Low-rank tensor completion methods have been advanced recently for modeling sparsely observed data with a multimode structure. However, low-rank priors may fail to interpret the model factors of general tensor objects. The most common method to address this drawback is to use regularizations together with the low-rank priors. However, due to the complex nature and diverse characteristics of real-world multiway data, the use of a single or a few regularizations remains far from efficient, and there are limited systematic experimental reports on the advantages of these regularizations for tensor completion. To fill these gaps, we propose a modified CP tensor factorization framework that fuses the l₂ norm constraint, sparseness (l₁ norm), manifold, and smooth information simultaneously. The factorization problem is addressed through a combination of Nesterov's optimal gradient descent method and block coordinate descent. Here, we construct a smooth approximation to the $l_1$ norm and TV norm regularizations, and then, the tensor factor is updated using the projected gradient method, where the step size is determined by the Lipschitz constant. Extensive experiments on simulation data, visual data completion, intelligent transportation systems, and GPS data of user involvement are conducted, and the efficiency of our method is confirmed by the results. Moreover, the obtained results reveal the characteristics of these commonly used regularizations for tensor completion in a certain sense and give experimental guidance concerning how to use them.

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Low-resolution (LR) document images may cause difficulties in reading or low recognition rates in computer vision. Thus, it is necessary to improve the resolution of an LR document image via some algorithms. In this study, a novel document image super-resolution (SR) method using structural similarity and Markov random field (MRF) is proposed. First, the non-local algorithm is utilised to find similar patches. Instead of using the Euclidian distance, a modified chi-square distance is proposed to measure the patch similarity because the bimodality characteristic of the document images can be better described by this modified chi-square distance. Finally, the structural similarity of similar patches is served as a constraint for the MRF-based SR method, which is proper to describe the neighbouring relationship between patches. The SR reconstruction for LR images of printed and handwritten documents are carried out by the proposed algorithm. Experimental results show that the reconstructed SR images obtain higher peak signal-to-noise ratio and structural similarity values than those of several state-ofthe-art SR methods and visually pleasant SR images can be produced as well.

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