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.

The development of sensor technology has resulted in the accumulation of extensive spatiotemporal traffic information, which holds great potential for predicting traffic patterns and improving traffic management strategies. Nevertheless, dealing with missing data poses a significant challenge for the intelligent traffic system (ITS). To address this issue, this study employs a nonconvex smoothly clipped absolute deviation (SCAD) penalty customized for tensors to surrogate tensor rank and incorporates the deep plug-and-play (PnP) prior into the low-rank tensor completion (LRTC) model. An efficient iterative framework is formulated to integrate these penalties into the alternating direction method of multipliers (ADMM) method. Moreover, two imputation methods, namely LRTC-SCAD and LRTC-SCAD-DeepPnP, are developed, affirming that the LRTC-SCAD method ensures convergence to the global optimum. We conduct simulated experiments using real-world traffic datasets, and our proposed methods outperform state-of-the-art imputation methods. For instance, on the Portland dataset, LRTC-SCAD achieves a noteworthy 9.86% improvement in mean absolute percentage error (MAPE) compared to the cutting-edge LRTC method while consuming only 28.26% of its total running time. Similarly, on the PeMS dataset, LRTC-SCAD-DeepPnP achieves an average 11.59% enhancement in MAPE, with visually compelling improvements in imputation results, further validating its efficacy in maintaining local consistency. The code is available athttps://github.com/peterchen96/LRTC_DeepPnP.

Deep denoisers have shown excellent performance in solving inverse problems in signal and image processing. In order to guarantee the convergence, the denoiser needs to satisfy some Lipschitz conditions like non-expansiveness. However, enforcing such constraints inevitably compromises recovery performance. This paper introduces a novel training strategy that enforces a weaker constraint on the deep denoiser called pseudo-contractiveness. By studying the spectrum of the Jacobian matrix, relationships between different denoiser assumptions are revealed. Effective algorithms based on gradient descent and Ishikawa process are derived, and further assumptions of strict pseudo-contractiveness yield efficient algorithms using half-quadratic splitting and forward-backward splitting. The proposed algorithms theoretically converge strongly to a fixed point. A training strategy based on holomorphic transformation and functional calculi is proposed to enforce the pseudo-contractive denoiser assumption. Extensive experiments demonstrate superior performance of the pseudo-contractive denoiser compared to related denoisers. The proposed methods are competitive in terms of visual effects and quantitative values.