On projection matrix optimization for compressive sensing systems

This paper considers the problem of designing the projection matrix \Phi for a compressive sensing (CS) system in which the dictionary \Psi is assumed to be given. The optimal projection matrix design is formulated in terms of finding those \Phi such that the Frobenius norm of the difference between the Gram matrix of the equivalent dictionary \Phi \Psi and the identity matrix is minimized. A class of the solutions is derived in a closed-form, which is a generalization of the existing results. More interestingly, it is revealed that this solution set is characterized by an arbitrary orthonormal matrix. This freedom is then used to further enhance the performance of the CS system by minimizing the coherence between the atoms of the equivalent dictionary. An alternating minimization-based algorithm is proposed for solving the corresponding minimization problem. Experiments are carried out and simulations show that the projection matrix obtained by the proposed approach significantly improves the signal recovery accuracy of the CS system and outperforms those by existing algorithms.