框架稀疏信号压缩感知的几点理论研究

Recovery of signals that are sparse in terms of a frame from compressed measurement is one of the important topics in Compressed Sensing in the past few years. The most common and theoretical-well-developed method for dealing with such problems is the Analysis model as well as its solver. Although well developed, most of the existing results on Analysis approach only consider linear measurements. In practical applications, due to experimental errors and lack of technical hardware such as the restriction of computer, the established mathematical models are often nonlinear measurements. Based on this, in this project, we will study recovery of frame-sparse signals from nonlinear measurements (especially quantization measurements and quasi-linear measurements). For different practical problems, we shall introduce an appropriate notion of generalized D-RIP that are satisfied by practical measurement systems, and then with the help of generalized D-RIP, we shall provide approximation errors for both l_1-Analysis model and its iterative solvers(such as iterative reweighted least squares algorithm, iterative soft/hard thresholding algorithm, projected gradient descent method). We hope that the developed results can provide effective algorithms and theoretical supports for engineering applications.

具有框架稀疏表示的信号压缩采样恢复是近些年压缩感知领域中的重点研究问题之一。处理该类问题较为普遍的且具有理论基础的方法是Analysis模型及其求解算法。虽然目前已取得关于Analysis模型收敛的大量理论研究成果，而大多数成果仅考虑线性测量模型。而在实际应用中，由于实验误差以及计算机硬件等技术手段的欠缺，所建立的数学模型经常是在非线性测量意义下的。基于此，本项目将考虑非线性测量（重点考虑量化测量以及拟线性测量）的框架稀疏信号恢复问题。对不同的实际应用问题，拟提出满足实际应用背景的泛D-RIP概念，然后借助泛D-RIP条件给出关于l_1-Analysis模型的逼近误差分析，并且研究迭代算法（包括迭代加权最小平方算法、软硬阈值迭代算法、投影梯度下降法等）的实现和其收敛性。希望研究成果能为工程应用提供高效的稀疏重构算法和理论指导。

本项目研究框架稀疏表示下的压缩感知问题1-Analysis模型优良变种的逼近恢复理论分析及其求解算法（包括投影梯度下降法等）的收敛性；此外考虑压缩采样下的信号不同成分分离问题，借助于约束等距属性条件和互不想干性条件得到关于非限制1-split-Analysis模型、lq分离分析DS模型及迭代求解算法的收敛性结果。结果改进了已有的理论结果，弥补了相关结果的不足，为实际数据处理问题提供具有可解释性理论的算法/模型。支持期间内，项目负责人在国际主流SCI期刊上总共发表相关学术论文5篇。