Plug and Play Splitting Techniques for Poisson Image Restoration

Abstract Plug and Play (PnP) methods achieve remarkable results in the framework of image restoration problems for Gaussian data. Nonetheless, the theory available for the Gaussian case cannot be extended to the Poisson case, due to the non-Lipschitz gradient of the fidelity function, the Kullback-Leibler functional, or the absence of closed-form solution for the proximal operator of such term, leading to employ iterative solvers for the inner subproblem. In this work we extend the idea of PIDSPLIT+ algorithm, exploiting the Alternating Direction Method of Multipliers, to PnP scheme: this allows to provide a closed-form solution for the deblurring step, with no need for iterative solvers. The convergence of the method is assured by employing a firmly non-expansive denoiser. The proposed method, namely PnPSplit $$^{+}$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mmultiscripts> <mml:mrow/> <mml:mrow/> <mml:mo>+</mml:mo> </mml:mmultiscripts> </mml:math> , is tested on different Poisson image restoration problems, showing remarkable performance even in presence of high noise level and severe blurring conditions.