Gradient-Step Denoiser

• The gradient-step denoiser is defined as x - ∇g₍σ₎(x), where g₍σ₎ is a learned potential, transforming denoising into a gradient descent step on a clear objective.
• The framework integrates into Plug-and-Play optimization, enabling convergence theory and practical applications in CT, tomography, and score-based point-cloud denoising.
• Architectural refinements using CNNs and smooth activations balance performance and convergence, ensuring the denoiser adheres to a controlled gradient update mechanism.

Searching arXiv for recent and foundational papers on gradient-step denoisers and closely related plug-and-play formulations. arXiv search query: "Gradient Step Denoiser plug-and-play"

A gradient-step denoiser is a denoising operator defined or interpreted through an explicit gradient update, most prominently in Plug-and-Play (PnP) inverse problems where the denoiser is constrained to have the form $D_\sigma(x)=x-\nabla g_\sigma(x)$ for a learned smooth potential $g_\sigma$. In that formulation, denoising is no longer a purely black-box replacement for a proximal map: it becomes the gradient part of an explicit objective, which enables convergence analysis for PnP schemes and links denoising to nonconvex regularization (Hurault et al., 2021). Closely related uses of the term also appear in gradient-domain CT restoration, score-based point-cloud denoising, diffusive denoising for tomography, and gradient-guided correction in broader inverse and data-cleaning pipelines, so the phrase denotes a family of closely connected constructions rather than a single universally fixed algorithm (Cascarano et al., 2021).

1. Conceptual scope and principal definitions

In the most specific and mathematically developed sense, the gradient-step denoiser is the operator

$D_\sigma(x)=x-\nabla g_\sigma(x),$

where $g_\sigma:\mathbb{R}^n\to\mathbb{R}$ is a learned scalar potential. This makes the denoiser a conservative vector field and allows the denoising action to be interpreted as one explicit gradient-descent step on the regularizer (Hurault et al., 2021).

A standard parametrization sets

$g_\sigma(x)=\frac12\|x-N_\sigma(x)\|_2^2,$

with $N_\sigma$ a smooth neural denoiser. Differentiation yields

$$D_\sigma(x)=N_\sigma(x)+J_{N_\sigma}(x)^\top\bigl(x-N_\sigma(x)\bigr),$$

so the deployed denoiser contains both the feed-forward network output and a Jacobian-transpose correction term (Herfeld et al., 11 Sep 2025).

A related but distinct usage appears in gradient-domain restoration. There, the unknown image $x\in\mathbb{R}^n$ is estimated from degraded data $b\simeq A x$ by minimizing

$\min_x \frac12\|A x-b\|_2^2+\lambda R(\nabla x),$

where $g_\sigma$0, and denoising acts directly on the gradient field rather than on the image itself (Cascarano et al., 2021).

A further related interpretation arises in score-based denoising of point clouds. For a noisy cloud sampled from a distribution $g_\sigma$1, the gradient field is

$g_\sigma$2

and denoising is implemented by iterative gradient ascent in point coordinates, pushing samples toward the underlying clean surface (Hu et al., 2022).

These formulations share one structural idea: denoising is performed by an explicit update along a learned or handcrafted gradient object, rather than by an unconstrained black-box image-to-image map. This suggests that “gradient-step denoiser” is best treated as a unifying operator class whose exact semantics depend on whether the gradient acts on an image prior, an image gradient field, a point-cloud log-density, or a task-specific loss.

2. Learned potentials and denoiser construction

The explicit-potential formulation was developed to preserve modern denoising performance while restoring variational interpretability. With $g_\sigma$3, the denoiser remains expressive because $g_\sigma$4 can be a conventional residual-type CNN, such as a DRUNet-style U-Net (Hurault et al., 2021).

Training is posed with additive Gaussian perturbations. One objective is

$g_\sigma$5

which directly aligns the regularizer gradient with the noise realization (Hurault et al., 2021). A later presentation gives the same relation in Monte-Carlo form and emphasizes that the network is trained so that the deployed operator exactly realizes $g_\sigma$6, rather than only approximating a denoiser implicitly (Herfeld et al., 11 Sep 2025).

