Optimal preconditioning for image deblurring with Anti-Reflective boundary conditions

Inspired by the theoretical results on optimal preconditioning stated by Ng, Chan, and Tang in the framework of Reflective boundary conditions (BCs), in this paper we present analogous results for Anti-Reflective BCs. Here a key technical difficulty is represented by the non-orthogonal character of the Anti-Reflective transform and indeed the proof proposed by Ng, Chan, and Tang does not work. Nevertheless, in both cases, the optimal preconditioner is the blurring matrix associated to the symmetrized Point Spread Function (PSF). The geometrical idea on which our proof is based is very simple and general, so it may be useful in the future to prove theoretical results for new proposed BCs. Numerical tests show that the optimal preconditioning strategy is effective when using both preconditioned conjugate gradient methods and recently introduced nonstationary preconditioned iterations.