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 $p$ -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 $\ell _{p}$ -norm minimization problem in GSC. By using the earned benchmark in sparse coding, the weighted $\ell _{p}$ -norm minimization is expected to obtain better performance than the three other norm minimization methods, i.e. , $\ell _{1}$ -norm, $\ell _{p}$ -norm and weighted $\ell _{1}$ -norm. To verify the feasibility of the proposed benchmark, we compare the weighted $\ell _{p}$ -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.

中文翻译：

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稀疏编码的基准：当组稀疏性遇到等级最小化时


稀疏编码在各种图像处理任务中取得了巨大的成功。然而，由于稀疏编码本质上是一个 NP 难题，因此缺少衡量图像块/组稀疏性的基准。这项工作试图从等级最小化的角度填补这一空白。我们首先设计了一个自适应字典来弥合基于组的稀疏编码（GSC）和排名最小化之间的差距。然后，我们表明，在设计的字典下，GSC和秩最小化问题是等价的，因此可以通过估计每个补丁组的奇异值来测量每个补丁组的稀疏系数。因此，我们获得了衡量每个补丁组稀疏性的基准，因为原始图像补丁组的奇异值可以通过奇异值分解（SVD）轻松计算。该基准可用于通过分析相应的秩最小化方法来评估稀疏编码中任何类型的范数最小化方法的性能。为此，我们利用四种著名的等级最小化方法来研究每个补丁组的稀疏性和加权 Schatten $p$范数最小化（WSNM）被发现是最接近每个补丁组的真实奇异值的。受上述秩最小化和 GSC 等价机制的启发，WSNM 可以转化为非凸加权$\ell_{p}$ - GSC 中的范数最小化问题。 通过使用稀疏编码中获得的基准，加权$\ell_{p}$ -范数最小化预计比其他三种范数最小化方法获得更好的性能， IE , $\ell_{1}$ -规范， $\ell_{p}$ - 范数和加权$\ell_{1}$ -规范。为了验证所提出的基准的可行性，我们比较了加权$\ell_{p}$ -针对稀疏编码中上述三种范数最小化方法的范数最小化。图像恢复应用（即图像修复和图像压缩感知恢复）的实验结果表明，所提出的方案是可行的，并且优于许多最先进的方法。