Gradient Restoration in Imaging & Optimization

• Gradient restoration is a technique that treats image gradients as primary signals to be restored, capturing local contrast, edge structure, and geometric information.
• It employs methods such as gradient amplification, graph Laplacian regularization, and learned priors to drive stable and robust reconstructions across various imaging tasks.
• By integrating classical enhancement with modern optimization and diffusion strategies, gradient restoration bridges explicit gradient manipulation with advanced restoration frameworks.

Searching arXiv for recent and foundational papers related to “gradient restoration” in image restoration, inverse problems, and graph/diffusion-based formulations. Gradient restoration is a family of restoration formulations in which gradients are treated as primary objects of reconstruction, regularization, or optimization guidance rather than as mere byproducts of intensity recovery. Across low-light enhancement, inverse problems, graph-based regularization, plug-and-play optimization, diffusion sampling, and multi-task adaptation, the common premise is that gradients encode local contrast, edge structure, geometric slope, or task relevance, and that restoring or managing those gradients can produce reconstructions that are more stable, more faithful to the underlying signal geometry, or more robust to degradation mismatch (Tanaka et al., 2018, Gharedaghi et al., 2023, Cai et al., 2024, Wu et al., 9 Jul 2025). The term therefore does not denote a single algorithm, but a technical viewpoint spanning explicit gradient-field manipulation, score-based priors, learned regularizing gradients, primal-dual data-consistency updates, and graph Laplacian models defined on gradients rather than raw intensities (Bigdeli et al., 2017, Fermanian et al., 2022, Li et al., 10 Nov 2025).

1. Conceptual scope and historical strands

A central strand defines gradient restoration literally: the gradient field of the unknown signal is enhanced, denoised, regularized, or reconstructed, and the final signal is obtained by integration or constrained optimization. In low-light enhancement, the underlying motivation is that the human visual system is more responsive to gradients than to absolute intensities; dark regions exhibit small gradients that are difficult to perceive, so the method amplifies gradients in dark regions and reconstructs an image under intensity-range constraints (Tanaka et al., 2018). In graph-based Retinex restoration, the illumination component is regularized in the gradient domain so that its gradient field becomes continuous and piecewise planar, suppressing noise-driven fluctuations before contrast enhancement (Gharedaghi et al., 2023).

A second strand uses “gradient restoration” in an optimization sense: restoration is driven by gradients of a posterior objective, a learned prior, or a task-specific fidelity term. In deep mean-shift priors, the residual of a denoising autoencoder estimates the gradient of a Gaussian-smoothed natural-image prior, enabling gradient-based Bayes risk minimization for deblurring, super-resolution, and demosaicing (Bigdeli et al., 2017). In plug-and-play gradient methods, a network is trained directly to model the gradient of a MAP regularizer, so that gradient descent can be used without appealing to Jacobian-symmetry assumptions of RED-style surrogates (Fermanian et al., 2022). In diffusion restoration, the interaction between prior and likelihood gradients becomes the object of analysis, and restoration quality depends on stabilizing those gradient dynamics (Wu et al., 9 Jul 2025).

A third strand relocates the notion from image-space gradients to other gradient-like objects. In multi-scenario adverse-weather restoration, gradients of the task loss with respect to model parameters determine binary parameter masks, so gradients restore the task-appropriate subset of the network without adding parameters (Guo et al., 2024). In point-cloud restoration, the gradient of the log-density defines a continuous field that drives noisy or sparse points toward the underlying surface (Chen et al., 2021). A plausible implication is that “gradient restoration” has become an umbrella term for methods in which restoring the signal and restoring the optimization geometry are deeply coupled.

