The Plug-and-Play framework has demonstrated that a denoiser can implicitly serve as the image prior for model-based methods for solving various inverse problems such as image restoration tasks. This characteristic enables the integration of the flexibility of model-based methods with the effectiveness of learning-based denoisers. However, the regularization strength induced by denoisers in the traditional Plug-and-Play framework lacks a physical interpretation, necessitating demanding parameter tuning. This paper addresses this issue by introducing the Constrained Plug-and-Play (CPnP) method, which reformulates the traditional PnP as a constrained optimization problem. In this formulation, the regularization parameter directly corresponds to the amount of noise in the measurements. The solution to the constrained problem is obtained through the design of an efficient method based on the Alternating Direction Method of Multipliers (ADMM). Our experiments demonstrate that CPnP outperforms competing methods in terms of stability and robustness while also achieving competitive performance for image quality.
Keywords: constrained formulation; discrepancy principle; image restoration; inverse problems; plug-and-play priors; regularization by denoising.