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As low-rank modeling has achieved great success in tensor recovery, many research efforts devote to defining the tensor rank. Among them, the recent popular tensor tubal rank, defined based on the tensor singular value decomposition (t-SVD), obtains promising results. However, the framework of the t-SVD and the tensor tubal rank are applicable only to three-way tensors and lack of flexibility to handle different correlations along different modes. To tackle these two issues, we define a new tensor unfolding operator, named mode-$k_1k_2$ tensor unfolding, as the process of lexicographically stacking the mode-$k_1k_2$ slices of an $N$-way tensor into a three-way tensor, which is a three-way extension of the well-known mode-$k$ tensor matricization. Based on it, we define a novel tensor rank, the tensor $N$-tubal rank, as a vector whose elements contain the tubal rank of all mode-$k_1k_2$ unfolding tensors, to depict the correlations along different modes. To efficiently minimize the proposed $N$-tubal rank, we establish its convex relaxation: the weighted sum of tensor nuclear norm (WSTNN). Then, we apply WSTNN to low-rank tensor completion (LRTC) and tensor robust principal component analysis (TRPCA). The corresponding WSTNN-based LRTC and TRPCA models are proposed, and two efficient alternating direction method of multipliers (ADMM)-based algorithms are developed to solve the proposed models. Numerical experiments demonstrate that the proposed models significantly outperform the compared ones.

We address the problem of restoring an original image from its blurry and noisy observation together with inaccurate information of the blurring process. For this purpose, we propose an enhanced regularized structured total least squares (RSTLS) model that can estimate the latent image and blur kernel simultaneously. In the proposed model, both the image and the blur kernel are characterized by a group-based low-rank prior, which assumes that a group of vectorized similar data patches can be well approximated by a low-rank matrix. We develop an alternating minimization algorithm to solve the proposed model efficiently. Numerical experiments demonstrate the effectiveness of our method in terms of both quantitative measures and visual quality.

In this paper, we study the problem of recovering a tensor with missing data. We propose a new model combining the total variation regularization and low-rank matrix factorization. A block coordinate decent (BCD) algorithm is developed to efficiently solve the proposed optimization model. We theoretically show that under some mild conditions, the algorithm converges to the coordinatewise minimizers. Experimental results are reported to demonstrate the effectiveness of the proposed model and the efficiency of the numerical scheme.

We propose a novel image decomposition method to decompose an image into its cartoon and texture components. To exploit the nonlocal self-similarity of cartoon-plus-texture images, we construct groups by stacking together similar image patches into 3-D arrays and consider group as the basic unit of decomposition. We decompose each group via a convex optimization model consisting of 3-D cartoon and texture priors. These priors characterize the local properties of the cartoon and texture components and the nonlocal similarity within each component in a unified and natural manner. We develop the alternating direction method of multipliers (ADMM) to efficiently solve the proposed model. For further improvement, we investigate an adaptive rule for the estimation of the regularization parameter. The proposed method is also extended to tackle noisy images. Numerical experiments confirm that the performance of the proposed method is competitive with some of the state-of-the-art schemes.