压缩感知中稀疏分解和重构精度改进的一种方法

An Improved Method for Sparse decomposition and Reconstruction in Compressed Sensing

稀疏分解、非相关观测和重构算法是压缩感知的三大要素,任何一个环节的设计优劣都对压缩感知的性能产生重大影响,稀疏分解是实现压缩感知的前提,现今使用的稀疏分解对大多数自然信号都不能做到理想的绝对稀疏,而是近似稀疏,这大大影响了压缩感知的重构性能。本文设计了一种可逆的阈值,并用其构造门限矩阵,从而门限矩阵可逆,将门限矩阵作用于信号经正交变换后的近似稀疏系数,可使系数更接近理想的绝对稀疏,而且门限矩阵对系数的处理过程是可逆的,即可由处理后的系数无损恢复原来的近似稀疏系数。重构算法采用贪婪算法中的OMP和CoSaMP,从理论上分析了在保证与CoSaMP同样的前提条件下,门限矩阵改进后的Co-SaMP重构误差明显减小,仿真实验用门限矩阵对OMP和CoSaMP的改进前后进行对比,验证了门限矩阵对重构精度有进一步的提高。

Sparse decomposition,incorrelate projection and reconstruction algorithm are the three elements of compressed sensing.Any aspect of the merits will significantly impact the performance of compressed sensing.Sparse decomposition is the precondition to achieve compressed sensing,now the sparse decomposition for most natural signals are not absolutely sparse,but approximately sparse,which will greatly influence the reconstruction property of compressed sensing.In this paper,we design an invertible thresholding function,and then a thresholding matrix is devised using this thresholding function,thus the thresholding matrix is invertible.Under this thresholding matrix,the approximately sparse coefficients through orthogonal transformation can be more close to absolutely sparse.Since the process of using thresholding matrix to deal with approximately sparse coefficients is inversible,we can recover exactly original approximately sparse coefficients.Reconstructions algorithm use orthogonal matching pursuit（OMP） and compressive sampling matching pursuit（CoSaMP） in greedy algorithms,the theoretical analysis showed that under the same precondition with compressive sampling matching pursuit,thresholding matrix can improve notablely reconstruction error of compressive sampling matching pursuit.The simulations are about improved performance comparison of compressive sampling matching pursuit and orthogonal matching pursuit using thresholding matrix,we find that the thresholding matrix can make accuracy of reconstruction better.