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

在计算机图形学和计算科学中，处理数字几何图形的基本计算工具对于医学可视化、大气数据分析和形状分割等广泛应用至关重要。为了满足这些需求，以前应用信号处理基础（如快速傅里叶变换）的尝试只取得了有限的成功：几何具有独特的特性，如不规则采样、拓扑和度量，使其不仅仅是另一种信号，而是计算机科学家必须面对的新挑战。独立于这些进步之外，谱图理论已经展示了任意图的拉普拉斯矩阵惊人的简单和强大的特性，表明特征值问题可以鲁棒地处理图的不规则性，并有助于互联网搜索引擎的发展。此外，谱图理论和微分几何之间的联系已经开始显现，不仅是相关的，而且在它们自己的权利上相当有见地。这个nsf资助的关于“特征几何”的项目？包括将这些光谱理论的发展带入计算领域。更准确地说，研究人员研究和开发新的工具，不仅用于分析和处理离散几何形状上定义的信号，还用于通过谱理论分析和处理形状本身。这些新颖的工具对网格本身包含的非均匀采样和不规则连接具有天然的鲁棒性，并在一些选定的应用（包括CAGD，脑成像和大气特征分析）中进行了测试。这项研究是一项真正的多学科和创新的努力，利用了图论、图形学、数值线性代数、应用几何和信号处理等技术。最后，特征结构在数学和物理领域的重要性保证了本项目开发的计算工具可能会产生广泛的影响。

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.