Zha Zhiyuan, Yuan Xin, Wen Bihan, Zhou Jiantao, Zhang Jiachao, Zhu Ce
IEEE Trans Image Process. 2020 Mar 10. doi: 10.1109/TIP.2020.2972109.
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 ℓp-norm minimization problem in GSC. By using the earned benchmark in sparse coding, the weighted ℓp-norm minimization is expected to obtain better performance than the three other norm minimization methods, i.e., ℓ1-norm, ℓp-norm and weighted ℓ1-norm. To verify the feasibility of the proposed benchmark, we compare the weighted ℓp-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.
稀疏编码在各种图像处理任务中取得了巨大成功。然而，由于稀疏编码本质上是一个NP难问题，因此缺少一个衡量图像块/组稀疏性的基准。这项工作试图从秩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，以弥合基于组的稀疏编码（GSC）和秩最小化之间的差距。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，因此每个图像块组的稀疏系数可以通过估计每个图像块组的奇异值来衡量。因此，我们得到了一个衡量每个图像块组稀疏性的基准，因为原始图像块组的奇异值可以通过奇异值分解（SVD）轻松计算出来。通过分析它们相应的秩最小化对应物，这个基准可以用于评估稀疏编码中任何一种范数最小化方法的性能。为此，我们利用四种著名的秩最小化方法来研究每个图像块组的稀疏性，发现加权Schatten p范数最小化（WSNM）最接近每个图像块组的真实奇异值。受上述秩最小化和GSC等价关系的启发，WSNM可以转化为GSC中的一个非凸加权ℓp范数最小化问题。通过在稀疏编码中使用得到的基准，加权ℓp范数最小化有望比其他三种范数最小化方法，即ℓ1范数、ℓp范数和加权ℓ1范数，获得更好的性能。为了验证所提出基准的可行性，我们在稀疏编码中将加权ℓp范数最小化与上述三种范数最小化方法进行了比较。在图像修复应用，即图像修复和图像压缩感知恢复方面的实验结果表明，所提出的方案是可行的，并且优于许多现有的先进方法。