2. Explicit reconstruction and regularization of gradient fields

The clearest explicit formulation appears in low-light image enhancement. Let the input grayscale image be

$$I:\Omega\subset\mathbb{Z}^2\to[0,255],\quad I(x,y)=I(\boldsymbol{x})=f(\boldsymbol{x}),$$

with discrete gradients

$$I_x(\boldsymbol{x})=f_h(\boldsymbol{x})=f(x+1,y)-f(x,y),\quad I_y(\boldsymbol{x})=f_v(\boldsymbol{x})=f(x,y+1)-f(x,y).$$

Dark-region enhancement is performed through an intensity-dependent weight

$L(\xi;\beta,\tau)= \begin{cases} \dfrac{\beta-1}{2\tau^2}\,\xi^2 - \dfrac{\beta-1}{\tau}\,\xi + \beta, & \xi \le \tau,\[6pt] 1, & \xi>\tau, \end{cases}$

where $\beta>1$ is the maximum gain and $\tau>0$ is the dark-region threshold. The enhanced gradients are

$$G'_x(\boldsymbol{x})=f_h(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),\quad G'_y(\boldsymbol{x})=f_v(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),$$

or equivalently $$G'(\boldsymbol{x})=w(\boldsymbol{x})\,\nabla I(\boldsymbol{x})$$ with $w(\boldsymbol{x})=L(I(\boldsymbol{x});\beta,\tau)$ (Tanaka et al., 2018). Reconstruction then solves a constrained least-squares integration problem,

$$\min_{J}\; \sum_{p\in\Omega}\Big(\,\big(J(p+\hat{\boldsymbol{x}})-J(p)-G'_x(p)\big)^2+\big(J(p+\hat{\boldsymbol{y}})-J(p)-G'_y(p)\big)^2\,\Big) \quad\text{s.t.}\quad J_{\min}\le J(p)\le J_{\max},$$

with $J_{\min}=0$ and $$I_x(\boldsymbol{x})=f_h(\boldsymbol{x})=f(x+1,y)-f(x,y),\quad I_y(\boldsymbol{x})=f_v(\boldsymbol{x})=f(x,y+1)-f(x,y).$$0 for 8-bit images (Tanaka et al., 2018). In this setting, gradient restoration consists of explicitly modifying a target gradient field and integrating it while preserving dynamic range.

Graph-based Retinex restoration generalizes this idea from direct gradient amplification to gradient-domain regularization. Under the model $$I_x(\boldsymbol{x})=f_h(\boldsymbol{x})=f(x+1,y)-f(x,y),\quad I_y(\boldsymbol{x})=f_v(\boldsymbol{x})=f(x,y+1)-f(x,y).$$1, reflectance is assumed piecewise constant and illumination continuous piecewise planar. The illumination prior is a Gradient Graph Laplacian Regularizer,

$$I_x(\boldsymbol{x})=f_h(\boldsymbol{x})=f(x+1,y)-f(x,y),\quad I_y(\boldsymbol{x})=f_v(\boldsymbol{x})=f(x,y+1)-f(x,y).$$2

where $$I_x(\boldsymbol{x})=f_h(\boldsymbol{x})=f(x+1,y)-f(x,y),\quad I_y(\boldsymbol{x})=f_v(\boldsymbol{x})=f(x,y+1)-f(x,y).$$3 is the forward-difference matrix and $$I_x(\boldsymbol{x})=f_h(\boldsymbol{x})=f(x+1,y)-f(x,y),\quad I_y(\boldsymbol{x})=f_v(\boldsymbol{x})=f(x,y+1)-f(x,y).$$4 is the Laplacian of a graph built on gradient samples (Gharedaghi et al., 2023). This regularizer acts directly on illumination gradients, encouraging them to be piecewise constant and smooth; integrating such gradients yields a piecewise planar illumination field. The restoration alternates between reflectance and illumination updates, solving sparse SPD linear systems by conjugate gradient with a diagonal preconditioner (Gharedaghi et al., 2023).

