  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 regularization. The
proposed algorithm is based on the iterative reweighted least squares (IRLS)
model, which is further accelerated by the preconditioned conjugate gradient
method. The convergence rate of the proposed algorithm is almost the same as
that of the traditional IRLS algorithms, that is, exponentially fast. Moreover,
with the specifically devised preconditioner, the computational cost for each
iteration 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 a practical application: 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.
