一种基于特征值分解的测量矩阵优化方法

测量矩阵是压缩感知中一个很重要的部分，为了减小测量矩阵与稀疏变换矩阵的互相干性，从而改善重建质量，本文首先通过测量矩阵和稀疏变换矩阵的乘积构造得到一个Gram矩阵，然后定义了一种基于Gram矩阵非对角线元素的整体互相干系数，推导出整体互相干系数与Gram矩阵特征值之间的关系。在此基础上，我们提出了一个最优化模型，在不改变Gram矩阵特征值和的前提下，让每个大于零的特征值的大小都为它们和的平均值，使得测量矩阵和稀疏变换矩阵的整体互相干系数达到最小，从而优化了测量矩阵的性能。将该方法用在一些已知的测量矩阵上，实验结果中矩阵的优化速度快，并且用优化矩阵所得的图像的PSNR有所提高，表明本文优化测量矩阵的方法在重建效果和优化速度方面都有一定的优势。 

An Optimization Method for Measurement Matrix Based on Eigenvalue Decomposition

Measurement matrix is a very important part in compressive sensing.In order to decrease the mutual coherence between the measurement matrix and sparse transformed matrix and improve the quality of reconstruction,a Gram matrix was constructed based on the product of the measurement matrix and sparse transformed matrix.Then a new global mutual coherent coefficient was defined based on off-diagonal elements of the Gram matrix.After deriving the relationship between the global mutual coherent coefficient and the eigenvalues of the Gram matrix,we proposed an optimization model,which could minimize the global mutual coherent coefficient of the given matrices by adjusting the eigenvalues above zero to the average value of the sum of these eigenvalues without changing the sum.The speed of optimizing matrix is fast and the PSNR of the picture is improved with the optimized measurement matrix from the experimental results.These showed that our proposed method had some advantages in terms of reconstruction effect and optimization speed.