Manifold-graph formulations make the same principle fully geometric. On a graph with sampling coordinates, local gradients are estimated by weighted least squares, assembled into a stacked vector $$I_x(\boldsymbol{x})=f_h(\boldsymbol{x})=f(x+1,y)-f(x,y),\quad I_y(\boldsymbol{x})=f_v(\boldsymbol{x})=f(x,y+1)-f(x,y).$$5, and coupled by a gradient-graph Laplacian:

$$I_x(\boldsymbol{x})=f_h(\boldsymbol{x})=f(x+1,y)-f(x,y),\quad I_y(\boldsymbol{x})=f_v(\boldsymbol{x})=f(x,y+1)-f(x,y).$$6

For planar signals $$I_x(\boldsymbol{x})=f_h(\boldsymbol{x})=f(x+1,y)-f(x,y),\quad I_y(\boldsymbol{x})=f_v(\boldsymbol{x})=f(x,y+1)-f(x,y).$$7, the paper shows that $$I_x(\boldsymbol{x})=f_h(\boldsymbol{x})=f(x+1,y)-f(x,y),\quad I_y(\boldsymbol{x})=f_v(\boldsymbol{x})=f(x,y+1)-f(x,y).$$8, so planar signals become zero graph-frequencies of the induced Laplacian (Chen et al., 2022). This is the precise technical sense in which GGLR promotes planar or piecewise planar reconstruction and avoids the staircase behavior of GLR and GTV on slowly varying signals.

3. Gradient restoration as prior modeling and learned optimization geometry

In Bayesian restoration, one major interpretation of gradient restoration is the recovery of a usable prior gradient. Deep mean-shift priors define a Gaussian-smoothed image prior

$$I_x(\boldsymbol{x})=f_h(\boldsymbol{x})=f(x+1,y)-f(x,y),\quad I_y(\boldsymbol{x})=f_v(\boldsymbol{x})=f(x,y+1)-f(x,y).$$9

and show that the optimal denoising autoencoder residual obeys

$L(\xi;\beta,\tau)= \begin{cases} \dfrac{\beta-1}{2\tau^2}\,\xi^2 - \dfrac{\beta-1}{\tau}\,\xi + \beta, & \xi \le \tau,\[6pt] 1, & \xi>\tau, \end{cases}$0

The residual is therefore proportional to the mean-shift or score vector field of the smoothed natural-image distribution (Bigdeli et al., 2017). Restoration then combines the data gradient with this learned prior gradient. For a linear observation model $L(\xi;\beta,\tau)= \begin{cases} \dfrac{\beta-1}{2\tau^2}\,\xi^2 - \dfrac{\beta-1}{\tau}\,\xi + \beta, & \xi \le \tau,\[6pt] 1, & \xi>\tau, \end{cases}$1 with Gaussian noise, a MAP-like update is

$L(\xi;\beta,\tau)= \begin{cases} \dfrac{\beta-1}{2\tau^2}\,\xi^2 - \dfrac{\beta-1}{\tau}\,\xi + \beta, & \xi \le \tau,\[6pt] 1, & \xi>\tau, \end{cases}$2

The paper further introduces stochastic prior-gradient evaluation via noise splitting, using

$L(\xi;\beta,\tau)= \begin{cases} \dfrac{\beta-1}{2\tau^2}\,\xi^2 - \dfrac{\beta-1}{\tau}\,\xi + \beta, & \xi \le \tau,\[6pt] 1, & \xi>\tau, \end{cases}$3

to keep DAE inputs in a regime where the learned score is reliable (Bigdeli et al., 2017). Here, gradient restoration means recovering the prior gradient field that pulls estimates toward the natural-image manifold.

