Collaborative Research: An Integrated Approach to Convex Optimization Algorithms

图像重建和特征提取一直是医学磁共振成像(MRI)和合成孔径雷达(SAR)等各种应用的重要方面。然而，这些程序涉及到挑战。不同的应用可能会在数据采集(采样)域、所需细节级别以及感兴趣特征的处理域中有所不同。数据采集通常是不规范的，而且有噪声。采样域和/或处理域可能不太适合基本问题。所有这些都使得问题不适定，需要各种正则化技术通过将问题描述为凸优化模型来研究问题。这个项目将开发一个研究这种凸优化模型的集成框架。该项目将为研究生提供通过参与研究进行培训的机会，并为他们在科学和工程领域的职业生涯做好准备。PI旨在提出一种系统的方法来评估这类模型中的各种正则化技术，对这些模型进行严格的数值分析，并开发求解这些模型的高效数值算法。具体地说，PI将解决以下技术问题：(1)必须对收集的数据施加什么约束，以构建对基本函数的数值稳健近似？(2)近似收敛的速度和意义有多快？(3)为保真度和正则化项开发的相应数值算法是否可行？(4)对原始数据的扰动容忍度有多好？该项目旨在为所有这些问题提供答案。

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.