Ordered Distributions and Wavelets on Two-Dimensional Manifolds

Vanderbilt University, Nashville TN

Investigators

Abstract

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.