Several architectural refinements are motivated by regularity requirements. In one nonconvex proximal formulation, $g_\sigma$7 is parameterized through a DRUNet-style U-net made $g_\sigma$8 by Softplus activations, then fine-tuned with a spectral-norm penalty on $g_\sigma$9 to enforce $D_\sigma(x)=x-\nabla g_\sigma(x),$0 (Hurault et al., 2022). In dental cone-beam CT reconstruction, a DRUNet base architecture with four cascaded U-Net blocks and Softplus activations is used so as to guarantee continuous differentiability, and the model is initialized from earlier gradient-step-denoiser weights (Tatachak et al., 27 May 2026).

A separate gradient-domain CT variant replaces image-domain denoising by denoising of noisy gradient patches $D_\sigma(x)=x-\nabla g_\sigma(x),$1, with $D_\sigma(x)=x-\nabla g_\sigma(x),$2, through a CNN trained to map noisy gradients back to clean gradients. In that setting the learned prior acts on vector-valued fields, and the denoiser is inserted into an HQS/PnP loop as a surrogate for a gradient-domain proximal map (Cascarano et al., 2021).

The explicit construction matters because it distinguishes gradient-step denoisers from arbitrary denoisers such as BM3D or DnCNN. In the former case the regularizer exists by design; in the latter case it is typically implicit, and neither symmetry nor conservative structure is guaranteed (Herfeld et al., 11 Sep 2025).

3. Plug-and-Play optimization and convergence theory

The main appeal of the explicit gradient-step denoiser is that it converts a PnP iteration into a standard first-order algorithm for an explicit energy. For a data-fidelity term $D_\sigma(x)=x-\nabla g_\sigma(x),$3 and regularizer weight $D_\sigma(x)=x-\nabla g_\sigma(x),$4, the target functional is

$D_\sigma(x)=x-\nabla g_\sigma(x),$5

The corresponding PnP update can be written as

$D_\sigma(x)=x-\nabla g_\sigma(x),$6

or equivalently

$D_\sigma(x)=x-\nabla g_\sigma(x),$7

which is precisely proximal-gradient descent on $D_\sigma(x)=x-\nabla g_\sigma(x),$8 (Hurault et al., 2021).

Under the assumptions that $D_\sigma(x)=x-\nabla g_\sigma(x),$9 is proper, lower semicontinuous and convex, $g_\sigma:\mathbb{R}^n\to\mathbb{R}$0 is $g_\sigma:\mathbb{R}^n\to\mathbb{R}$1 with $g_\sigma:\mathbb{R}^n\to\mathbb{R}$2-Lipschitz gradient, $g_\sigma:\mathbb{R}^n\to\mathbb{R}$3 is bounded below, and $g_\sigma:\mathbb{R}^n\to\mathbb{R}$4, the iterates satisfy three central properties: $g_\sigma:\mathbb{R}^n\to\mathbb{R}$5 is nonincreasing and converges, $g_\sigma:\mathbb{R}^n\to\mathbb{R}$6, and every cluster point is stationary in the sense that $g_\sigma:\mathbb{R}^n\to\mathbb{R}$7 (Hurault et al., 2021). With an additional Kurdyka–Łojasiewicz assumption, full convergence and finite-length behavior can also be established (Herfeld et al., 11 Sep 2025).

A stronger structural result is the proximal characterization. If $g_\sigma:\mathbb{R}^n\to\mathbb{R}$8 and $g_\sigma:\mathbb{R}^n\to\mathbb{R}$9 is $g_\sigma(x)=\frac12\|x-N_\sigma(x)\|_2^2,$0-Lipschitz with $g_\sigma(x)=\frac12\|x-N_\sigma(x)\|_2^2,$1, then

$g_\sigma(x)=\frac12\|x-N_\sigma(x)\|_2^2,$2

is $g_\sigma(x)=\frac12\|x-N_\sigma(x)\|_2^2,$3-strongly convex, $g_\sigma(x)=\frac12\|x-N_\sigma(x)\|_2^2,$4 is injective, and there exists a possibly nonconvex scalar function $g_\sigma(x)=\frac12\|x-N_\sigma(x)\|_2^2,$5 such that

$g_\sigma(x)=\frac12\|x-N_\sigma(x)\|_2^2,$6

This reformulation enables convergence results not only for PnP-PGD but also for PnP-ADMM and Douglas–Rachford-type schemes, again with stationarity guarantees for the explicit composite objective (Hurault et al., 2022).

