依据列相关性优化高斯测量矩阵

Optimize Measurement of Gaussian Matrix Based on Column Correlation

为了提高信号重建的精度以及稀疏度适用范围,提出了一种新的测量矩阵优化方法,减小测量矩阵和稀疏变换矩阵的相关性;首先,由测量矩阵和稀疏变换矩阵的乘积构造Gram矩阵;根据Gram矩阵的维数,计算互相关函数的下确界即Welch界;其次,由Welch界确定阈值,收缩Gram矩阵中大于阈值的非对角元;然后,由新得的Gram矩阵和稀疏变换矩阵反解出测量矩阵,迭代更新,从而达到减小相关性,优化测量矩阵的目的;实验结果表明:依据Welch界优化测量矩阵,能快速降低压缩感知矩阵相关性的最大值,提高OMP算法的性能,例如在误差率为10-0.9时,原高斯随机矩阵需要23个观测值,算法优化后只需16个观测值,相对于Elad、Zhao等观测矩阵优化方法,文中提出的算法具有更小的重构误差,性能和稳定性也略有提升。

In order to improve the accuracy of signal reconstruction and the application range of sparsity, a new method of measuring ma- trix optimization is proposed. First, multiply measurement matrix and sparse transformation matrix to construct a Gram matrix, and calcu- late the minimum value of mutual coherence, that is the Welch bound; Secondly, set a threshold based on Welch bound and reduce the ele- ments of the non--diagonal of the Gram matrix; Third, produce new projection matrix from inverse solution of new Gram matrix and sparse transformation matrix iteratively, so as to achieve the purpose of reduction the mutual coherence and optimizing the measurement matrix. Experiments in the last show： measurement matrix based on the Welch optimization can rapidly reduce the maximum value of the compressed sensing correlation matrix and improve the performance of OMP algorithm, such as when error rate is 10-0.9, the original Gauss random ma- trix need 23 observations, but our optimized matrix only 16 observations. On the whole, when compared with the optimization method of Elad and Zhao, the reconstruction error of the algorithm in this paper is smaller, the performance and stability are slightly improved.