Learning Cocoercive Conservative Denoisers via Helmholtz Decomposition for Poisson Inverse Problems
This addresses a specific bottleneck in imaging for Poisson inverse problems, offering an incremental improvement over existing plug-and-play methods.
The paper tackles the challenge of applying plug-and-play methods with deep denoisers to Poisson inverse problems, where standard assumptions like non-expansiveness are violated, by proposing a cocoercive conservative denoiser that improves denoising and enables global convergence, outperforming related methods in visual quality and quantitative metrics.
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. 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.