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