A Benchmark for Sparse Coding When Group Sparsity Meets Rank Minimization

Abstract:

Sparse coding excels in image processing. Since sparse coding is NP-hard, there is no image patch/group sparsity benchmark. This rank minimization-based work fills the gap. We first design an adaptive dictionary to bridge group-based sparse coding (GSC) and rank minimization.

Then, we show that under the designed dictionary, GSC and rank minimization problems are equivalent, so the sparse coefficients of each patch group can be measured by estimating their singular values.

The singular value decomposition (SVD) can easily compute the singular values of the original image patch groups, giving us a benchmark to measure each patch group’s sparsity. Through rank minimization analysis, this benchmark can evaluate any norm minimization method in sparse coding.

To study the sparsity of each patch group, we use four well-known rank minimization methods, and the weighted Schatten l p -norm minimization (WSNM) is closest to the real singular values. WSNM can be translated into a non-convex weighted l p-norm minimization problem in GSC using the rank minimization-GSC equivalence regime.

Weighted l p -norm minimization is expected to outperform the other three norm minimization methods in sparse coding by using the earned benchmark. To test the benchmark, we compare the weighted l p-norm minimization to the three sparse coding norm minimization methods. The proposed scheme works for image inpainting and image compressive sensing recovery, according to experimental results.

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