[Paper Review] Gradient Step Denoiser for convergent Plug-and-Play

Samuel Hurault, Arthur Leclaire+1 more|arXiv (Cornell University)|Oct 7, 2021

Sparse and Compressive Sensing TechniquesEngineering53 references19 citations

TL;DR

This paper proposes a novel Plug-and-Play (PnP) framework, Gradient Step Denoiser (GS-PnP), that replaces the traditional denoiser in PnP-HQS with a gradient descent step on a deep neural network parameterized functional. By ensuring the denoiser satisfies the proximal mapping condition via a learnable functional, the method guarantees convergence to a stationary point of an explicit global functional, even for non-convex data-fidelity terms, while achieving state-of-the-art performance in deblurring, super-resolution, and inpainting.

ABSTRACT

Plug-and-Play methods constitute a class of iterative algorithms for imaging problems where regularization is performed by an off-the-shelf denoiser. Although Plug-and-Play methods can lead to tremendous visual performance for various image problems, the few existing convergence guarantees are based on unrealistic (or suboptimal) hypotheses on the denoiser, or limited to strongly convex data terms. In this work, we propose a new type of Plug-and-Play methods, based on half-quadratic splitting, for which the denoiser is realized as a gradient descent step on a functional parameterized by a deep neural network. Exploiting convergence results for proximal gradient descent algorithms in the non-convex setting, we show that the proposed Plug-and-Play algorithm is a convergent iterative scheme that targets stationary points of an explicit global functional. Besides, experiments show that it is possible to learn such a deep denoiser while not compromising the performance in comparison to other state-of-the-art deep denoisers used in Plug-and-Play schemes. We apply our proximal gradient algorithm to various ill-posed inverse problems, e.g. deblurring, super-resolution and inpainting. For all these applications, numerical results empirically confirm the convergence results. Experiments also show that this new algorithm reaches state-of-the-art performance, both quantitatively and qualitatively.

Motivation & Objective

• To address the lack of convergence guarantees in Plug-and-Play (PnP) methods when using generic deep denoisers.
• To develop a PnP framework that maintains high image restoration performance while ensuring theoretical convergence.
• To enable convergence for non-convex data-fidelity terms, such as in ill-posed inverse problems like deblurring and super-resolution.
• To provide a method where the denoiser is explicitly tied to a functional, allowing interpretability and numerical control.
• To demonstrate that a learnable denoiser can achieve state-of-the-art performance without sacrificing convergence guarantees.

Proposed method

• The method replaces the proximal operator of the regularizer in PnP-HQS with a gradient descent step on a functional parameterized by a deep neural network.
• The denoiser is constructed such that its update rule corresponds to the gradient of a learnable functional, ensuring it satisfies the condition ∇g(x) = x − D(x).
• The algorithm is formulated as a half-quadratic splitting scheme where the data-fidelity term is minimized via a proximal step with large step size τ, followed by a denoising step via gradient descent on the learned functional.
• A backtracking line search is used to adaptively adjust the step size τ, improving stability and convergence.
• The denoiser is trained end-to-end to minimize the overall restoration objective, while preserving convergence guarantees through the functional structure.
• The method is applied to various inverse problems, including deblurring, super-resolution, and inpainting, using the same framework across tasks.

Experimental results

Research questions

• RQ1 Can a PnP method based on gradient descent steps on a learnable functional achieve convergence guarantees for non-convex data-fidelity terms?
• RQ2 Can such a method maintain state-of-the-art image restoration performance while ensuring convergence?
• RQ3 Does the proposed denoiser, derived from a functional, satisfy the necessary conditions for convergence in the non-convex setting?
• RQ4 How robust is the algorithm to initialization and hyperparameter choices, particularly σ and λ?
• RQ5 Can the method outperform existing PnP schemes like DPIR and RED in both convergence and restoration quality?

Key findings

• The proposed GS-PnP algorithm converges to a stationary point of an explicit global functional, even when the data-fidelity term is non-convex, under mild assumptions on the denoiser.
• The residual ||x_{k+1} − x_k||² reaches ∼10⁻¹³ in GS-PnP, indicating strong convergence, while DPIR shows no convergence and residual remains around 10⁻⁴.
• For deblurring on CBSD10 with 10 kernels, GS-PnP achieves a PSNR of 32.45 dB with σ/ν = 2, outperforming RED and other baselines.
• The method is robust to noisy initializations: even with σ_init = 60/255, the PSNR remains stable and visually consistent.
• In inpainting and super-resolution tasks, GS-PnP achieves state-of-the-art PSNR and SSIM scores, demonstrating generalization across diverse inverse problems.
• The ablation study confirms that the backtracking procedure stabilizes convergence and enables large initial τ values, improving robustness.