Tensor completion via joint reweighted tensor Q-nuclear norm for visual data recovery

Recently, the transform-based tensor nuclear norm methods have achieved encouraging results for low-rank tensor completion (LRTC) under the tensor singular value decomposition (t-SVD) framework. Among them, the tensor Q-nuclear norm, which uses a data-dependent matrix Q as transform, is more flexible than that of using fixed transform when handling different types of data. However, it only describes the spectral correlations and ignores the spatial dimensions' information. Besides, it disregards the necessity for unbalanced singular value penalty, which may lead to the loss of primary information and inadequate sparsity of singular values in the recovery results. To overcome the above defects, this paper presents a new definition of tensor rank, called tensor joint Q-rank, via the proposed tensor decomposition, i.e., the mode -k Q-T-SVD. In addition, we adopt a joint reweighted tensor Q-nuclear norm (JRTQN) as its non-convex relaxation, with a novel reweighted strategy and data-dependent transforms Qk (k = 1, 2, 3) along each mode. What is more, based on the low-rank assumption, we provide a method to choose Qk by maximizing the variance of singular value distribution. Then, we propose a JRTQN-TC model, solved via the alternating direction multipliers method and the theoretical convergence is guaranteed. Extensive experiments carried out on color image and video recovery, multispectral image inpainting, face image completion and CT and MRI image restoration demonstrate the highly competitive performance of the proposed method quantitatively and visually, compared with the related methods.