The distinction between fixed-point convergence and objective convergence is important. In gradient-domain CT PnP, a general fixed-point theorem is established under “bounded denoisers” and slowly increasing $g_\sigma(x)=\frac12\|x-N_\sigma(x)\|_2^2,$7, without requiring an explicit objective; the limit satisfies

$g_\sigma(x)=\frac12\|x-N_\sigma(x)\|_2^2,$8

This is a weaker but still operationally useful guarantee than stationarity of an explicit scalar energy (Cascarano et al., 2021). By contrast, the explicit-potential GS-PnP line of work targets stationary points of a known global functional (Hurault et al., 2021).

4. Tomographic and CT instantiations

Medical imaging provides several concrete instantiations of gradient-step denoisers. In CT image enhancement, the gradient-domain PnP formulation introduces an auxiliary field $g_\sigma(x)=\frac12\|x-N_\sigma(x)\|_2^2,$9 and solves

$N_\sigma$0

The denoising step updates $N_\sigma$1 through either an internal prior or an external learned prior, while the image step solves

$N_\sigma$2

When $N_\sigma$3 is a convolution, the $N_\sigma$4-update is solvable by FFT (Cascarano et al., 2021).

The internal prior is Total Variation with

$N_\sigma$5

whose proximal map is vector soft-thresholding. The external prior is a deep CNN trained directly on the gradient domain rather than on images, which the authors contrast with state-of-the-art image-domain approaches (Cascarano et al., 2021).

Reported CT results include up to $N_\sigma$6–$N_\sigma$7 dB PSNR and $N_\sigma$8–$N_\sigma$9 SSIM over image-domain PnP using IRCNN/ICNN, superior edge sharpness on low-contrast circles and thin vessels, robustness at $$D_\sigma(x)=N_\sigma(x)+J_{N_\sigma}(x)^\top\bigl(x-N_\sigma(x)\bigr),$$0–$$D_\sigma(x)=N_\sigma(x)+J_{N_\sigma}(x)^\top\bigl(x-N_\sigma(x)\bigr),$$1 ADU, fixed-point attainment typically within $$D_\sigma(x)=N_\sigma(x)+J_{N_\sigma}(x)^\top\bigl(x-N_\sigma(x)\bigr),$$2–$$D_\sigma(x)=N_\sigma(x)+J_{N_\sigma}(x)^\top\bigl(x-N_\sigma(x)\bigr),$$3 iterations, and clearly better segmentation Jaccard index on epidural bleeding when pre-denoised by gradient PnP (Cascarano et al., 2021).

In dental cone-beam CT reconstruction, the explicit GS framework is used directly in the image domain. With

$$D_\sigma(x)=N_\sigma(x)+J_{N_\sigma}(x)^\top\bigl(x-N_\sigma(x)\bigr),$$4

the reconstruction step is

$$D_\sigma(x)=N_\sigma(x)+J_{N_\sigma}(x)^\top\bigl(x-N_\sigma(x)\bigr),$$5

The deployed denoiser is

$$D_\sigma(x)=N_\sigma(x)+J_{N_\sigma}(x)^\top\bigl(x-N_\sigma(x)\bigr),$$6

Using simulated XCAT phantoms reconstructed by FBP, denoising alone improves PSNR from $$D_\sigma(x)=N_\sigma(x)+J_{N_\sigma}(x)^\top\bigl(x-N_\sigma(x)\bigr),$$7 dB to $$D_\sigma(x)=N_\sigma(x)+J_{N_\sigma}(x)^\top\bigl(x-N_\sigma(x)\bigr),$$8 dB on a test set of $$D_\sigma(x)=N_\sigma(x)+J_{N_\sigma}(x)^\top\bigl(x-N_\sigma(x)\bigr),$$9 simulated images, and full GS-PnP reconstruction on $x\in\mathbb{R}^n$0 simulated jaw acquisitions improves PSNR from $x\in\mathbb{R}^n$1 dB for conjugate gradient to $x\in\mathbb{R}^n$2 dB at $x\in\mathbb{R}^n$3 (Tatachak et al., 27 May 2026).

