S³-PnP: Self-Supervised Score Priors for Plug-and-Play Denoising in Scientific Imaging

Scientific imaging has long faced an awkward reality: for modalities like hyperspectral imaging (HSI) and magnetic resonance imaging (MRI), clean training labels are nearly impossible to obtain at scale. Traditional supervised learning approaches struggle here, and plug-and-play ADMM (PnP-ADMM) schemes that rely on external pretrained diffusion priors—while sidestepping the label problem—introduce a new distribution mismatch: the learned prior may deviate from the target scientific image distribution, and may also fail to align with the denoiser input distribution induced by ADMM dual feedback.

Where the Problem Lies: Dual Distribution Mismatch

The classic PnP-ADMM approach takes a diffusion model pretrained on large-scale natural images as a denoising prior and plugs it into ADMM iterations for image restoration. There are two levels of problems.

The first is target distribution mismatch. The statistical properties of natural images differ significantly from hyperspectral and MRI images. Forcing a natural image prior onto scientific imaging means the prior itself is inaccurate.

The second is more subtle—ADMM intermediate-state distribution mismatch. The dual variable in ADMM feeds back into the denoiser input, so the distribution the denoiser actually faces is not the clean image distribution, but an intermediate-state distribution distorted by dual feedback. A pretrained prior has never seen such inputs, so denoising performance naturally degrades.

These two mismatches together explain why directly transferring pretrained diffusion priors into scientific imaging PnP frameworks often underperforms expectations.

The iterative structure of PnP-ADMM deserves elaboration. When ADMM (alternating direction method of multipliers) solves problems of the form $\min_x f(x)+g(x)$, it introduces an auxiliary variable $z$ and a dual variable $u$, and alternates three steps: the $x$ update (data fidelity term), the $z$ update (regularization term, i.e., the denoising step), and the $u$ update (dual ascent). In the PnP framework, the $z$ update is replaced by a single denoiser call, i.e., $z^{k+1}=D_\sigma(x^{k+1}+u^k)$. The key lies in the term $x^{k+1}+u^k$: the dual variable $u$ accumulates constraint violations from previous iterations and continuously feeds them back to the denoiser input, so the denoiser does not see the original noisy image, but an intermediate state superimposed with dual feedback. This is precisely the mechanism behind "intermediate-state distribution mismatch"—there is a systematic shift between the input distribution the denoiser actually faces and the clean image distribution it saw during training, and this shift evolves dynamically with iteration rounds.

S³-PnP's Approach: Learning Scores from Observed Intermediate States

The S³-PnP framework proposed in this work has as its core operation learning a score prior directly on the observation-driven intermediate states of ADMM, requiring neither clean training targets nor external pretrained weights.

Specifically, the learned score function is used to correct the input to the analytical prior update, combining the observation-driven correction term with analytical regularization. This means the prior is not "borrowed" from outside, but "grown" from the current observation and the intermediate states that naturally emerge during ADMM iterations. Since the intermediate states themselves contain both the distribution information of the target modality and the influence of dual feedback, the dual mismatch is alleviated at its source.

From a methodological perspective, this design couples self-supervised learning more tightly with the iterative structure of classical optimization algorithms—rather than learning a prior first and then plugging it into optimization, it learns the prior synchronously during optimization.

Score prior is the core concept for understanding this paper's method. The score function is defined as the gradient of the log probability density, $\nabla_x \log p(x)$, which points in the direction of steepest increase in probability density within the data distribution, and can therefore serve as a force field for "pushing noisy samples back toward high-probability regions" during denoising. Diffusion models realize generation and denoising precisely by training a network to approximate the score function $\nabla_x \log p_\sigma(x)$ at various noise levels. There is a Tweedie equation connection between classical denoisers and scores: the optimal denoiser can be written as $D(x)=\frac{1}{\sigma^2}(x+\sigma^2\nabla_x\log p_\sigma(x))$, which explains why learning a score is equivalent to learning a denoising prior. What makes S³-PnP unique is that it does not rely on large-scale clean images to train the score network, but directly estimates the score from the observation-driven intermediate states produced by ADMM iterations, thereby making the prior "fit" the data distribution along the actual optimization trajectory.

