  In recent works several authors have proposed the use of precise boundary
conditions (BCs) for blurring models and they proved that the resulting choice
(Neumann or reflective, anti-reflective) leads to fast algorithms both for
deblurring and for detecting the regularization parameters in presence of
noise. When considering a symmetric point spread function, the crucial fact is
that such BCs are related to fast trigonometric transforms.
  In this paper we combine the use of precise BCs with the Total Variation (TV)
approach in order to preserve the jumps of the given signal (edges of the given
image) as much as possible. We consider a classic fixed point method with a
preconditioned Krylov method (usually the conjugate gradient method) for the
inner iteration. Based on fast trigonometric transforms, we propose some
preconditioning strategies which are suitable for reflective and
anti-reflective BCs. A theoretical analysis motivates the choice of our
preconditioners and an extensive numerical experimentation is reported and
critically discussed. The latter shows that the TV regularization with
anti-reflective BCs implies not only a reduced analytical error, but also a
lower computational cost of the whole restoration procedure over the other BCs.
