Theory and Algorithms for Exploiting Sparsity in Signal Processing Applications

摘要这项研究考察了在需要计算稀疏解的信号处理问题中出现的理论、算法和计算问题。在许多信号处理应用中，对解向量的稀疏约束自然出现。脑成像技术，如脑磁图和脑电信号，具有大延迟扩展的稀疏通信信道，高分辨率谱分析，到达方向估计和压缩感知等都是几个例子。本文所考虑的稀疏贝叶斯学习(SBL)技术的推广和扩展将拓宽应用领域，并对现有的常用最大后验概率(MAP)方法提供非常有力的补充，在某些情况下甚至超过它们。研究者研究稀疏源恢复问题的扩展和推广，以极大地拓宽应用领域。这项工作中的一个关键考虑是开发一个严格的框架来处理稀疏框架中的依赖关系。受具有稀疏但局部结构的应用程序的启发，该研究考虑了单一测量情况下的矢量内依赖关系，以及多测量环境中所需的矢量内依赖关系等。这项研究还包括发展多用户通信理论和稀疏信号恢复问题之间的联系，以阐明稀疏信号恢复可能具有的稳定性，并加深对次优源恢复方法的局限性的理解。为了处理非平稳环境，该研究开发了利用应用程序固有的稀疏结构的在线自适应算法。研究还包括在几个重要应用领域对生成的算法进行评估。工作级别声明在推荐的支持级别上，PI和co-PI将尽一切努力满足项目的原始工作范围和水平。

AbstractThis research examines theoretical, algorithmic, and computational issues that arise in signal processing problems where there is a need to compute sparse solutions. There are numerous signal-processing applications where sparsity constraint on the solution vector naturally arises. Brain imaging techniques such as MEG and EEG, sparse communication channels with large delay spread, high-resolution spectral analysis, direction of arrival estimation and compressed sensing are a few examples. The generalization and extension of the sparse Bayesian learning (SBL) techniques considered in this research will broaden the application domain and provide a very powerful complement to the existing maximum a posteriori (MAP) methods commonly used and in some cases even surpass them.The investigators study extensions and generalizations of the sparse source recovery problem to greatly broaden the application domain. A key consideration in the work is developing a rigorous framework to deal with dependency in the sparsity framework. Motivated by applications with sparse but local structure, the research considers intra-vector dependency in the single measurement case, as well as intra-vector dependency as required in the multiple measurement contexts, among others. The research also includes the development of connections between multi-user communication theory and the sparse signal recovery problem to shed light on the stability with which sparse signal recovery is possible and to develop an understanding of the limits of suboptimal source recovery methods. To deal with non-stationary environments, the research develops on-line adaptive algorithms that exploit the inherent sparse structure of the application. The research also includes evaluation of the resulting algorithms in several important application domains.Level of Effort StatementAt the recommended level of support, the PI and co-PI will make every attempt to meet the original scope and level of effort of the project.