An Improved Method for Sparse decomposition and Reconstruction in Compressed sensing

Sparse decomposition, incorrelate projection and reconstruction algorithm are the three elements of compressed sensing. Any aspect of the merits will significantly impact the performance of compressed sensing. Sparse decomposition is the precondition to achieve compressed sensing, now the sparse decomposition for most natural signals are not absolutely sparse, but approximately sparse, which will greatly influence the reconstruction property of compressed sensing. In this paper, we design an invertible thresholding function, and then a thresholding matrix is devised using this thresholding function，thus the thresholding matrix is invertible. Under this thresholding matrix, the approximately sparse coefficients through orthogonal transformation can be more close to absolutely sparse. Since the process of using thresholding matrix to deal with approximately sparse coefficients is inversible, we can recover exactly original approximately sparse coefficients. Reconstructions algorithm use orthogonal matching pursuit(OMP) and compressive sampling matching pursuit(CoSaMP) in greedy algorithms, the theoretical analysis showed that under the same precondition with compressive sampling matching pursuit, thresholding matrix can improve notablely reconstruction error of compressive sampling matching pursuit. The simulations are about improved performance comparison of compressive sampling matching pursuit and orthogonal matching pursuit using thresholding matrix, we find that the thresholding matrix can make accuracy of reconstruction better.