Mathematical Sciences: A Unified Approach to Discrete Data Representation and Wavelet Analysis

9505460 Chui Four problem areas concerning discrete data representation and wavelet analysis are addressed in this project. The first area is to establish a unified scheme for constructing wavelets based only on nested sequences of subspaces but not on any two-scale relation. The second area is the study of multivariate wavelets and frames with matrix dilations. The third area is on representation of discrete data by radial basis functions (RBF's). The fourth area is the investigation of adaptive wavelets, based on spline techniques. A particular interest is in modeling by spline and radial basis functions and in constructing and analyzing the corresponding wavelets. In view of the fact that most modeling functions, including all the interesting standard radial basis functions, do not satisfy a two-scale (or refinement) relation, a unified scheme for the construction of wavelets that does not rely on such a relation is developed. This constitutes the first area of the proposed research. The second area is centered around multivariate wavelets, where the main concern is matrix dilation. For instance, the boundedness of the affine operator and that of the corresponding Littlewood-Paley energy functions, for a finite family of square-integrable functions with zero mean, will be investigated and sufficient conditions that guarantee lower bounds will be established. The third area is centered on the study of localized cosine Riesz bases and their corresponding duals. The duals, or bi-orthogonal bases, must facilitate computational efficiency. For this and other reasons, spline techniques, such as knot insertion and knot removal, will be investigated. The final area concerns the modeling of discrete data. For high dimensions, radial basis functions seem to be the most efficient tool, and the emphasis will be on various linear and nonlinear problems involving these functions. The central theme of this rese arch proposal is the development of mathematical tools for analysis and synthesis of discrete data in one or higher dimensions, in an Euclidean space or on a closed manifold such as the unit ball. This proposed research is a unified approach to represent any discrete data set in an effective way so that "wavelet" analysis can be readily performed. Both spline and radial basis functions will be used for data representation. It is well known that spline functions provide a very powerful tool for data modeling due to their most desirable properties such as flexibility for adaptive implementation, computational efficiency, and localization capability. On the other hand, radial basis functions are more powerful for handling higher-dimensional data sets, particularly when the data information is taken randomly. Wavelets will be constructed and wavelet algorithms will be developed by using both spline functions and radial basis functions. In addition, for analyzing nonstationary data that change in shape, the investigators will look into the use of localized cosine transforms by using splines or radial basis functions as envelopes (or windows). Adaptive algorithms will also be developed. One of the main reasons for wavelets to be considered as a very powerful tool for data analysis is their fast algorithms. The research will be focused on constructing wavelets using both spline and radial basis functions for effective implementation and fast computation. Software development will be carried out concurrently, and one of the main goals is to ensure that the algorithms can also be implemented in hardware. Industrial standards in the hardware prototype development will also be kept in mind.

9505460 Chui Four problem areas concerning discrete data representation and wavelet analysis are addressed in this project. The first area is to establish a unified scheme for constructing wavelets based only on nested sequences of subspaces but not on any two-scale relation. The second area is the study of multivariate wavelets and frames with matrix dilations. The third area is on representation of discrete data by radial basis functions (RBF's). The fourth area is the investigation of adaptive wavelets, based on spline techniques. A particular interest is in modeling by spline and radial basis functions and in constructing and analyzing the corresponding wavelets. In view of the fact that most modeling functions, including all the interesting standard radial basis functions, do not satisfy a two-scale (or refinement) relation, a unified scheme for the construction of wavelets that does not rely on such a relation is developed. This constitutes the first area of the proposed research. The second area is centered around multivariate wavelets, where the main concern is matrix dilation. For instance, the boundedness of the affine operator and that of the corresponding Littlewood-Paley energy functions, for a finite family of square-integrable functions with zero mean, will be investigated and sufficient conditions that guarantee lower bounds will be established. The third area is centered on the study of localized cosine Riesz bases and their corresponding duals. The duals, or bi-orthogonal bases, must facilitate computational efficiency. For this and other reasons, spline techniques, such as knot insertion and knot removal, will be investigated. The final area concerns the modeling of discrete data. For high dimensions, radial basis functions seem to be the most efficient tool, and the emphasis will be on various linear and nonlinear problems involving these functions. The central theme of this rese arch proposal is the development of mathematical tools for analysis and synthesis of discrete data in one or higher dimensions, in an Euclidean space or on a closed manifold such as the unit ball. This proposed research is a unified approach to represent any discrete data set in an effective way so that "wavelet" analysis can be readily performed. Both spline and radial basis functions will be used for data representation. It is well known that spline functions provide a very powerful tool for data modeling due to their most desirable properties such as flexibility for adaptive implementation, computational efficiency, and localization capability. On the other hand, radial basis functions are more powerful for handling higher-dimensional data sets, particularly when the data information is taken randomly. Wavelets will be constructed and wavelet algorithms will be developed by using both spline functions and radial basis functions. In addition, for analyzing nonstationary data that change in shape, the investigators will look into the use of localized cosine transforms by using splines or radial basis functions as envelopes (or windows). Adaptive algorithms will also be developed. One of the main reasons for wavelets to be considered as a very powerful tool for data analysis is their fast algorithms. The research will be focused on constructing wavelets using both spline and radial basis functions for effective implementation and fast computation. Software development will be carried out concurrently, and one of the main goals is to ensure that the algorithms can also be implemented in hardware. Industrial standards in the hardware prototype development will also be kept in mind.