Collaborative Research: An Integrated Approach to Convex Optimization Algorithms

图像重建和特征提取在医学共振成像（MRI）和合成孔径雷达（SAR）等领域有着重要的应用。然而，这些程序涉及挑战。不同的应用可能在数据采集（采样）域、所需的细节级别和感兴趣特征的处理域方面有所不同。数据采集通常是在规定和噪声。采样域和/或处理域可能不太适合潜在问题。所有这些都使得问题不适定，各种正则化技术是必要的研究问题，制定他们作为凸优化模型。本计画将发展一个整合的架构来研究这种凸最佳化模式。该项目将通过参与研究为研究生提供培训机会，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些模型中的各种正则化技术，对模型进行严格的数值分析，并开发求解模型的有效数值算法。具体而言，PI将解决以下技术问题：（1）为了构建基本函数的数值稳健近似，必须对收集的数据施加哪些约束？(2)近似收敛的速度有多快，在什么意义上收敛？(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如何？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.