Ordered Distributions and Wavelets on Two-Dimensional Manifolds

二维流形上的有序分布和小波

• 批准号: 0505756
• 负责人: Douglas Hardin
• 金额: --
• 依托单位: Vanderbilt University
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2005
• 资助国家: 美国
• 起止时间: 2005-07-15 至 2009-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0505756&HistoricalAwards=false
• 关键词: Ordered; Distributions; Wavelets; Two; Dimensional

This project has two primary goals: (1) the determination of the asymptotic behavior of minimum energy configurations of points restricted to a manifold in Euclidean space, and (2) the construction and application of wavelets on manifolds. This project focuses on two-dimensional manifolds (surfaces) because of their importance in applications to computer graphics, biological membranes, and materials science. The connection between geometrical and analytical properties of minimum energy configurations on a surface and the geometrical properties of that surface will be investigated and algorithms for the rapid generation of well-distributed point sets on surfaces will be developed. The second goal of this project is to develop the theory of a new class of wavelets generated from "refinable macroelements" for the efficient representation of surfaces and data on surfaces. Applications to geometric modeling, computer graphics, and multiscale methods in scientific computing will be investigated.The main objective of this project is to develop effective methods for discrete representations of two-dimensional surfaces and data defined on these surfaces. The recent development of the field of wavelets and multiresolution analysis has provided tools and a unifying framework for efficiently representing large classes of data arising in science and engineering. The goals of the first component of this project are to develop the theory of a class of "nonuniform" wavelets on surfaces and to develop multiscale high-performance applications to computer graphics and scientific computing. The goal of the second component of this project is the investigation of geometrical and analytical properties of minimum energy (or "ground state") configurations of large numbers of points distributed on a surface and interacting via a pairwise repulsive interaction. The research on minimum energy points and its usefulness in discretizing manifolds will be of significance to methods for data sampling, best-packing problems, and geometric design. The development of fast algorithms for generating uniformly distributed points is of significance in computational complexity theory. Furthermore, the investigation of the ordering of ground state configurations of particles on curved surfaces will improve understanding of the physics of membranes and films.

该项目有两个主要目标：（1）确定限于欧几里得空间中歧管的点的最小能量配置的渐近行为，以及（2）小波在歧管上的构建和应用。 该项目的重点是二维流形（表面），因为它们在计算机图形，生物膜和材料科学方面的应用很重要。 将研究表面上最小能量构型与该表面的几何特性的几何和分析性能之间的联系，并将开发在表面上快速生成良好分布点集的算法。 该项目的第二个目标是开发从“可改善宏观”生成的新一类小波的理论，以有效地表示表面上的表面和数据。 将研究科学计算中的几何建模，计算机图形和多尺度方法的应用。该项目的主要目的是开发有效的方法，以分散代表二维表面和在这些表面上定义的数据。 小波和多分析分析领域的最新发展提供了工具和一个统一的框架，用于有效地代表科学和工程中出现的大量数据。 该项目的第一个组成部分的目标是在表面上开发一类“不均匀”小波的理论，并为计算机图形和科学计算开发多尺度的高性能应用。 该项目的第二个组成部分的目的是研究大量分布在表面上并通过成对排斥相互作用相互作用的大量点的几何和分析性能（或“基态”）配置的研究。 对最低能量点及其在离散流形中的有用性的研究对数据采样，最包装问题和几何设计的方法具有重要意义。 在计算复杂性理论中，用于生成均匀分布点的快速算法的开发具有重要意义。 此外，研究粒子在弯曲表面上的基态构型的顺序将提高人们对膜和膜物理的理解。