  Alternating direction method of multiplier (ADMM) is a widely used algorithm
for solving constrained optimization problems in image restoration. Among many
useful features, one critical feature of the ADMM algorithm is its modular
structure which allows one to plug in any off-the-shelf image denoising
algorithm for a subproblem in the ADMM algorithm. Because of the plug-in
nature, this type of ADMM algorithms is coined the name "Plug-and-Play ADMM".
Plug-and-Play ADMM has demonstrated promising empirical results in a number of
recent papers. However, it is unclear under what conditions and by using what
denoising algorithms would it guarantee convergence. Also, since Plug-and-Play
ADMM uses a specific way to split the variables, it is unclear if fast
implementation can be made for common Gaussian and Poissonian image restoration
problems.
  In this paper, we propose a Plug-and-Play ADMM algorithm with provable fixed
point convergence. We show that for any denoising algorithm satisfying an
asymptotic criteria, called bounded denoisers, Plug-and-Play ADMM converges to
a fixed point under a continuation scheme. We also present fast implementations
for two image restoration problems on super-resolution and single-photon
imaging. We compare Plug-and-Play ADMM with state-of-the-art algorithms in each
problem type, and demonstrate promising experimental results of the algorithm.
