Tensor Krylov subspace methods with an invertible linear transform product applied to image processing

Abstract

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.