Theoretical Guarantees and Experimental Validation

Under assumptions of ideal scores and quadratic data fidelity, the authors analyze the convergence and reconstruction error of the corrected ADMM algorithm. This provides some theoretical support for the method's reliability, although the ideal score assumption can only be approximately satisfied in practice.

Experiments cover three tasks: reference-guided hyperspectral restoration, single-image hyperspectral denoising, and MRI denoising. According to the paper, S³-PnP achieves consistent improvements over competing methods on these tasks. Multi-task validation also shows that the method is not limited to a single modality.

The ideal score assumption is a common simplification in this type of theoretical analysis. It assumes that in each iteration, the learned score function exactly matches the true score of the real data distribution, i.e., $s_\theta(x)=\nabla_x\log p(x)$ holds without error. Based on this assumption, the authors can derive convergence conditions and reconstruction error upper bounds for the corrected ADMM. In practice, however, the score network is estimated from finite data, so approximation error is inevitable, and the error varies with noise level and iteration position. Therefore, the theoretical guarantee provides qualitative support that "the algorithm is feasible under the optimal denoiser," rather than a strict practical performance bound. Evaluating the practical value of such methods also requires considering experimental sensitivity to score estimation error and stability under different ADMM penalty parameters.

Significance for Scientific Imaging Denoising

What makes this work noteworthy is that it directly addresses a frequently overlooked issue in the PnP framework: the interaction between the denoiser input distribution and dual feedback. Many previous works focus only on the transferability of the prior itself, whereas S³-PnP incorporates ADMM iteration dynamics into the prior learning process, bringing the approach closer to the actual optimization trajectory.

For fields with scarce annotations such as hyperspectral and MRI, a self-supervised scheme requiring neither clean labels nor external pretrained weights lowers the deployment barrier. Of course, there is still a gap from the paper to actual scientific imaging pipelines. The degree to which the ideal score assumption can be relaxed, computational overhead, and robustness to different ADMM parameter settings are all directions that need further validation.

Hyperspectral imaging (HSI) collects data across hundreds of contiguous narrow bands. A single data cube is far larger than an ordinary RGB image, and adjacent bands are highly correlated. Its physical imaging process limits the acquisition of clean labels—there is no "noise-free hyperspectral reference" in real scenes, and supervision signals can usually only be constructed through synthetic noise or physical simulation, which is precisely the fundamental reason supervised learning routes are constrained. Magnetic resonance imaging (MRI) is similar: fully sampled acquisition of clean images takes too long, and clinical practice generally uses undersampling for acceleration, causing training data to naturally contain reconstruction artifacts; the complex-valued nature of k-space data and its noise distribution also differ from natural images. The commonality of these two modalities is that the target modality's data distribution is highly specialized and clean samples are difficult to collect at scale, so self-supervised prior learning methods that require no external labels have direct deployment value.

One-line Summary

S³-PnP self-supervises score priors from ADMM observation intermediate states, resolving dual distribution mismatch in scientific imaging.

Paragraph Summary

Clean labels are scarce in scientific imaging, making traditional supervised learning hard to apply, while PnP-ADMM schemes relying on external pretrained diffusion priors face a dual distribution mismatch: the pretrained prior does not match the target scientific image distribution, and ADMM dual feedback causes the denoiser input distribution to deviate from the clean image distribution. The S³-PnP framework proposes learning a score prior directly on ADMM observation-driven intermediate states, requiring neither clean training labels nor external pretrained weights. The learned score is used to correct the input to the analytical prior update, combining observation-driven correction with analytical regularization. Under assumptions of ideal scores and quadratic data fidelity, the authors analyze the convergence and reconstruction error of the corrected ADMM, and report consistent improvements on three tasks: reference-guided hyperspectral restoration, single-image hyperspectral denoising, and MRI denoising, indicating that this self-supervised approach has practical significance in annotation-scarce scientific imaging scenarios.