  Group sparse representation (GSR) based method has led to great successes in
various image recovery tasks, which can be converted into a low-rank matrix
minimization problem. As a widely used surrogate function of low-rank, the
nuclear norm based convex surrogate usually leads to over-shrinking problem,
since the standard soft-thresholding operator shrinks all singular values
equally. To improve traditional sparse representation based image compressive
sensing (CS) performance, we propose a generalized CS framework based on GSR
model, which leads to a nonconvex nonsmooth low-rank minimization problem. The
popular L_2-norm and M-estimator are employed for standard image CS and robust
CS problem to fit the data respectively. For the better approximation of the
rank of group-matrix, a family of nuclear norms are employed to address the
over-shrinking problem. Moreover, we also propose a flexible and effective
iteratively-weighting strategy to control the weighting and contribution of
each singular value. Then we develop an iteratively reweighted nuclear norm
algorithm for our generalized framework via an alternating direction method of
multipliers framework, namely, GSR-AIR. Experimental results demonstrate that
our proposed CS framework can achieve favorable reconstruction performance
compared with current state-of-the-art methods and the robust CS framework can
suppress the outliers effectively.
