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A Relaxed Gradient Step Denoiser for Splitting Methods in Poisson Inverse Problems

This paper introduces \gsdsplit, a novel splitting method for Poisson inverse problems that replaces the standard denoiser in Plug&Play frameworks with a relaxed Gradient Step Denoiser based on a learned convex potential, thereby ensuring convergence guarantees while achieving stable, low-sensitivity reconstructions across various noise levels and blur operators.

Original authors: Alessandro Benfenati

Published 2026-07-30

📖 4 min read🧠 Deep dive

CC BY 4.0

Original authors: Alessandro Benfenati

Original paper licensed under CC BY 4.0 (http://creativecommons.org/licenses/by/4.0/). ✨ This is an AI-generated explanation of the paper below. It is not written or endorsed by the authors. For technical accuracy, refer to the original paper. Read full disclaimer

Imagine you are trying to listen to a friend whispering a secret across a crowded, noisy room. The friend's voice is the "clean image," but the crowd's chatter is "noise," and the distance between you blurs the words. In the world of science, this is called an "inverse problem": we have the messy, blurry result (the whisper we heard) and we want to figure out what the original message was. This happens everywhere, from telescopes trying to see distant stars through atmospheric haze to doctors looking at blurry scans inside a human body.

Usually, scientists have two main tools to fix this mess. First, they use math to describe how the noise behaves. Sometimes the noise is like static on an old TV (Gaussian noise), which is easy to handle. But in many real-world situations, like counting individual photons of light in a microscope, the noise behaves differently—it's "Poisson noise," which gets wilder when the signal is weak. Second, they use a "denoiser," which is like a smart filter that knows what a clean picture should look like. In modern times, these filters are often trained by artificial intelligence to spot patterns and remove the static. The challenge is that when you mix the tricky Poisson noise with these powerful AI filters, the math can get shaky, and the solution might not actually converge to the right answer.

This paper introduces a new method called GSDSplit+ to solve exactly that problem. Think of the researchers as mechanics trying to tune a high-performance engine (the image restoration algorithm) that keeps stalling when using a specific type of fuel (Poisson noise). They take an existing, reliable engine design called PnPSplit+ and swap out its standard "denoising" part. Instead of using a generic black-box AI filter, they install a custom-built "Gradient Step Denoiser."

Here is the clever part: this new denoiser isn't just guessing; it's built on a mathematical "potential" (imagine a smooth, bowl-shaped landscape) that the AI has learned. The denoiser works by taking a step down this landscape to find the cleanest image. The authors trained this AI to ensure the landscape is perfectly smooth and shaped just right so that the steps it takes are guaranteed to lead to a solution, not get stuck in a loop. They call this property "firm nonexpansiveness," which is a fancy way of saying the AI is very well-behaved and won't make wild, unpredictable jumps.

The researchers found that while their new method, GSDSplit+, is slightly slower to compute than the older version (because it has to calculate these smooth steps carefully), it is incredibly robust. In their tests, they messed with the "tuning knobs" of the algorithm. When they accidentally set the knobs to the wrong values, the old method crashed or produced terrible images. The new GSDSplit+ method, however, kept working well, producing clear, stable pictures even when the settings weren't perfect. It's like a car that might be a bit heavier, but it won't skid on ice when you hit a patch of black ice unexpectedly.

The paper confirms through numerical experiments that this new approach works for different types of blurs (like motion blur or out-of-focus lenses) and various levels of noise. While the older method can produce slightly sharper images if you tune it perfectly, the new method is much more forgiving if you don't know the perfect settings. The authors suggest that this trade-off—slightly more computer time for much greater stability—makes GSDSplit+ a very promising tool for fixing images in fields like astronomy and medical imaging, where getting the right answer is more important than getting it in the split second.

Technical Summary: A Relaxed Gradient Step Denoiser for Splitting Methods in Poisson Inverse Problems

Problem Statement
Image restoration in scientific fields such as microscopy, astronomy, and medical imaging often involves recovering a clean image $x^\star$ from observed data $g$ corrupted by linear blur $H$ , background $b$ , and Poisson noise. The standard approach formulates this as a Maximum A Posteriori (MAP) estimation problem, minimizing a Kullback–Leibler (KL) data fidelity term plus a regularization term. While Plug-and-Play (PnP) methods have achieved significant success in Gaussian noise settings by replacing proximal operators with learned denoisers, their application to Poisson inverse problems is challenging. The non-quadratic nature of the KL fidelity function complicates convergence analysis, particularly when integrating with splitting methods like PnPSplit+ which rely on specific properties of the denoiser, such as Firm Non-Expansiveness (FNE).

