Bridge the Gap Between Group Sparse Coding and Rank Minimization via Adaptive Dictionary Learning

Both sparse coding and rank minimization have led to great successes in various image processing tasks. Though the underlying principles of these two approaches are similar, no theory is available to demonstrate the correspondence. In this paper, starting by designing an adaptive dictionary for each group of image patches, we analyze the sparsity of image patches in each group using the rank minimization approach. Based on this, we prove that the group-based sparse coding is equivalent to the rank minimization problem under our proposed adaptive dictionary. Therefore, the sparsity of the coefficients for each group can be measured by estimating the singular values of this group. Inspired by our theoretical analysis, four nuclear norm like minimization methods including the standard nuclear norm minimization (NNM), weighted nuclear norm minimization (WNNM), Schatten $p$-norm minimization (SNM), and weighted Schatten $p$-norm minimization (WSNM), are employed to analyze the sparsity of the coefficients and WSNM is found to be the closest solution to the singular values of each group. Based on this, WSNM is then translated to a non-convex weighted $\ell_p$-norm minimization problem in group-based sparse coding, and in order to solve this problem, a new algorithm based on the alternating direction method of multipliers (ADMM) framework is developed. Experimental results on two low-level vision tasks: image inpainting and image compressive sensing recovery, demonstrate that the proposed scheme is feasible and outperforms state-of-the-art methods.