A Benchmark for Sparse Coding: When Group Sparsity Meets Rank Minimization

Sparse coding has achieved a great success in various image processing tasks. However, a benchmark to measure the sparsity of image patch/group is missing since sparse coding is essentially an NP-hard problem. This work attempts to fill the gap from the perspective of rank minimization. We firstly design an adaptive dictionary to bridge the gap between group-based sparse coding (GSC) and rank minimization. Then, we show that under the designed dictionary, GSC and the rank minimization problems are equivalent, and therefore the sparse coefficients of each patch group can be measured by estimating the singular values of each patch group. We thus earn a benchmark to measure the sparsity of each patch group because the singular values of the original image patch groups can be easily computed by the singular value decomposition (SVD). This benchmark can be used to evaluate performance of any kind of norm minimization methods in sparse coding through analyzing their corresponding rank minimization counterparts. Towards this end, we exploit four well-known rank minimization methods to study the sparsity of each patch group and the weighted Schatten <inline-formula> <tex-math notation="LaTeX">$p$ </tex-math></inline-formula>-norm minimization (WSNM) is found to be the closest one to the real singular values of each patch group. Inspired by the aforementioned equivalence regime of rank minimization and GSC, WSNM can be translated into a non-convex weighted <inline-formula> <tex-math notation="LaTeX">$\ell _{p}$ </tex-math></inline-formula>-norm minimization problem in GSC. By using the earned benchmark in sparse coding, the weighted <inline-formula> <tex-math notation="LaTeX">$\ell _{p}$ </tex-math></inline-formula>-norm minimization is expected to obtain better performance than the three other norm minimization methods, <italic>i.e.</italic>, <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>-norm, <inline-formula> <tex-math notation="LaTeX">$\ell _{p}$ </tex-math></inline-formula>-norm and weighted <inline-formula> <tex-math notation="LaTeX">$\ell _{1}$ </tex-math></inline-formula>-norm. To verify the feasibility of the proposed benchmark, we compare the weighted <inline-formula> <tex-math notation="LaTeX">$\ell _{p}$ </tex-math></inline-formula>-norm minimization against the three aforementioned norm minimization methods in sparse coding. Experimental results on image restoration applications, namely image inpainting and image compressive sensing recovery, demonstrate that the proposed scheme is feasible and outperforms many state-of-the-art methods.