Collaborative Research: CPA-G&V: Eigengeometry: Geometric Spectral Computing for Computer Graphics and Computational Science

合作研究：CPA-G

• 批准号: 0811313
• 负责人: Yiying Tong
• 金额: $ 21.23万
• 依托单位: Michigan State University
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2008
• 资助国家: 美国
• 起止时间: 2008-07-01 至 2012-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0811313&HistoricalAwards=false
• 关键词: Collaborative; Research; CPA; G& Eigengeometry

Tong, Yiying (0811313)Desbrun, Mathieu (0811373) AbstractBasic computational tools to process digital geometry in computer graphics and computational science are crucially needed for a wide range of applications including medical visualizations, atmospheric data analysis, and shape segmentation. Previous attempts to apply signal processing foundations such as the Fast Fourier Transform in order to fulfill these needs have only led to limited success: geometry has distinctive properties such as irregular sampling, topology, and metric, making it not just another signal, but a new challenge that computer scientists must face. Independently of these advances, spectral graph theory has shown surprisingly simple and powerful properties of the Laplacian matrices of arbitrary graphs, demonstrating that eigenvalue problems can robustly handle graph irregularity and help in the development of Internet search engines. Moreover, connections between spectral graph theory and differential geometry have started to appear as not only relevant, but quite insightful in their own rights.This NSF-funded project on ``Eigengeometry?? involves bringing these spectral theoretical developments into the realm of computing. More precisely, the investigators study and develop novel tools for the analysis and processing of not only signals defined over discrete geometric shapes, but of the shapes themselves via spectral theory. These novel tools, naturally robust to non-uniform sampling and irregular connectivity that meshes inherently contain, are tested on a few selected applications (covering CAGD, brain imaging, and analysis of atmospherical features). This research is a truly multidisciplinary and innovative effort drawing upon techniques from graph theory, graphics, numerical linear algebra, applied geometry, and signal processing. Finally, the importance of eigenstructures in mathematical and physical fields promises that the computing tools developed in this project are likely to have a broad impact.

Tong，Yiying（0811313）Desbrun，Mathieu（0811373）摘要基本计算工具，用于处理计算机图形和计算科学中数字几何形状的摘要计算工具，对于广泛的应用，包括医疗可视化，大气数据分析和形状分割。以前尝试应用信号处理基础（例如快速傅立叶变换）以满足这些需求的尝试只会导致成功：几何形状具有独特的属性，例如不规则采样，拓扑，拓扑和指标，不仅使它成为另一个信号，而且是计算机科学家必须面对的新挑战。与这些进步无关，光谱图理论表明，任意图的拉普拉斯矩阵的简单简单和强大的属性，表明特征值问题可以牢固地处理图形不规则性并有助于开发Internet搜索引擎。此外，光谱图理论与差异几何形状之间的连接开始不仅相关，而且对自己的权利很有见识。这本NSF资助的项目``eigengemementry''？涉及将这些光谱理论发展带入计算领域。更确切地说，研究人员研究并开发了新颖的工具，用于分析和处理不仅在离散的几何形状上定义的信号，而且还通过光谱理论来定义了形状。这些新颖的工具自然稳健至不均匀的采样和固有地包含的不规则连接性，在一些选定的应用程序（涵盖CAGD，脑成像和大气特征的分析）上进行了测试。这项研究是一项真正的多学科和创新的努力，借助图理论，图形，数值线性代数，应用几何和信号处理的技术。最后，特征结构在数学和物理领域的重要性承诺，该项目中开发的计算工具可能会产生广泛的影响。