Analyzing the group sparsity based on the rank minimization methods

Abstract Sparse coding has achieved a great success in various image processing studies. However, there is not any benchmark to measure the sparsity of image patch/group because sparse discriminant conditions cannot keep unchanged. This paper analyzes the sparsity of group based on the strategy of the rank minimization. Firstly, an adaptive dictionary for each group is designed. Then, we prove that group-based sparse coding is equivalent to the rank minimization problem, and thus the sparse coefficients of each group are measured by estimating the singular values of each group. Based on that measurement, the weighted Schatten p-norm minimization (WSNM) has been found to be the closest solution to the real singular values of each group. Thus, WSNM can be equivalently transformed into a non-convex ℓp-norm minimization problem in group-based sparse coding. Experimental results on two applications: image in painting and image compressive sensing (CS) recovery show that the proposed scheme outperforms many state-of-the-art methods