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Sparse coding has achieved a great success in various image processingstudies. However, there is not any benchmark to measure the sparsity of imagepatch/group because sparse discriminant conditions cannot keep unchanged. Thispaper analyzes the sparsity of group based on the strategy of the rankminimization. Firstly, an adaptive dictionary for each group is designed. Then,we prove that group-based sparse coding is equivalent to the rank minimizationproblem, and thus the sparse coefficient of each group is measured byestimating the singular values of each group. Based on that measurement, theweighted Schatten $p$-norm minimization (WSNM) has been found to be the closestsolution to the real singular values of each group. Thus, WSNM can beequivalently transformed into a non-convex $\ell_p$-norm minimization problemin group-based sparse coding. To make the proposed scheme tractable and robust,the alternating direction method of multipliers (ADMM) is used to solve the$\ell_p$-norm minimization problem. Experimental results on two applications:image inpainting and image compressive sensing (CS) recovery have shown thatthe proposed scheme outperforms many state-of-the-art methods.

Analyzing the group sparsity based on the rank minimization methods