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 -nuclear norm, which uses a data-dependent matrix 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- Q-T-SVD. In addition, we adopt a joint reweighted tensor -nuclear norm (JRTQN) as its non-convex relaxation, with a novel reweighted strategy and data-dependent transforms along each mode. What is more, based on the low-rank assumption, we provide a method to choose 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.

中文翻译：

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通过联合重加权张量 Q-核范数进行张量补全，用于视觉数据恢复

最近，基于变换的张量核范数方法在张量奇异值分解（t-SVD）框架下的低秩张量完成（LRTC）方面取得了令人鼓舞的结果。其中，张量核范数采用与数据相关的矩阵作为变换，在处理不同类型的数据时比使用固定变换更加灵活。然而，它只描述了光谱相关性而忽略了空间维度的信息。此外，它忽略了不平衡奇异值惩罚的必要性，这可能会导致恢复结果中原始信息的丢失和奇异值的稀疏性不足。为了克服上述缺陷，本文通过提出的张量分解，即模式QT-SVD，提出了一种新的张量秩定义，称为张量联合Q-rank。此外，我们采用联合重加权张量核范数（JRTQN）作为其非凸松弛，并采用新颖的重加权策略和沿每种模式的数据相关变换。更重要的是，基于低秩假设，我们提供了一种通过最大化奇异值分布方差来进行选择的方法。然后，我们提出了一种JRTQN-TC模型，通过交替方向乘子法求解，保证了理论收敛性。在彩色图像和视频恢复、多光谱图像修复、人脸图像补全以及CT和MRI图像恢复方面进行的大量实验在定量和视觉上证明了该方法与相关方法相比具有很强的竞争力。