RUI: Development of Sparsity-inducing Dual Frames and Algorithms with Applications, II

San Francisco State University, San Francisco CA

Investigators

Abstract

This award supports the ongoing research program of the Principal Investigator in the general area of signal processing. Signal processing is concerned with making signals easier to handle or better in quality. It is widely used in modern society, such as in cell-phone communications. In this project, the principal novelties introduced to signal processing are developments of the notion of "sparsity-inducing dual frames" ("sparse duals") and advanced ideas of recovering underlying signals from much smaller "samples" of the signal. Sparsity-inducing dual frames are a set of signal components used in obtaining the sparsest (the most concise) decomposition of a signal. The substance of sparse duals lies in the fact that they have the capacity to greatly advance the effectiveness of signal recovery through a small number of samples of a signal. The objective is to advance the art of new signal-detection techniques. This study has the potential to impact information technology, ranging from simplified sampling devices to radar systems, commercial imaging techniques, geographic survey, mapping, wireless communication, surveillance systems, medical image requisitions and combinations, and a number of signal requisition applications. Topics proposed for study include the development of tail-minimization techniques in the sparse-dual-based l-one analysis approach and applications, together with the development of a soft-thresholding-infused null-space tuning algorithm with feedbacks for more effective denoising and signal smoothness. Several themes will be developed. One theme aims at directly reducing the tail coefficients of sparse frame expansions. This is motivated by the observation that the signal recovery error bound is directly proportional to the "tails" of the signal coefficients. Another theme is to construct sparsity-inducing dual frames that minimize the tail coefficients for a class of s-sparse signals and nearly s-sparse signals. Such sparse duals are clearly ultimate analysis operators in the analysis approach. A third theme is designed to understand an equivalent l-one synthesis problem derived from a sparse-dual-based l-one analysis formulation. The new synthesis problem decouples the product of the sensing matrix and the frame matrix, and the kernel of the new sensing matrix is smaller. A final theme is aimed at infusing soft thresholding into the iterative "Null Space Tuning Algorithm with Feedbacks" (NST+FB). The NST+FB algorithm is known to converge in finitely many steps. The Principal Investigator will enrich the NST+FB algorithm with a soft-thresholding mechanism for enhanced denoising and smoothness, for applications such as image processing and radar imaging.