基于迭代支撑集检测的稀疏信号重构算法的研究和拓展

Compressive sensing is of great practical values in many fields such as medical imaging, wireless communications, and therefore recently becomes a hot topic in either academia or industry. Its significant advantage lies in the capability to reconstruct a true sparse signal using a sampling rate which is much lower than the classical Shannon-Nyquist sampling rate. The design of sparse signal reconstruction algorithms usually considers two aspects: the first one is low samping requirement and the second is fast reconstruction. However, most of the current existing algorithms fail to satisfy both simultaneously, even though they might be very good in terms of one single aspect. In order to achieve both, I propose a so called "iterative support detection " (ISD, for short) algorithm for the specific fast decaying sparse signals，since it is easy for us to detect the partial support information of the true signal from it. It achieves good numerical results and has been published in SIAM Journal on Imaging Sciences. The support detection makes use of the fast decaying property and adopts the threshold method. The detected partial support information of the true signal helps reduce the sampling requirement and improve the reconstruction quality. In this project, we will further improve ISD by considering more features and geometrical structures of sparse signals arising from various fields, in order to significantly improve the quality of support detection and further reduce the sampling requirement and accelerate the reconstruction. We would also like to extend ISD from the reconstruction of a single sparse vector signal to the reconstruction of multiple vectors, and to the reconstruction of low rank matrices and tensors. Finally, we will perform its theoretical analysis and practical performance comparison with other state of the art algorithms.

压缩感知由于其在医学成像，无线通信等众多领域的重大应用价值，成为近些年研究的一个热点问题。压缩感知的优势在于能够用远比经典的Nyquist采样率低的采样数据，准确重构出真实的未知稀疏信号。信号重构算法的设计需要考虑两个方面：一是较低的采样需求；第二是较快的重构速度。现有算法大都无法同时满足这两个方面。为了同时取得上述要求，针对快速递减信号，我提出了基于迭代支撑集检测的算法(简称ISD)，取得了很好的效果，并发表于SIAM 会刊。利用快速递减特点，真实信号的支撑集检测采用了阀值的方法，检测到的部分支撑集信息帮助降低采样需要和提高信号的重构效果。在本项目中，我们将更加深入研究如何利用不同领域稀疏信号的特点和非零元素间的几何关系，显著提高支撑集的检测效果，进一步降低ISD算法的采样需求并提高重构速度；并将其从单一向量重构推广到多向量重构和低秩矩阵和张量重构；并对其做深入的理论分析和性能比较。

压缩感知由于其在医学成像，无线通信等众多领域的重大应用价值，成为近些年研究的一个热点问题。压缩感知的优势在于能够用远比经典的Nyquist采样率低的采样数据，准确重构出真实的未知稀疏信号。信号重构算法的设计需要考虑两个方面：一是较低的采样需求；第二是较快的重构速度。现有算法大都无法同时满足这两个方面。为了同时取得上述要求，针对快速递减信号，我提出了基于迭代支撑集检测的算法(简称ISD)，取得了很好的效果，并发表于SIAM 会刊。利用快速递减特点，真实信号的支撑集检测采用了阀值的方法，检测到的部分支撑集信息帮助降低采样需要和提高信号的重构效果。在本项目中，我们深入了研究如何利用不同领域稀疏信号的特点和非零元素间的几何关系，显著提高支撑集的检测效果，进一步降低ISD算法的采样需求并提高重构速度；并将其从单一向量重构推广到多向量重构和低秩矩阵和张量重构；并和现有的算法进行了性能比较。压缩感知是一种信号收集的方法，同时我们也考虑信号处理和信号的理解，即模式识别。相应的，我们将基于迭代支撑集检测启发的稀疏优化算法应用到图像处理和模式识别领域，取得了较现有方法更好的效果和计算效率。