Optimized Projection Matrix for Compressed Sensing

Sparse signals can be reconstructed from far fewer samples than those that were required by the Shannon sampling theorem, if compressed sensing (CS) is employed. Traditionally, a random Gaussian (rGauss) matrix is used as a projection matrix in CS. Alternatively, optimization of the projection matrix is considered in this paper to enhance the quality of the reconstruction in CS. Bringing the multiplication of the projection matrix and the sparsifying basis to be near an equiangular tight frame (ETF) is a good idea proposed by some previous works. Here, a low-rank Gram matrix model is introduced to realize this idea. Also, an algorithm is presented via a computational method of the low-rank matrix nearness problem. Simulations show that the proposed method is better than some other methods in optimizing the projection matrix in terms of image denoising via sparse representation.