CAREER: Subdivision Schemes and Wavelets: New Tools, New Settings

Rensselaer Polytechnic Institute, Troy NY

Investigators

Abstract

The proposed research will focus on three partly related directions in multiresolution methods: (I) subdivision schemes and wavelets in the geometric setting, (II) the advantage of redundant systems and (III) ideal data representation in higher dimensions. Subdivision and multiresolution methods in the geometric setting is a recently launched research area. Prompted by the extremely high interests in applications such as terrain modeling and 3-D scanning technology, this research direction has attracted the attention of both computer scientists and applied mathematicians. Existing subdivision schemes for surface generation do not promise well-defined curvature everywhere in a resulted surface. In various applications it is desirable to work with curvature smooth surfaces. In the regular setting the proposer recently constructed a family of C^2 subdivision schemes with a small interpolating stencil. Underlying this construction is a variety of tools from optimization, wavelet theory and multivairate interpolation theory. This project will further refine these tools in order to resolve the original open problem and such that these tools can be applied to other settings as well. For various applications in image reconstruction/processing, it was found that highly redundant wavelet-like image representations significantly outperform the standard non-redundant orthogonal wavelet transforms used ubiquitously in the image coding community. On the one hand, there is currently no complete theory for explaining the fundamental advantages of redundancy. On the other hand, researchers have just started to realize the fundamental limits of wavelet transforms in higher dimensions and have proposed a wide variety of novel data representation schemes that address these fundamental limits. Research activity under this project seeks to develop mathematical theory and computational tools in the general area of adaptive methods of representing and analyzing images and other multidimensional data. The educational plan proposes a new way of teaching computational science. The project will develop teaching materials for our numerical computing courses that better suit the needs, interests and mathematical ability of computer science students. The goal of this project is to develop a set of electronic lecture notes for our future senior numerical computing course. These lecture notes will add two new dimensions to any of the existing textbooks: (i) An interactive computational environment (based on QPE's such as Matlab or Octave) will be integrated into the notes so that a student can easily reproduce any computational result presented and derive any new computational experiments. (ii) Applications areas such as computer graphics, computer vision, data-mining and computational fluid dynamics will be used in various places of this notes to illustrate the role of numerical methods in industry.