Low-Rank Tensor Completion and Total Variation Minimization for Color Image Inpainting

Low-rank (LR) and total variation (TV) are two most frequent priors that occur in image processing problems, and they have sparked a tremendous amount of researches, particularly for moving from scalar to vector, matrix or even high-order based functions. However, discretization schemes used for TV regularization often ignore the difference of the intrinsic properties, so it will lead to the problem that local smoothness cannot be effectively generated, let alone the problem of blurred edges. To address the image inpainting problem with corrupted data, in this paper, the color images are naturally considered as three-dimensional tensors, whose prior of smoothness can be measured by varietal TV norm along different dimensions. Specifically, we propose incorporating Shannon total variation (STV) and low-rank tensor completion (LRTC) into the construction of the final cost function, in which a new nonconvex low-rank constraint, namely truncated $\gamma $ -norm, is involved for closer rank approximation. Moreover, two methods are developed, i.e., LRRSTV and LRRSTV-T, due to the fact that LRTC can be represented by tensor unfolding and tensor decomposition. The final solution can be achieved by a practical variant of the augmented Lagrangian alternating direction method (ALADM). Experiments on color image inpainting tasks demonstrate that the proposed methods perform better then the state-of-the-art algorithms, both qualitatively and quantitatively.