The project will be focused on specific problems in nonlinear n-term approximation from large redundant systems, which is commonly referred to as highly nonlinear approximation. Redundant systems have already had a serious impact in signal and image processing and upon Approximation Theory in general. The research community is currently developing two forms of redundant systems. The first deals with libraries of bases such as the wavelet packets and cosine packets. The second major thrust has been to study dictionaries such as the Gabor functions and neural networks. The focus of our investigations is toward dictionaries of a different nature, which fall in two categories: (i) Dictionaries consisting of shifts and dilates of infinitely smooth, rapidly decaying functions which include the Gaussian, radial partial fractions, or more general anisotropic systems consisting of certain transformations of such functions. (ii) Collections of piecewise polynomials generated by multilevel dyadic partitions of the pace, multilevel nested triangulations, or simplex partitions. The primary goal of project is to understand nonlinear and highly nonlinear processes. In particular, we must develop an understanding of the nature of the smoothness conditions (specified by specific spaces) which govern the rate of approximation in a collection of important metrics. The second major effort of the project is to develop algorithms which are grounded in fundamental analysis, capable of achieving the rate of the best approximation (i.e. compression), and are practical to implement.
This project is actively coordinating its investigations with collaborators involved in the analysis and presentation of data for a wide range of application fields, including digital elevation maps and associated imagery arising from geographical information systems (GIS), signal and image analysis, computer aided geometric design (CAGD), graphical rendering and visualization of large scale numerical simulations. In each case, the objective is to analyze complicated data and signals in the context of prescribed dictionaries and to decompose them into basic elements which are natural for the physical system under investigation. The amplitudes associated with the base elements, whether they be wavelets, a multiresolution redundant system, or anisotropic systems, provide inherent information about the data and ideally lead to the maximal entropy encoding of classes of signals (data). This encoded representation of data is important for the storage, transmission, fast query, visual display, correlation, or registration against data from other modalities. The primary motivation of this project is to make substantial contributions to the underlying theoretical foundation for these investigations.
