Patch-based weighted SCAD prior for compressive sensing

The nuclear norm-based convex surrogate of the rank function has been widely used in compressive sensing (CS) to exploit the sparsity of nonlocal similar patches in an image. However, this method treats different singular values equally and thus may produce a result far from the optimum one. In order to alleviate the limitations of the nuclear norm, different singular values should be treated differently. The reason is that large singular values can be used to retrieve substantial contents of an image, while small ones may contain noisy information. In this paper, we propose a model via non-convex weighted Smoothly Clipped Absolute Deviation (SCAD) prior. Our motivation is that SCAD shrinkage behaves like a soft shrinkage operator for small enough inputs, whereas for large enough ones, it leaves the input intact and behaves like hard shrinkage. For moderate input values, SCAD makes a good balance between soft shrinkage and hard shrinkage. Numerically, the alternating direction method of multiplier (ADMM) is adopted to split the original problem into several sub-problems with closed-form solutions. We further analyze the convergence of the proposed method under mild conditions. Various experimental results demonstrate that the proposed model outperforms many existing state-of-the-art CS methods.