Learning Cocoercive Conservative Denoisers via Helmholtz Decomposition for Poisson Imaging Inverse Problems

Deliang Wei, Peng Chen, Haobo Xu, Jiale Yao, Fang Li, Tieyong Zeng

Keywords: Poisson inverse problems, plug-and-play methods, cocoercive conservative, Helmholt decomposition, global convergence

TL;DR: This paper studies convergent plug-and-play methods with cocoercive conservative denoisers for Poisson inverse problems.

Abstract: Plug-and-play (PnP) methods with deep denoisers have shown impressive results in imaging problems. They typically require strong convexity or smoothness of the fidelity term and a (residual) non-expansive denoiser for convergence. These assumptions, however, are violated in Poisson inverse problems, and non-expansiveness can hinder denoising performance. To address these challenges, we propose a cocoercive conservative (CoCo) denoiser, which may be (residual) expansive, leading to improved denoising performance. By leveraging the generalized Helmholtz decomposition, we introduce a novel training strategy that combines Hamiltonian regularization to promote conservativeness and spectral regularization to ensure cocoerciveness. We prove that CoCo denoiser is a proximal operator of a weakly convex function, enabling a restoration model with an implicit weakly convex prior. The global convergence of PnP methods to a stationary point of this restoration model is established. Extensive experimental results demonstrate that our approach outperforms closely related methods in both visual quality and quantitative metrics.

Primary Area: Optimization (e.g., convex and non-convex, stochastic, robust)

Submission Number: 17852