Tensor completion is a challenging problem with various applications, especially in recovering incomplete visual data. Considering a color image or gray video as a three-dimensional tensor, many related models based on the low-rank prior of the tensor have been proposed. However, the low-rank prior may not be enough to recover the original tensor from the observed incomplete tensor. In this paper, we propose a tensor completion method to recover color images and gray videos by exploiting both the low-rank and sparse prior of the observed tensor. Specifically, the tensor completion task can be formulated as a low-rank minimization problem with a sparse regularizer. The low-rank property is depicted by the tensor truncated nuclear norm based on tensor singular value decomposition which is a better approximation of tensor tubal rank than tensor nuclear norm. While the sparse regularizer is imposed by a ℓ₁-norm in a discrete cosine transformation domain, which can better employ the local sparse property of the incomplete data. To solve the optimization problem, we employ an alternating direction method of multipliers in which we only need to solve several subproblems which have closed-form solutions. Substantial experiments on real-world images and videos show that the proposed method has better performances than the existing state-of-the-art methods.

中文翻译：

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变换域中具有稀疏正则化的低秩张量补全

张量完成是各种应用程序中的一个具有挑战性的问题，尤其是在恢复不完整的视觉数据方面。考虑将彩色图像或灰度视频作为三维张量，已经提出了许多基于张量低秩先验的相关模型。然而，低秩先验可能不足以从观察到的不完整张量中恢复原始张量。在本文中，我们提出了一种张量完成方法，通过利用观察到的张量的低秩和稀疏先验来恢复彩色图像和灰色视频。具体来说，张量完成任务可以被表述为具有稀疏正则化器的低秩最小化问题。低秩属性由基于张量奇异值分解的张量截断核范数描述，这是比张量核范数更好的张量管秩近似。虽然稀疏正则化器是由ℓ _{1 -}范数在离散余弦变换域中，可以更好地利用不完整数据的局部稀疏性。为了解决优化问题，我们采用乘法器的交替方向方法，其中我们只需要解决几个具有封闭形式解的子问题。对真实世界图像和视频的大量实验表明，所提出的方法比现有的最先进方法具有更好的性能。