  Inspired by the theoretical results on optimal preconditioning stated by Ng,
R.Chan, and Tang in the framework of Reflective boundary conditions (BCs), in
this paper we present analogous results for Anti-Reflective BCs, where an
additional technical difficulty is represented by the non orthogonal character
of the Anti-Reflective transform and indeed the technique of Ng, R.Chan, and
Tang can not be used. 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 boundary conditions. Computational results show that the
preconditioning strategy is effective and it is able to give rise to a
meaningful acceleration both for slightly and highly non-symmetric PSFs.
