本文的主要目的是为三阶张量恢复开发张量核规范的框架表示。在文献中，张量核范数可以通过使用基于离散傅立叶变换矩阵的张量奇异值分解来计算，并且张量补全可以通过张量核范数的最小化来实现，即张量核范数从所有傅立叶变换的矩阵额叶切片。通过对原始张量的管应用离散傅里叶变换，可以获得这些傅里叶变换的矩阵额叶切片。在本文中，我们建议采用每个管的框架表示，以便可以构造框架变换的张量。由于基于框架的冗余性，每个管的表示都被稀疏表示。当原始张量的矩阵切片高度相关时，我们期望来自所有小帧变换的矩阵正面切片的矩阵秩的相应总和将较小，并且可以更好地执行所得张量完成。所提出的最小化模型是凸的，并且可以获得全局最小化器。测试了几种类型的多维数据（视频，多光谱图像和磁共振成像数据）的数值结果，结果表明，该方法优于其他方法。
The main aim of this paper is to develop a framelet representation of the tensor nuclear norm for third-order tensor recovery. In the literature, the tensor nuclear norm can be computed by using tensor singular value decomposition based on the discrete Fourier transform matrix, and tensor completion can be performed by the minimization of the tensor nuclear norm which is the relaxation of the sum of matrix ranks from all Fourier transformed matrix frontal slices. These Fourier transformed matrix frontal slices are obtained by applying the discrete Fourier transform on the tubes of the original tensor. In this paper, we propose to employ the framelet representation of each tube so that a framelet transformed tensor can be constructed. Because of framelet basis redundancy, the representation of each tube is sparsely represented. When the matrix slices of the original tensor are highly correlated, we expect the corresponding sum of matrix ranks from all framelet transformed matrix frontal slices would be small, and the resulting tensor completion can be performed much better. The proposed minimization model is convex and global minimizers can be obtained. Numerical results on several types of multi-dimensional data (videos, multispectral images, and magnetic resonance imaging data) have tested and shown that the proposed method outperformed the other testing methods.
