Weighted tensor Golub-Kahan-Tikhonov-type methods applied to image processing using a t-product

Abstract: 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.