Performance of Three Preconditioners for Image Deblurring Problem in Primal-Dual Formulation

In this paper, we consider the generalized saddle point linear system of equations which is obtained from discretizing the Euler Lagrange equations associated with image debulrring problem. This system is ill-conditioned and is of huge size. Moreover, the (2,2) block of the coefficient matrix of this system contains summation of two terms. One of these terms is a product of a Toepelitz matrix with Toepelitz blocks (BTTB) and its transpose. This structure needs a prconditioner to speed up the iterative method such us the minimal residual (MINRES) method. Hence, we devolve three block diagonal preconditioners which are of Murphy, Golub and Wathen [MGW] type for this saddle point system. The first preconditioner is based on approximating the product of the (BTTB) matrix and its transpose by a symmetric BTTB matrix while in the second one, we approximate the BTTB matrix by the Strang circulant. In the the third preconditioner, we approximate the BTTB matrix by the optimal circulant. We investigate the efficiency of these three preconditioners by several numerical computations in term of CPU time, iteration numbers and the quality of the reconstruction images.