A Non-Convex Approach for Signal and Image Processing

随着数字革命增加了磁共振成像和雷达等传感方法生成的数据量，更好、更快、更便宜地处理数据的需求已成为许多研究的焦点，尤其是通过压缩传感 (CS) 的研究。然而，计算机科学也并非没有问题，其中大部分问题是随着计算机科学从理论走向实践而出现的。该理论是针对凸问题发展起来的，但许多实际应用需要处理非凸问题的能力，而这些问题不容易像数字传感系统所需的那样快速解决。该研究项目重点关注特定的非凸模型以及相关的数值算法，该研究项目完成后将推动非凸优化领域的发展。将进行理论研究，以建立保证性能的条件，这将有助于工程师和科学家设计实验，以更有效的方式获取数据并恢复有用的信息。由于计算机科学的深远影响，开发的工具将具有广泛的适用性，特别是在本项目涉及的医学成像和地理空间信息领域。此外，研究者还将把研究成果融入到本科生和研究生课程中，并开发新的理论和应用并重的跨学科课程，包括机器学习和医学影像，作为招生的跳板。压缩感知（CS）可以从不相干的线性系统中精确地恢复稀疏信号（大多数元素为零），其中任何两个测量值的相关性尽可能小。稀疏性和不相干性是CS中的两个重要假设，但许多实际问题都是相干的，常规方法效果不佳。为了克服一致性障碍，研究人员和合作者研究了一种新颖的非凸模型，该模型比 CS 中最先进的方法具有优势。该项目的目标是解决算法计算和理论方面的关键挑战，建立精确恢复的新标准，并证明其在典型问题中的适用性。因此，本研究有三个目标：（1）开发有效的算法来解决非凸最小化问题，使用凸优化和动力系统技术来设计算法并分析收敛性； （2）寻找可以量化凸和非凸方法成功的条件，例如相干性和最小分离； (3)针对医学图像重建和高光谱图像分类两类实际问题进行数值实验，验证了该方法在精度和效率方面的优势。总体而言，该项目将促进计算数学的理论理解和算法开发，并提供新的视角，以在广泛的应用中实现基于 CS 的数据恢复。

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.