This paper discusses weighted tensor Golub-Kahan-type bidiagonalization processes using the t-product. This product was introduced in [M. E. Kilmer and C. D. Martin, Factorization strategies for third order tensors, Linear Algebra Appl., 435 (2011), pp.~641--658]. A few steps of a bidiagonalization process with a weighted least squares norm are carried out to reduce a large-scale linear discrete ill-posed problem to a problem of small size. The weights are determined by symmetric positive definite (SPD) tensors. Tikhonov regularization is applied to the reduced problem. An algorithm for tensor Cholesky factorization of SPD tensors is presented. The data is a laterally oriented matrix or a general third order tensor. The use of a weighted Frobenius norm in the fidelity term of Tikhonov minimization problems is appropriate when the noise in the data has a known covariance matrix that is not the identity. We use the discrepancy principle to determine both the regularization parameter in Tikhonov regularization and the number of bidiagonalization steps. Applications to image and video restoration are considered.

翻译：本文讨论使用 t 产品 的 加权 Exgor Golub- Kahan 类型 色化过程 。 该产品在 [M. E. E. Kilmer 和 C. D. Martin, 第三顺序 的量化策略, Linear Algebra Appl., 435 (2011), pp. ~ 641- 658] 中引入了该产品 。 实施了一个具有加权最小方位规范的拖线化过程的几个步骤, 以减少大规模线性离散错误问题, 解决小问题 。 权重由对称正数定数( SPD) 温度决定 。 Tikhonov 正规化适用于减少的问题 。 数据是面向横向的矩阵或一般第三顺序 。 在 Tikhoonov 的忠诚术语中使用加权的Frobenius 规范, 最大限度地减少问题是合适的。 当数据中的噪音有一个已知的共性矩阵, 而不是身份 。 我们使用差异原则来确定 Tikhonov 图像 的恢复 格式化步骤 。