Some Problems on Analyses and Applications of Centroidal Voronoi Tessellations

质心Voronoi曲面细分分析及应用的几个问题

• 批准号: 0609575
• 负责人: Lili Ju
• 金额: $ 12.28万
• 依托单位: University South Carolina Research Foundation
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-08-01 至 2009-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0609575&HistoricalAwards=false
• 关键词: Some; Problems; Analyses; Applications; Centroidal

Centroidal Voronoi tessellations (CVTs) are special Voronoi tessellations having the property that the generators of the Voronoi tessellation are also the centers of mass, with respect to a given density function, of the corresponding Voronoi cells. CVTs have been very usedful in a wide range of research fields, including image and data analysis, vector quantization, computer graphics, resource optimization, cell biology, numerical solution of partial differential equations, optimal control, mobile sensing networks, and so on. This project aims at further analysis on CVTs' properties and algorithms, and the broadening of applications for which CVTs and related concepts can be used as a basis for more efficient and accurate treatments.In the theoretical aspects, the convergence and acceleration schemes of popular algorithms for computing CVTs and generalization of CVTs in other metric settings and with hierarchical structures will be studied. In the application aspects, the investigator will develop and implement robust CVT-based mesh generation and optimization algorithms, and then incorporate them in adaptive computations of numerical partial differential equations using finite element methods or finite volume methods, and in the solution of some challenging physical problems on the sphere and other surfaces such as geophysical flows, in light of the high-quality CVT-based surface meshing.Also considered will be the cortical surface-flattening techniques based on the CVT methodology, which are very important to brain-imaging data analysis, including quantitative mapping of functional variability and construction of probabilistic brain-surface atlases. The proposed research will offer new insight into a number of outstanding theoretical issues and lead to renovation of computational algorithms for diverse important applications in science and engineering. The software resulting from this project will be actively disseminated, so that it can be used not only by researchers in the scientific computing area, but also by practitioners in a much broader community for application to problems in interdisciplinary sciences.

质心Voronoi Tessellations（CVT）是具有特殊的voronoi镶嵌物，其特性具有相应的Voronoi细胞的给定密度函数，而voronoi tessellation的发电机也是质量中心。 CVT在广泛的研究字段中非常有用，包括图像和数据分析，矢量量化，计算机图形，资源优化，细胞生物学，部分微分方程的数值解决方案，最佳控制，移动传感网络等。 This project aims at further analysis on CVTs' properties and algorithms, and the broadening of applications for which CVTs and related concepts can be used as a basis for more efficient and accurate treatments.In the theoretical aspects, the convergence and acceleration schemes of popular algorithms for computing CVTs and generalization of CVTs in other metric settings and with hierarchical structures will be studied. 在应用程序方面，研究人员将使用有限元方法或有限的体积方法来开发和实施基于CVT的基于CVT的网状生成和优化算法，然后将它们纳入数值偏微分方程的自适应计算中，并在诸如地球体上的其他表面上遇到一些挑战性的物理问题，例如，在诸如地球体上遇到了较高的效果，以示为较高的效果，以示地地理位置。基于CVT方法论的皮质表面压纹技术，对于大脑成像分析非常重要，包括功能变异性的定量映射和概率脑表面图谱的构建。 拟议的研究将为许多出色的理论问题提供新的见解，并导致对科学和工程中各种重要应用的计算算法进行翻新。 该项目产生的软件将被积极传播，因此，科学计算领域的研究人员不仅可以使用它，而且可以由更广泛的社区中的从业者使用，以应用于跨学科科学的问题。