadening diversity and inclusion efforts in their own universities.The project focuses on shapes that have some common underlying annotation framework on top of the signature, which is usually additional structural or geometric information from the original embedding. The research consists of two major components. In the first, the investigators are initiating a principled study of algorithms and approaches to develop a unified framework which leverages multiple signatures for shape comparison. The goal of this phase is to provide theoretical results as well as empirical evaluations on a variety of data sets and signatures. The second major component of the project studies inverse problems, which aim to reconstruct shapes from a combination of signatures. Such problems are notoriously difficult for geometric or topological signatures, as they are necessarily lossy and remove certain types of information. During the course of the project, the investigators are also developing a shape signatures toolkit that enables computation of a range of signatures and distances, adding to the software both existing notions of distance and new ones developed over the course of the project.This project is jointly funded by the Algorithmic Foundations Core Program and by the Established Program to Stimulate Competitive Research (EPSCoR).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.

数据分析的一个基本方面是比较数据集的能力，以便测量（不）相似性并量化数据中存在的模式。 然而，数据通常太大且复杂，无法对其进行整体分析，因此使用不同的技术来汇总数据，以便使用更小、更易于管理的表示形式。 该项目通过数学的视角研究数据比较问题，使用几何和拓扑特征来简明地表示这些形状。该项目将考虑存在于较大几何或拓扑空间中的各种不同类型的形状数据（例如，GIS 轨迹、点集、网格、3D 扫描或图形），并考虑代数、几何和图形签名类别可以用来简洁地表示这些形状。 该项目主要利用新兴但快速发展的拓扑数据分析领域，其中同源或同伦等拓扑工具与几何测量相结合，创建用于分析数据形状的强大分析工具。研究生和本科生将紧密融入该项目，并将特别努力让来自代表性不足群体的学生参与其中。研究团队的其他努力包括规划一个专注于该领域女性的研讨会，以及扩大她们自己大学的多样性和包容性努力。该项目重点关注在签名之上有一些共同的底层注释框架的形状，即通常来自原始嵌入的附加结构或几何信息。该研究由两个主要部分组成。 首先，研究人员正在启动对算法和方法的原则性研究，以开发一个利用多个签名进行形状比较的统一框架。此阶段的目标是提供理论结果以及对各种数据集和签名的实证评估。该项目的第二个主要组成部分研究逆问题，旨在根据签名组合重建形状。 众所周知，此类问题对于几何或拓扑签名来说非常困难，因为它们必然是有损的并且会删除某些类型的信息。 在项目过程中，研究人员还开发了一个形状签名工具包，可以计算一系列签名和距离，将现有的距离概念和项目过程中开发的新概念添加到软件中。该项目是由算法基础核心计划和刺激竞争研究既定计划 (EPSCoR) 共同资助。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力价值和更广泛的影响进行评估，被认为值得支持审查标准。

项目成果

Aggregating community maps

• DOI: 10.1145/3557915.3560961
• 发表时间: 2022-11
• 期刊: Proceedings of the 30th International Conference on Advances in Geographic Information Systems
• 作者: E. Chambers;M. Duchin;Ranthony A. C. Edmonds;Parker B. Edwards;JN Matthews;Anthony E. Pizzimenti;Chanel Richardson;Parker Rule;Ari Stern

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

Topological Simplification of Nested Shapes

• DOI: 10.1111/cgf.14611
• 发表时间: 2022-08
• 期刊: Computer Graphics Forum
• 影响因子: 2.5
• 作者: Dan Zeng;E. Chambers;D. Letscher;T. Ju

Distances Between Immersed Graphs: Metric Properties

• DOI: 10.1007/s44007-022-00037-8
• 发表时间: 2023-01
• 期刊: La Matematica
• 作者: M. Buchin;E. Chambers;Pan Fang;Brittany Terese Fasy;Ellen Gasparovic;E. Munch;C. Wenk

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