The Order-p Tensor Linear Complementarity Problem for Images Deblurring

In this paper, we first study the equivalence between the third order tensor linear complementarity problem under the framework of t-product and the least squares problem under the t-product with nonnegative constraints, and based on their equivalence, apply the third order tensor linear complementarity problem to the t-product Arnoldi–Tikhonov regularization method for grayscale images deblurring. Secondly, we extend the definition of the third order tensor linear complementarity problem under the t-product to the order-p ( \(p>3\) $p>3$ ) tensor linear complementarity problem, propose a fixed point iterative method for solving the order-p ( \(p>3\) $p>3$ ) tensor linear complementarity problem, and prove that the equivalence between the third order tensor linear complementarity problem and the least squares problem under the t-product with nonnegative constraints also holds at the pth ( \(p>3\) $p>3$ ) order. Finally, we establish the tensor t-product model for color images deblurring with the within-channel and the cross-channel blurring, and propose the t-product Arnoldi–Tikhonov regularization method for this model. Moreover, we apply the fourth order tensor linear complementarity problem to solve the t-product Arnoldi–Tikhonov regularization method with nonnegative constraints.