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产生的数据量，更好、更快、更便宜地处理数据的需求一直是许多研究的重点，最值得注意的是通过压缩传感（CS）。然而，CS并非没有问题，其中大部分问题都是随着CS从理论走向实践而出现的。该理论是用凸问题开发的，但许多实际应用需要处理非凸问题的能力，这些问题不容易像数字传感系统所要求的那样快速解决。本研究项目的重点是一个特定的非凸模型沿着相关的数值算法，完成后将推进非凸优化领域。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和恢复有用的信息。由于CS的深远影响，开发的工具将具有广泛的适用性，特别是在本项目所涉及的医学成像和地理空间信息领域。此外，研究人员将把研究结果纳入本科和研究生课程，并将开发新的跨学科课程，重点放在理论和应用上，包括机器学习和医学成像，这将作为学生招聘的跳板。压缩感知（CS）可以从非相干线性系统中精确恢复稀疏信号（大多数元素为零），其中任何两个测量值的相关性尽可能小。稀疏性和非相干性是压缩感知的两个重要假设，但许多实际问题是相干的，传统的方法不能很好地解决。为了克服相干性障碍，研究者和合作者研究了一种新的非凸模型，该模型比CS中最先进的方法具有优势。这个项目的目标是解决关键的挑战，关于算法的计算和理论方面，建立新的标准，准确的恢复，并证明其在原型问题的适用性。因此，本研究有三个目标：（1）发展有效的算法来解决非凸极小化问题，利用凸优化和动力系统中的技术来设计算法和分析收敛性：（2）寻找可以量化凸和非凸方法成功的条件，例如，一致性和最小分离;（3）对医学图像重建和高光谱图像分类两类真实的问题进行数值实验，验证了该方法在精度和效率方面的优势。总的来说，该项目将推进计算数学的理论理解和算法发展，并提供一个新的视角，使基于CS的数据恢复在广泛的应用。

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.