k-稀疏恢复限制等距常数上界及压缩感知算法研究

The goal of Compressive Sensing(CS) is to recover the high-dimensional signals from a very limited number of measurements. It is a brand new theory in information science. (l1)-minimization can reconstruct sparse signals with a small or zero error provided that the RIP constants of the sensing matrices satisfy a certain bound. However, it is difficult to verify the RIP for a given sensing matrix. A general technique for avoiding verifying the RIP directly is to show that the randomly generating sensing matrix satisfis RIP with high probability. Hence, the study of the new bounds for the RIP constants is always a hot issue of CS. Moreover, the non-sparsity of practical signals also confine the quality of the recovered signals obtained by CS theory. Based on the null space of the sensing matrix, a unified approach for studying the upper bounds of RIP constants is proposed, which will provide a new way for capturing the new bounds of RIP constants. Further, the correlation between the CS algorithm and the sparse representation method is exploited. By investigating the natures of the peak transform and tree structure of the wavelet coefficients, new algorithms for the sparse representation and CS non-linear reconstruction of practical signals are established, which can improve the performance of CS algorithms and the quality of recovered signals. It offers new theoretical foundations for extending the practical applications of CS.

压缩感知(CS)的目标是以较少的测量数据实现高维信号的恢复，是信息科学全新的理论。 (l1)约束极小化能准确或以很小的误差重构稀疏信号，只要测量矩阵的RIP常数满足一定的上界条件。然而，直接验证一个感知矩阵的RIP是困难的。为避免直接验证RIP，一个普遍的技巧是去表明随机生成的感知矩阵高概率地满足RIP。因此，RIP常数新界的研究一直是CS的一个热点。另外，实际信号的非稀疏性也制约了CS对信号的重构质量。本课题以感知矩阵的零空间为切入点，提出了研究RIP常数上界的统一方案，为获取RIP常数新界提供了一条新的途径。进一步，考虑了CS恢复算法与稀疏表示的相关性，通过研究峰值变换和小波树的特性，建立起信号稀疏表示和CS非线性重构的新算法，从而提高CS的算法性能和重构质量，为扩大CS的实际应用提供新的理论依据。

压缩感知(CS)的目标是以较少的测量数据实现高维信号的恢复，是信息科学全新的理论和热点问题。本项目着重研究压缩理论相关4个方面的理论和应用问题，包括：(1) RIP常数上界研究。提出求解限制等距常数上界的统一方法。 (2) 信号的稀疏表示方法。提出基于小波域维纳滤波器和小波模极大值线的信号稀疏表示压缩感知算法；在基于冗余字典的稀疏表示及其应用方面，提出了ASeDiL及MSeDiL字典训练算法，提高了字典稀疏表示能力及字典训练速度；提出了紧支撑双正交多小波对称扩张的显表达式方法、具有对称性和较高消失矩的三带小波紧框架对称扩充和参数化构造方案，以及矩阵值多分辨率分析和对应的矩阵值小波框架研究理论。(3) 压缩感知稀疏恢复算法方面研究。在OMP算法的基础上，提出了4种贪婪类稀疏恢复算法。针对压缩感知只能对一维信号进行处理的问题，提出了基于自适应秩原子分解算法，将稀疏重构拓展到可以直接对二维信号进行重构。(4) 低秩矩阵重构及其应用。作为压缩感知在二维上的拓展，针对大稀疏噪声、混合噪声及算法运算速度方面，深入研究了低秩矩阵重构理论，提出了相关的矩阵重构模型和RGLRMD、RASRFA优化求解算法。并应用到TFT-LCD面板和手机触摸屏缺陷检测，以及视频修复，取得了较好的应用结果。通过本项目的研究，提高了压缩感知相关理论和算法性能以及对信号、图像的重构质量，为扩大CS的实际应用提供新的理论依据。