t-Product Golub–Kahan–Tikhonov Regularization with Extrapolation Applied to Ill-Posed Problems in Image and Video Restoration

Large-scale multilinear discrete ill-posed problems formulated within the t- product framework arise in a wide range of scientific and engineering applica- tions, yet their numerical solution remains highly challenging. Among existing approaches, an effcient Golub–Kahan bidiagonalization method with Tikhonov regularization was proposed by Reichel et al. [1]. However, selecting an appro- priate regularization parameter in the Tikhonov scheme is notoriously diffcult, typically requiring computationally expensive procedures, while the method’s inherent smoothing effect can blur sharp features in the reconstructed solution. In this work, we propose a novel extrapolation-based technique that estimates a parameter-free solution from a finite set of Tikhonov-regularized solutions, treated as functions of predefined regularization parameters. The proposed approach eliminates the need for parameter tuning and preserves sharp details more effectively. Extensive numerical experiments on image and video restoration tasks demonstrate that the zero-parameter extrapolated solution consistently outperforms the standard Tikhonov method, achieving higher reconstruction accuracy while avoiding the computational burden of automatic parameter selection.