Learning Cocoercive Conservative Denoisers via Helmholtz Decomposition for Poisson Inverse Problems

TL;DR

This paper introduces a novel training strategy for deep denoisers in Poisson inverse problems, leveraging Helmholtz decomposition to ensure cocoerciveness and conservativeness, leading to improved convergence and denoising performance.

Contribution

It proposes a cocoercive conservative denoiser trained via Helmholtz decomposition, enabling convergence guarantees and better performance in Poisson inverse problems.

Findings

Outperforms related methods in visual quality.

Ensures convergence of PnP methods to stationary points.

Demonstrates effectiveness on imaging tasks.

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. 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.