A related tomography solver, DDGM, combines repeated gradient-descent steps on $x\in\mathbb{R}^n$4 with injected Gaussian noise and a CNN denoiser: $x\in\mathbb{R}^n$5 Both $x\in\mathbb{R}^n$6 and $x\in\mathbb{R}^n$7 decay exponentially, and high accuracy is reported with as few as $x\in\mathbb{R}^n$8 denoising steps. On the Fly Hemibrain validation set, DDGM attains MSE $x\in\mathbb{R}^n$9 and SSIM $b\simeq A x$0, compared with DDRM at $b\simeq A x$1 and $b\simeq A x$2, and DPS at $b\simeq A x$3 and $b\simeq A x$4 (Luther et al., 2023).

These CT and tomography variants differ in whether denoising acts on $b\simeq A x$5, on $b\simeq A x$6 through an explicit potential, or through a denoiser coupled to stochastic gradient dynamics. What they share is a split between a data-consistency move and a denoising move, with the latter encoded as a principled gradient-based operator.

5. Gradient fields and iterative denoising of point clouds

In point-cloud denoising, the gradient-step idea is reformulated in terms of score estimation on geometric data. A noisy point cloud $b\simeq A x$7 is treated as samples from a noisy distribution $b\simeq A x$8, and the denoising direction is the score

$b\simeq A x$9

Denoising then proceeds by gradient ascent,

$\min_x \frac12\|A x-b\|_2^2+\lambda R(\nabla x),$0

so that each point moves toward the underlying clean surface (Hu et al., 2022).

For dynamic point clouds, temporal correspondence is incorporated by dividing each frame into overlapping patches, aligning patches to adjacent frames via rigid-motion iterations driven by the neighboring frame’s gradient field, and averaging scores across time. With $\min_x \frac12\|A x-b\|_2^2+\lambda R(\nabla x),$1, the temporal-averaged score is

$\min_x \frac12\|A x-b\|_2^2+\lambda R(\nabla x),$2

and denoising updates each point by $\min_x \frac12\|A x-b\|_2^2+\lambda R(\nabla x),$3 (Hu et al., 2022).

A second development addresses instability in iterative score ascent through momentum. The heavy-ball update is

$\min_x \frac12\|A x-b\|_2^2+\lambda R(\nabla x),$4

or, in the leaky-averaging form used in the paper,

$\min_x \frac12\|A x-b\|_2^2+\lambda R(\nabla x),$5

On Gaussian noise with $\min_x \frac12\|A x-b\|_2^2+\lambda R(\nabla x),$6K points, the momentum method improves CD/P2M from $\min_x \frac12\|A x-b\|_2^2+\lambda R(\nabla x),$7 to $\min_x \frac12\|A x-b\|_2^2+\lambda R(\nabla x),$8, and with $\min_x \frac12\|A x-b\|_2^2+\lambda R(\nabla x),$9K points from $g_\sigma$00 to $g_\sigma$01; inference time decreases from $g_\sigma$02 min to $g_\sigma$03 min for $g_\sigma$04K-point clouds and from $g_\sigma$05 min to $g_\sigma$06 min for $g_\sigma$07K-point clouds (Zhao et al., 2022).

A third line focuses on uniformity. Existing gradient-based denoisers are reported to lack uniformity despite promising quantitative denoising scores, so GPCD++ inserts an ultra-lightweight refinement network, UniNet, after each backbone score step. UniNet approximates the gradient of a joint log-likelihood term $g_\sigma$08 capturing local point interactions and outputs per-point refinements $g_\sigma$09, yielding

$g_\sigma$10

Its parameter count is on the order of $g_\sigma$11 M, corresponding to an overhead of $g_\sigma$12–$g_\sigma$13 over the backbone (Xu et al., 2022).

