From many fewer acquired measurements than that suggested by the Nyquist sampling theory, compressive sensing (CS) theory demonstrates that a signal can be reconstructed with high probability when it exhibits sparsity in a certain domain. Recent CS methods have employed analytical sparsifying transforms such as wavelets, curvelets, and finite differences. In this paper, we propose a novel algorithm for image recovery, which minimizes a linear combination of three terms corresponding to least square data fitting, adaptive dictionary, and Hessian Schatten‐norm regularization. We split the problem into some subproblems which turn the minimization task into much simpler. Numerical experiments are conducted on several test images with a variety of sampling patterns and ratios in both noiseless and noise scenarios. The results demonstrate the superior performance of the proposed algorithm.

中文翻译：

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Hessian Schatten-norm和自适应字典，用于图像恢复

通过比Nyquist采样理论建议的少得多的获得的测量结果，压缩感测（CS）理论表明，当信号在特定域中表现出稀疏性时，可以以较高的概率重建信号。最近的CS方法采用了解析稀疏变换，例如小波，曲线小波和有限差分。在本文中，我们提出了一种新颖的图像恢复算法，该算法可最小化对应于最小二乘数据拟合，自适应字典和Hessian Schatten-norm正则化的三个项的线性组合。我们将问题分为几个子问题，这些子问题使最小化任务变得更加简单。在无噪声和有噪声的情况下，对具有多种采样模式和比率的几张测试图像进​​行了数值实验。