Alternative Arnoldi process for ill-conditioned tensor equations with application to image restoration

Abstract

This paper is concerned with developing an iterative tensor Krylov subspace method to solve linear discrete ill-posed systems of equations with a particular tensor product structure. We use the well-known Frobenius inner product for two tensors and the n-mode matrix-product of a tensor with a matrix to define tensor QR decomposition and alternative Arnoldi algorithms. Moreover, we illustrate how the tensor alternative Arnoldi process can be exploited to solve ill-posed problems by recovering blurry color images and videos in conjunction with the Tikhonov regularization technique, to derive approximate regularized solutions. We also review a generalized cross-validation technique for selecting the regularization parameter in the Tikhonov regularization. Theoretical properties of this method are demonstrated and applications including image deblurring and video processing are investigated. Numerical examples compare the proposed method with several other methods and illustrate the potential superiority of mentioned methods.