  A standard model for image reconstruction involves the minimization of a
data-fidelity term along with a regularizer, where the optimization is
performed using proximal algorithms such as ISTA and ADMM. In plug-and-play
(PnP) regularization, the proximal operator (associated with the regularizer)
in ISTA and ADMM is replaced by a powerful image denoiser. Although PnP
regularization works surprisingly well in practice, its theoretical convergence
-- whether convergence of the PnP iterates is guaranteed and if they minimize
some objective function -- is not completely understood even for simple linear
denoisers such as nonlocal means. In particular, while there are works where
either iterate or objective convergence is established separately, a
simultaneous guarantee on iterate and objective convergence is not available
for any denoiser to our knowledge. In this paper, we establish both forms of
convergence for a special class of linear denoisers. Notably, unlike existing
works where the focus is on symmetric denoisers, our analysis covers
non-symmetric denoisers such as nonlocal means and almost any convex
data-fidelity. The novelty in this regard is that we make use of the
convergence theory of averaged operators and we work with a special inner
product (and norm) derived from the linear denoiser; the latter requires us to
appropriately define the gradient and proximal operators associated with the
data-fidelity term. We validate our convergence results using image
reconstruction experiments.