Uniformity is quantified by a metric

$g_\sigma$14

where smaller values indicate more uniform local point distributions. On the PU dataset with $g_\sigma$15K points and $g_\sigma$16 noise, PSR gives CD/P2M $g_\sigma$17 and Uni $g_\sigma$18, while GPCD++(PSR) gives $g_\sigma$19 and $g_\sigma$20, which the paper summarizes as an order-of-magnitude uniformity improvement (Xu et al., 2022).

The point-cloud literature therefore extends the gradient-step denoiser away from explicit image priors and toward learned score fields on geometric manifolds, but the iterative mechanism remains recognizably the same: denoising is the accumulation of controlled gradient moves.

6. Generalizations, misconceptions, and limitations

Several recent works broaden the gradient-step viewpoint beyond classical imaging PnP. In multi-task inverse solvers, the denoiser residual is interpreted through the Tweedie–Miyasawa relation,

$g_\sigma$21

and a linear inverse problem is solved by iterative gradient ascent on the log-posterior: $g_\sigma$22 End-to-end multi-task fine-tuning across six inverse problems plus denoising produces an average PSNR improvement of $g_\sigma$23 dB at $g_\sigma$24 iterations relative to the baseline BF-CNN solver, while reducing the required iterations (Bled et al., 4 Dec 2025).

Outside imaging, DenoGrad defines noise through input-loss gradients of a reference deep model,

$g_\sigma$25

and corrects only samples with $g_\sigma$26 using normalized input updates

$g_\sigma$27

Across $g_\sigma$28 datasets, the method is reported to achieve the lowest KL-divergence or average correlation-matrix difference in $g_\sigma$29 cases and to improve average $g_\sigma$30 for interpretable models by $g_\sigma$31–$g_\sigma$32 points in the “train:denoised $g_\sigma$33 test:noisy” setting (Alonso-Ramos et al., 13 Nov 2025).

Diffusion-guided inpainting supplies another related gradient-step construction. GradPaint computes a coherence loss on the model’s denoised estimate $g_\sigma$34, backpropagates it through the diffusion network to obtain $g_\sigma$35, and updates the latent by

$g_\sigma$36

This remains a gradient-guided denoising step, but the gradient is now with respect to an inference-time coherence objective rather than a learned explicit prior (Grechka et al., 2023).

Several misconceptions recur in this area. First, a gradient-step denoiser is not identical to an arbitrary denoiser inserted into PnP; the explicit-potential construction is precisely what enables convergence to stationary points of a known functional (Hurault et al., 2021). Second, fixed-point convergence is not the same as convergence to a global optimum; DDGM explicitly remarks that no strict global-optimum convergence proof is offered beyond intuitive arguments (Luther et al., 2023). Third, the explicit construction is not computationally free: because $g_\sigma$37 must be computed, runtime is reported to be roughly doubled relative to a plain feed-forward denoiser (Herfeld et al., 11 Sep 2025).

Limitations are application-dependent. For large-hole inpainting, GS-PnP and Prox-PnP are reported not to hallucinate plausible textures across very wide gaps, which is described as an expected limitation of denoising priors rather than generative models (Herfeld et al., 11 Sep 2025). In dental CBCT, increasing $g_\sigma$38 from $g_\sigma$39 to $g_\sigma$40 smoothly suppresses noise, but small low-attenuation regions lose contrast at the highest $g_\sigma$41 (Tatachak et al., 27 May 2026). In point clouds, moving points independently toward the surface can create clustering or holes unless inter-point interactions are modeled explicitly (Xu et al., 2022).

Taken together, these developments show that the gradient-step denoiser is less a single algorithm than a rigorous design principle: encode denoising as a controlled gradient update, ideally one tied to an explicit potential or score field, then interleave that update with data-consistency steps. In the most developed PnP formulations, this yields both state-of-the-art restoration quality and convergence guarantees to fixed points or stationary points of explicit nonconvex energies (Hurault et al., 2022).