This paper discusses several transform-based methods for solving linear discrete ill-posed problems for third order tensor equations based on a tensor-tensor product defined by an invertible linear transform. Linear transform-based tensor-tensor products were first introduced in Kernfeld et al. (2015) [16]. These tensor-tensor products are applied to derive Tikhonov regularization methods based on Golub-Kahan-type bidiagonalization and Arnoldi-type processes. GMRES-type solution methods based on the latter process also are described. By applying only a fairly small number of steps of these processes, large-scale problems are reduced to problems of small size. The number of steps required by these processes and the regularization parameter are determined by the discrepancy principle. The data tensor is a general third order tensor or a tensor defined by a laterally oriented matrix. A quite general regularization tensor can be applied in Tikhonov regularization. Applications to color image and video restorations illustrate the effectiveness of the proposed methods.

本文讨论了基于可逆线性变换定义的张量-张量积的几种基于变换的方法，用于求解三阶张量方程的线性离散不适定问题。基于线性变换的张量-张量积首先在Kernfeld等人中介绍。（2015）[16]。这些张量-张量积用于基于Golub-Kahan型双对角化和Arnoldi型过程推导Tikhonov正则化方法。还描述了基于后一过程的GMRES型求解方法。通过仅采用这些过程中相当少的步骤，就可以将大规模的问题减少为小尺寸的问题。这些过程所需的步骤数和正则化参数由差异原理确定。数据张量是一般的三阶张量或由横向定向矩阵定义的张量。可以在Tikhonov正则化中应用一个相当笼统的正则张量。彩色图像和视频恢复的应用说明了所提出方法的有效性。