A Non-Convex Approach for Signal and Image Processing

University Of Texas At Dallas, Richardson TX

Investigators

Abstract

As the digital revolution increases the amount of data generated by sensing methodology such as magnetic resonance imaging and radar, the need to process the data better, faster, and cheaper has been the focus of much research, most notably through work with compressive sensing (CS). However, CS is not without its problems, most of which have emerged as CS has moved from the theoretical to the practical. The theory was developed with convex problems, but many practical applications require the ability to process nonconvex problems that are not easy to solve as quickly as digital sensing systems require. This research project focuses on a particular nonconvex model along with associated numerical algorithms, which when completed will advance the field of nonconvex optimization. Theoretical investigations will be performed to establish conditions for guaranteed performance, which will help engineers and scientists devise experiments to acquire data and recover useful information in a more effective manner. The tools developed will have broad applicability due to the profound impacts of CS, specifically in the fields of medical imaging and geospatial information that are addressed in this project. Furthermore, the investigator will incorporate results of the research into undergraduate and graduate courses and will develop new interdisciplinary courses with focus on both the theory and application, including machine learning and medical imaging, which will serve as a springboard for student recruitment. Compressive sensing (CS) can exactly recover a sparse signal (most elements being zero) from incoherent linear systems, in which any two measurements have as little correlation as possible. Sparsity and incoherence are two important assumptions in CS, but many practical problems are coherent, and conventional methods do not work well. To overcome the coherency barrier, the investigator and collaborators investigate a novel nonconvex model that has advantages over the state-of-the-art methods in CS. The goal of this project is to address key challenges regarding both computational and theoretical aspects of the algorithms, to establish new criteria for exact recovery, and to demonstrate its applicability in prototypical problems. As such, this research is organized with three objectives: (1) Developing efficient algorithms to solve the nonconvex minimization problem, using techniques in convex optimization and dynamical systems to design algorithms and analyze convergence; (2) Searching for conditions that can quantify the success of both convex and nonconvex methods, for example, coherence and minimum separation; (3) Conducting numerical experiments in two types of real problems, medical image reconstruction and hyperspectral image classification, to demonstrate the advantages of the method in terms of accuracy and efficiency. Overall, this project will advance theoretical understanding and algorithmic developments in computational mathematics and provide a new perspective to enable CS-based data recovery in a wide spectrum of applications.