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曲面细分（CVT）是特殊的Voronoi曲面细分，其具有这样的性质：相对于给定的密度函数，Voronoi曲面细分的生成元也是对应的Voronoi单元的质心。 CVTs在图像和数据分析、矢量量化、计算机图形学、资源优化、细胞生物学、偏微分方程数值解、最优控制、移动的传感网络等领域有着广泛的应用。以及拓宽应用，其中CVT和相关概念可用作更有效和更准确治疗的基础。在理论方面，将研究用于计算CVTs的流行算法的收敛和加速方案以及在其他度量设置和具有分层结构的CVTs的推广。 在应用方面，研究人员将开发和实现强大的基于CVT的网格生成和优化算法，然后将其纳入使用有限元方法或有限体积方法的数值偏微分方程的自适应计算中，并解决球体和其他表面上的一些具有挑战性的物理问题，如地球物理流动，同时，还将考虑基于CVT方法的皮层表面平坦化技术，这对于脑成像数据分析，包括功能变异性的定量映射和概率脑表面图谱的构建都是非常重要的。 拟议的研究将提供新的见解，一些突出的理论问题，并导致计算算法在科学和工程中的各种重要应用的革新。 将积极传播这一项目产生的软件，以便不仅科学计算领域的研究人员可以使用，而且更广泛的社区的从业人员也可以使用，将其应用于解决跨学科科学的问题。