CIF: Small: Collaborative Research: Design and Analysis of Novel Compressed Sensing Algorithms via Connections with Coding Theory

压缩感知（CS）是信号处理和统计学的一个快速发展的领域，有可能从根本上改变模拟信号转换为数字信号的方式。 其主要思想是使用专门的采样和重建过程从非常少量的测量中获取稀疏信号。 由于CS的一个有前途的应用是医学成像，因此CS系统的改进也有望推进现实世界的医疗保健应用。 在本项目中，研究人员将研究纠错码（ECC）和CS之间的基本联系，并利用ECC的最新进展来设计改进的CS测量和重建系统。特别是，二进制线性码的线性规划（LP）解码和LP重建之间的联系将用于开发CS算法和测量矩阵的设计和分析的非渐近理论。 该项目的第一部分将集中在CS重建问题的新松弛，允许非凸正则化和迭代求解。 该项目的第二部分将侧重于应用伪码字理论，该理论最初是为了理解二进制线性码的迭代和LP解码而开发的，以实现CS迭代重建算法的非渐近分析。 该项目的第三部分将侧重于利用额外的信号结构（即，超出稀疏性），其存在于诸如血管造影的高对比度成像应用中。

Compressed sensing (CS) is a rapidly advancing area of signal processing and statistics that has the potential to radically change the way that analog signals are transformed into digital signals. The main idea is to acquire a sparse signal from a very small number of measurements using a specialized sampling and reconstruction process. Since one promising application of CS is medical imaging, improvements in CS systems are also expected to advance real-world healthcare applications. In this project, the investigators will study the fundamental connection between error-correcting codes (ECC) and CS and leverage recent advances in ECC to design improved CS measurement and reconstruction systems.In particular, the connection between linear-programming (LP) decoding of binary linear codes and LP reconstruction will be used to develop a non-asymptotic theory for the design and analysis of CS algorithms and measurement matrices. The first part of the project will focus on novel relaxations of the CS reconstruction problem that allow non-convex regularization and iterative solution. The second part of the project will focus on applying the theory of pseudo-codewords, which was originally developed to understand iterative and LP decoding of binary linear codes, to achieve a non-asymptotic analysis of iterative reconstruction algorithms for CS. The third part of the project will focus on exploiting additional signal structure (i.e., beyond sparsity) that exists in high-contrast imaging applications such as angiograms.