NURBS-Wavelets for Surface Design, Analysis, Visualization, and Applications.

摘要

The central theme of this research program is 'splines and wavelets,' with applications in two specific areas: signal processing and computer graphics. New mathematical theories on affine and nonstationary wavelets under the tight frame structure were developed. While the author's theory on affine wavelets is quite complete, including arbitrary matrix dilation, vanishing moment recovery, oversampling theory, and approximation duals, his theory of nonstationary wavelets is a pioneering idea that opens up a new research area in computational mathematics. In particular, since his nonstationary consideration applies to all B-splines on arbitrary nested knot sequences, his wavelet theory extends the well-established theory of spline functions, and his new spline- wavelets with minimum support extend the current powerful spline tool box and are certainly compatible with the IGES and STEPS standards for the CAD/CAM industry. In applications to signal processing, he also has developed mathematical theories on localized cosines and Gabor frames for time-frequency localization as well as balanced multi-wavelets and the new notion of ARM-lets for processing scalar-valued signals without the need of pre-processing. In computer graphics, the author has introduced a direct approach to treating extraordinary points in surface subdivisions, based on refinable bivariate splines with small supports. The 22 papers completed over the duration of the funding period can be divided into the following three categories: (1) the development of mathematical theory and methods that will have direct application to the advancement of the fields of signal processing, image analysis, and computer graphics, including NURBS and nonstationary wavelets, affine wavelets, localized cosines, Gabor frames, and multi-wavelets; (2) signal and image processing; and (3) computer-aided surface design. Research publications produced under this grant, invited addresses, and conferences are listed.