Projection matrix optimization based on SVD for compressive sensing systems

Sparse signals can be sensed with a reduced number of projections and then reconstructed if compressive sensing (CS) is employed. Traditionally, the projection matrix is chosen as a random matrix, but a projection sensing matrix that is optimally designed for a certain class of signals can further improve the reconstruction accuracy or further reduce the necessary number of measurement samples. This paper considers the problem of designing the projection matrix Φ for a compressive sensing system in which the dictionary Ψ is assumed to be given. The optimal projection matrix design is formulated in terms of finding those Φ such that the Frobenius norm of the difference between the Gram matrix of the equivalent dictionary ΦΨ and the identity matrix is minimized. A novel algorithm based on SVD for optimal projection matrix searching is proposed to solve the corresponding minimization problem. Simulation results reveal that the signal recovery performance of sensing matrix obtained by proposed algorithm surpasses that of other standard sensing matrix designs.