Methodology
The paper proposes GSDSplit+, a structured variant of the PnPSplit+ algorithm designed for Poisson data. The core innovation is the replacement of the generic PnP denoising block with a Relaxed Gradient Step Denoiser (GSD).

Algorithmic Framework: The method utilizes an Alternating Direction Method of Multipliers (ADMM) scheme. Unlike standard PnP where the denoiser is a black box, GSDSplit+ defines the denoiser as:
$D_{\vartheta, \alpha}(x) = x - \alpha \nabla \psi_\vartheta(x)$
Here, $\psi_\vartheta$ is a learned convex potential parameterized by an Input Convex Neural Network (ICNN), and $\alpha$ is a relaxation parameter. This structure ensures the denoising residual is an explicit gradient field, bridging splitting schemes with fixed-point formulations like RED-PRO.
Convergence Conditions: The convergence of GSDSplit+ relies on the denoiser being Firmly Non-Expansive (FNE). The authors establish that if $\psi_\vartheta$ is convex and $L$ -smooth, the operator $D_{\vartheta, \alpha}$ is FNE provided that $\alpha L \leq 1$ . This allows the use of networks with smoothness constants $L > 1$ by adjusting the relaxation parameter $\alpha$ .
Training Strategy: The ICNN is trained using a residual Gaussian denoising loss. To satisfy the theoretical requirements, the training includes:
- A smoothed activation function (a differentiable approximation of LeakyReLU) to ensure the network remains convex and differentiable.
- A smoothness-oriented fine-tuning phase using a spectral penalty on the Hessian of the potential. This penalty encourages the Lipschitz constant $L$ of the gradient to be close to 1, facilitating the selection of a valid $\alpha$ .

Key Contributions

Introduction of GSDSplit+: Embedding an ICNN-based Gradient Step Denoiser into the PnPSplit+ framework for Poisson image restoration.
Theoretical Convergence: Deriving a convergence result that links the FNE property of the denoiser to the convexity and smoothness of the learned potential and the relaxation parameter $\alpha$ .
Tailored Architecture and Training: Implementing a specific ICNN architecture with a custom smooth activation and a two-stage training strategy (standard denoising followed by Hessian spectral penalty) to promote the necessary smoothness conditions.
Empirical Validation: Providing evidence that the trained denoiser operates in an FNE-compatible regime on validation data when $\alpha$ is chosen appropriately (e.g., $\alpha = 0.9$ ).

Results
Numerical experiments were conducted on the DIV2K and BSD500 datasets with Gaussian blur and Poisson noise.

Reconstruction Quality: When both GSDSplit+ and the baseline PnPSplit+ (using a standard CNN denoiser) are tuned with their optimal ADMM parameters ( $\gamma$ ), PnPSplit+ achieves slightly higher Peak Signal-to-Noise Ratio (PSNR) and SSIM.
Robustness to Hyperparameters: The primary advantage of GSDSplit+ is its stability. It is substantially less sensitive to the choice of the ADMM parameter $\gamma$ . While PnPSplit+ performance degrades significantly (approx. 13.29 dB PSNR drop) when $\gamma$ is overestimated, GSDSplit+ shows minimal degradation (approx. 1.50 dB drop).
Generalization: The method demonstrates robustness across different blur operators (motion blur, out-of-focus) and varying Poisson noise levels.
Computational Cost: The robustness comes at a computational price. GSDSplit+ requires calculating the gradient of the ICNN at every iteration, making it slower than PnPSplit+, which uses a standard CNN forward pass.

Significance and Claims
The paper claims that GSDSplit+ offers a theoretically grounded alternative for Poisson inverse problems by explicitly connecting the denoiser's structure to convergence guarantees. While it does not outperform the optimally tuned PnPSplit+ in terms of peak reconstruction quality, it provides stable reconstructions even when the ADMM parameter is suboptimal or overestimated. This reduced sensitivity to hyperparameter tuning is presented as a significant practical benefit. The authors modestly note that the method operates effectively even outside the strictly empirically supported FNE range, though convergence is not guaranteed in those cases. Future work is suggested to extend this strategy to RED-PRO formulations and investigate adaptive strategies for the relaxation parameter $\alpha$ .

Technical Summary: A Relaxed Gradient Step Denoiser for Splitting Methods in Poisson Inverse Problems

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