Sensing Matrix Optimization Based on Equiangular Tight Frames With Consideration of Sparse Representation Error

摘要

This paper deals with the sensing matrix optimization problem for compressed sensing (CS) systems. Traditionally, the optimal sensing matrix is designed such that the Gram of the equivalent dictionary defined as the product of the sensing matrix and the dictionary is as close to a target Gram with some proper properties as possible. In this study, the sensing matrix is designed to make the equivalent dictionary approximate to a certain target frame. In addition, to avoid the sparse representation error (SRE) to be amplified in the measurement domain, a penalty term related to the SRE is included in the design criterion. An alternating minimization algorithm is proposed to solve the optimum sensing matrix problem, where the target frame is taken as the relaxed equiangular tight frame, which is constructed with a new method with the purpose of reducing the mutual coherence and maintaining the tightness of the frame, then the solution of the optimal sensing matrix is derived analytically with the target frame fixed. Experiments are carried out with synthetic data and real images, which demonstrate promising performance of the proposed algorithms and superiority of the CS system designed with the optimized sensing matrix to existing ones in terms of signal reconstruction accuracy.