Fast iteratively reweighted least squares algorithms for analysis-based sparse reconstruction.

In this paper, we propose a novel algorithm for analysis-based sparsity reconstruction. It can solve the generalized problem by structured sparsity regularization with an orthogonal basis and total variation (TV) regularization. The proposed algorithm is based on the iterative reweighted least squares (IRLS) framework, and is accelerated by the preconditioned conjugate gradient method. The proposed method is motivated by that, the Hessian matrix for many applications is diagonally dominant. The convergence rate of the proposed algorithm is empirically shown to be almost the same as that of the traditional IRLS algorithms, that is, linear convergence. Moreover, with the specifically devised preconditioner, the computational cost for the subproblem is significantly less than that of traditional IRLS algorithms, which enables our approach to handle large scale problems. In addition to the fast convergence, it is straightforward to apply our method to standard sparsity, group sparsity, overlapping group sparsity and TV based problems. Experiments are conducted on practical applications of compressive sensing magnetic resonance imaging. Extensive results demonstrate that the proposed algorithm achieves superior performance over 14 state-of-the-art algorithms in terms of both accuracy and computational cost.