Recovering Signals Sparse in a Frame: Theory and Applications

University Of North Carolina At Wilmington, Wilmington NC

Investigators

Abstract

During the last two decades, there has been an explosive growth of the development of high dimensional data sensing, representation, and recovery. Problems such as medical imaging in the form of MRI or CT, hyperspectral satellite imaging, radar imaging, and remote sensing can involve large dimensional data where classically designed methods become impractical. The challenges of high dimensional data require techniques such as compressed sensing, which is able to extract the nonlinear low-dimensional structure of the signal. The investigator aims to further advance the scope of compressed sensing with an emphasis on signals that are sparsely represented by a collection of atoms, or a frame. This award will support the development of mathematical techniques for this problem as well as its applications on imaging while providing theoretical guarantees. The analysis in this project will apply broadly to signal processing problems like phase retrieval, one-bit sensing, and many data science problems in general. Compressed sensing aims to accurately and stably recover sparse signals from drastically undersampled measurements. The investigator will perform a theoretical analysis for recovering signals that are sparsely synthesized in a frame or dictionary from very few measurements and conduct numerical experiments on imaging applications where subsampled Fourier or convolution measurements are used. This is motivated by (1) numerous practical applications in MRI, MIMO radar, tomography, remote sensing, etc., where sparsity is exploited; (2) the benefit of robust signal acquisition and expansion in redundant frames; and (3) the lack of theoretical work on the synthesis-sparse sensing problem. The expected outcomes of this project are building a framework of this synthesis-sparse sensing problem, analyzing the effectiveness of randomized sensing, and improving image restoration quality with provable guarantees. This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.