New Reweighted `1-minimization Algorithms for Compressive Sampling Recovery

Abstract

A recent compression method which overlooks the classical Shannon-Nyquist theorem is called compressive sampling, also known as compressed sensing. The reconstruction of this new compression method is proved to be done with high probability of success by performing `1-minimization problem. The `1-minimization reconstruction has been developed to the reweighted algorithm which recovers closely approximate sparse solutions. However, there is no rule that automatically selects the appropriate weighting values. This paper proposes the enhancements of reweighted `1-minimization by indicating the choice of weighting functions and the suggestion to find the weighting values. 
In reconstruction process, the approximate `1-minimization might recover the fault signal by shifting the zero solutions to the other values. Thus, the hard selective reweighted (HSR) algorithm is designed to increase the importance of zero candidates by selecting the near-zero solutions whose numbers are equal to a number of original zero entries scaled by greater weighting value. In general, the locations of zero entries are not known so that the HSR algorithm could not apply to the real-world problems. This problem is coped with by the second proposed automatic adaptive reweighted (AAR) algorithm which is used to predict the locations of zero entries without knowing a number of original entries. The idea is to find the smallest frequency bin of solutions which contains empty member then set it to be the threshold and the solutions which are close to zero and the others scaled by larger and smaller weighting values, respectively. The numerical results show comparatively that HSR and AAR algorithms outperform `1-minimization. Furthermore, both of these algorithms are demonstrated to be applied to manmade and magnetic resonance imaging (MRI) images.