  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.
