An Efficient Sparse Optimization Algorithm for Weighted $\ell _{0}$ Shearlet-Based Method for Image Deblurring

Sparsity is one of the key concepts that allows the signal recovery at a significantly lower subsample rate than required by the Nyquist-Shannon sampling theorem. By using a multistate transform, such as wavelets and shearlets system, the sparse representation of signals can be obtained. To further exploit the sparsity of the reconstructed signal, a generalized gradient regularizer is introduced to the proposed model. Motivated by the idea of iterative support detection (ISD), an optimization algorithm framework for image deblurring is given. The algorithm aims to solve a reweighted
$\ell _{0}$
-minimization problem in split Bregman framework, and the weights used for the next iteration are decided by an ISD process. The advantage of this process is that it forms an iterative-feedback mechanism, which improves the effectiveness for solution searching. A series of experiments are presented to demonstrate the availability of the proposed framework. Experimental results show that this method yields significant improvement in peak signal-to-noise ratio when compared to other counterparts. However, the numerical experiments also show that more computing time is required due to the utilization of the redundant multiscale system.