Collaborative Research: An Integrated Approach to Convex Optimization Algorithms

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Image reconstruction and feature extraction have been important aspects in various applications such as medical resonance imaging (MRI) and synthetic aperture radar (SAR). However, these procedures involve challenges. Different applications may vary in data acquisition (sampling) domains, levels of detail required, and processing domains for the features of interest. The data acquisition is usually under-prescribed and noisy. The sampling domains and/or processing domains may not be well suited for the underlying question. All of these make the problems ill-posed, and various regularization techniques are necessary to study the problems by formulating them as convex optimization models. This project will develop an integrated framework of investigating such convex optimization models. The project will provide graduate students with opportunities for training through research involvement and will prepare them for careers in science and engineering. The PIs aim to propose a systematic way of evaluating various regularization techniques in such models, conduct a rigorous numerical analysis of the models, and develop efficient numerical algorithms of solving the models. Specifically, the PIs will address the following technical questions: (1) What constraints must be placed on the collected data in order to construct a numerically robust approximation to the underlying function? (2) How quickly and in what sense does the approximation converge? (3) Are the corresponding numerical algorithms developed for the fidelity and regularization terms viable? (4) How well are perturbations from the original data tolerated? The project aims to provide answers to all of these questions.