Optimized Sensing Matrix Design Based on Parseval Tight Frame and Matrix Decomposition

Recent efforts have shown that the reconstruction performance could be improved with optimized sensing matrix according to a given dictionary for a compressed sensing (CS) system. The existed optimizing conditions are mainly used to address the worst-case performance of CS recovery. Considering the quality of a sensing matrix with respect to the mean squared error (MSE) performance of the Oracle estimator, Chen et al. proposed the sensing matrix based on Parseval tight frame, which exhibits superior performance in relation to other existed designs. However, the equivalent sensing matrix under this design framework couldn’t achieve the optimal mutual coherence. In light of the matrix decomposition theory, the bigger the smallest singular value, the stronger non-correlation of the columns of the matrix have. We further optimize the sensing matrix combining with the matrix decomposition theory, so as to achieve the optimal statistical reconstruction and the optimal mutual coherence performance at the same time. Through the approximate QR decomposition and the mean singular value decomposition (SVD), we adjust the singular values of the sensing matrix, so as to reduce the correlation of the matrix. A great number of experiments show that the proposed optimized sensing matrix realizes the minimum of the reconstructed error compared to other designs in the literature with different sparse recovery algorithms.