An efficient algorithm for designing projection matrix in compressive sensing based on alternating optimization

This paper considers the problem of optimally designing the projection matrix Φ for a certain class of signals which can be sparsely represented by a specified dictionary Ψ. The optimal projection matrix is proposed to minimize the distance between the Gram matrix of the equivalent dictionary ΦΨ and a set of relaxed Equiangular Tight Frames (ETFs). An efficient method is derived for the optimal projection matrix design with a given Gram matrix. In addition, an extension of projection matrix design is derived for the scenarios where the signals cannot be represented exactly sparse in a specified dictionary. Simulations with synthetic data and real images demonstrate that the obtained projection matrix significantly improves the signal recovery accuracy of a system and outperforms those obtained by the existing algorithms.