Dictionary Learning and Sensing Matrix Optimization for Compressed Sensing

This paper deals with dictionary learning and optimal sensing matrix design for compressed sensing (CS) systems. An improved version of the method of optimal directions (MOD) is proposed, which can overcome the problem with matrix inversion. The optimal sensing matrix design problem is defined as to find those sensing matrices that minimize a Frobenius norm-based difference between the Gram of the equivalent dictionary and the identity matrix. The solution set is characterized, which is a generalization of the existing results. A numerical algorithm is derived to find the best sensing matrix among the solution set. Simulation results are carried out, which show that the proposed algorithm for sensing matrix optimization can significantly improve the signal recovery accuracy of CS systems.