The theory of filtering and transforms is fundamental for the compression, transmission, and analysis of signals from biomedical, geological, or entertainment sources, in audio, image, or video format. Because many natural signals are modeled more accurately by nonlinear models than by linear models, it is especially important to pursue the design, implementation and evaluation of nonlinear transforms. This research advances the state-of-the-art multiresolution analysis tools by bringing powerful mathematical tools from algebraic geometry to bear upon the design of systems for linear and nonlinear signal analysis. The use of Groebner bases fundamentally improves one's ability to construct discrete-time systems, even when the corresponding design equations are nonlinear. This research will lead to a deeper understanding of how to successfully apply Groebner bases in practice, and what their limitations are.
The investigator will also organize an Internet website, `A Signal Processing Exercise and Lab Bank', devoted to signal processing education. The website will be an evolving collection of innovative exercises, projects and demos contributed by educators and researchers, that illustrate both basic theory and its application to real-world signals, and which can be updated to reflect the progress in signal processing techniques. It will facilitate the integration of signal processing research and education, and will likewise help in modernizing SP curriculums, by giving educators access to a collection of introductory exercises illustrating recent and ongoing research topics. By drawing together existing resources, and by providing a point for the dissemination of future innovative exercises and labs, the website will enhance the NSF's existing investment in signal processing education and research.
