  In most state-of-the-art image restoration methods, the sum of a
data-fidelity and a regularization term is optimized using an iterative
algorithm such as ADMM (alternating direction method of multipliers). In recent
years, the possibility of using denoisers for regularization has been explored
in several works. A popular approach is to formally replace the proximal
operator within the ADMM framework with some powerful denoiser. However, since
most state-of-the-art denoisers cannot be posed as a proximal operator, one
cannot guarantee the convergence of these so-called plug-and-play (PnP)
algorithms. In fact, the theoretical convergence of PnP algorithms is an active
research topic. In this letter, we consider the result of Chan et al. (IEEE
TCI, 2017), where fixed-point convergence of an ADMM-based PnP algorithm was
established for a class of denoisers. We argue that the original proof is
incomplete, since convergence is not analyzed for one of the three possible
cases outlined in the paper. Moreover, we explain why the argument for the
other cases does not apply in this case. We give a different analysis to fill
this gap, which firmly establishes the original convergence theorem.
