A Relaxed Gradient Step Denoiser for Splitting Methods in Poisson Inverse Problems

Abstract

Plug and Play methods combine classical variational models with learned denoisers and have achieved strong results in imaging inverse problems. Their convergence has been widely studied for Gaussian data, whereas Poisson models require additional care because of the nonquadratic Kullback-Leibler fidelity. This work introduces GSDSplit+ , a splitting method built upon PnPSplit+, in which the generic denoising block is replaced by a relaxed Gradient Step Denoiser. The resulting method retains explicit Poisson fidelity updates. The denoiser is defined through the gradient of a learned convex potential parameterized by an Input Convex Neural Network. Its architecture and training promote both blind denoising capability and a smoothness condition sufficient for firm nonexpansiveness. Empirical smoothness estimates are used to select a relaxation parameter compatible with the convergence assumptions of PnPSplit+. Numerical experiments provide evidence that the trained denoiser operates in an FNE-compatible regime on the considered validation data. GSDSplit+ also yields stable reconstructions outside the empirically supported range and shows substantially lower sensitivity to the ADMM parameter, while remaining effective for different blur operators and Poisson noise levels.