Multi-dimensional images can be viewed as tensors and have often embedded a low-rankness property that can be evaluated by tensor low-rank measures. In this paper, we first introduce a tensor low-rank and sparsity measure and then propose low-rank and sparsity models for tensor completion, tensor robust principal component analysis, and tensor denoising. The resulting tensor recovery models are further solved by the augmented Lagrangian method with a convergence guarantee. And its augmented Lagrangian subproblem is computed by the proximal alternative method, in which each variable has a closed-form solution. Numerical experiments on several multi-dimensional image recovery applications show the superiority of the proposed methods over the state-of-the-art methods in terms of several quantitative quality indices and visual quality.

多维图像可以被视为张量，并且通常嵌入可以通过张量低秩度量来评估的低秩属性。在本文中，我们首先引入张量低秩和稀疏性度量，然后提出用于张量补全、张量鲁棒主成分分析和张量去噪的低秩和稀疏模型。由此产生的张量恢复模型通过具有收敛保证的增强拉格朗日方法进一步求解。其增广拉格朗日子问题采用近端替代法计算，其中每个变量都有一个封闭式解。对多个多维图像恢复应用的数值实验表明，所提出的方法在多个定量质量指数和视觉质量方面优于最先进的方法。