DGUNet relocates the same idea into deep unfolding. Classical proximal gradient descent uses

$L(\xi;\beta,\tau)= \begin{cases} \dfrac{\beta-1}{2\tau^2}\,\xi^2 - \dfrac{\beta-1}{\tau}\,\xi + \beta, & \xi \le \tau,\[6pt] 1, & \xi>\tau, \end{cases}$4

but when the degradation operator is unknown or too complex, the true gradient $L(\xi;\beta,\tau)= \begin{cases} \dfrac{\beta-1}{2\tau^2}\,\xi^2 - \dfrac{\beta-1}{\tau}\,\xi + \beta, & \xi \le \tau,\[6pt] 1, & \xi>\tau, \end{cases}$5 is unavailable. DGUNet replaces it with a learned surrogate:

$L(\xi;\beta,\tau)= \begin{cases} \dfrac{\beta-1}{2\tau^2}\,\xi^2 - \dfrac{\beta-1}{\tau}\,\xi + \beta, & \xi \le \tau,\[6pt] 1, & \xi>\tau, \end{cases}$6

followed by an informative proximal mapping module (Mou et al., 2022). The paper explicitly interprets this as a “gradient estimation strategy” that restores reliable gradients for the data-fidelity term under unknown degradations. This suggests that, in unfolding literature, gradient restoration concerns not only signal gradients but the restoration of optimization directions themselves.

PnP-ReG makes this restoration of optimization geometry more explicit. If a denoiser $L(\xi;\beta,\tau)= \begin{cases} \dfrac{\beta-1}{2\tau^2}\,\xi^2 - \dfrac{\beta-1}{\tau}\,\xi + \beta, & \xi \le \tau,\[6pt] 1, & \xi>\tau, \end{cases}$7 is interpreted as the proximal operator of $L(\xi;\beta,\tau)= \begin{cases} \dfrac{\beta-1}{2\tau^2}\,\xi^2 - \dfrac{\beta-1}{\tau}\,\xi + \beta, & \xi \le \tau,\[6pt] 1, & \xi>\tau, \end{cases}$8, then the first-order optimality condition yields

$L(\xi;\beta,\tau)= \begin{cases} \dfrac{\beta-1}{2\tau^2}\,\xi^2 - \dfrac{\beta-1}{\tau}\,\xi + \beta, & \xi \le \tau,\[6pt] 1, & \xi>\tau, \end{cases}$9

This identity supervises a network $\beta>1$0 that directly models the regularizing gradient $\beta>1$1 via the loss

$\beta>1$2

jointly with denoiser supervision (Fermanian et al., 2022). The resulting restoration step is

$\beta>1$3

Unlike RED-style constructions, the method does not rely on Jacobian symmetry of a practical denoiser, and instead learns the regularizing gradient directly (Fermanian et al., 2022).

4. Primal-dual, plug-and-play, and diffusion formulations

Primal-dual hybrid gradient methods provide another precise formulation of gradient restoration: the restoration is driven by explicit primal and dual gradient-like updates, while the prior term is implemented by a denoiser or generative model. In the PDHG-based plug-and-play framework,

$\beta>1$4

with saddle form

$\beta>1$5

the updates are

$\beta>1$6

The proximal of $\beta>1$7 is replaced by a time-dependent denoiser derived from a flow-matching model, and the framework admits $\beta>1$8, $\beta>1$9, and squared $\tau>0$0 fidelity terms with closed-form dual proximals (Li et al., 10 Nov 2025). The method supports denoising, super-resolution, deblurring, and inpainting, and reports that $\tau>0$1 and $\tau>0$2 fidelity outperform squared $\tau>0$3 under non-Gaussian noise (Li et al., 10 Nov 2025). In this setting, gradient restoration is the combination of principled primal-dual data-consistency updates and a learned generative prior.

Diffusion-based restoration often injects a likelihood gradient into each reverse step. Decoupled data consistency with diffusion purification separates these roles: a reconstruction phase takes gradient steps on the likelihood,

$\tau>0$4

and a refinement phase performs prior-only diffusion purification from a noised version of $\tau>0$5 (Li et al., 2024). The method argues that embedded likelihood gradients are computationally expensive and difficult to reconcile with accelerated samplers, whereas decoupling allows independent control over data consistency and prior enforcement (Li et al., 2024).

A later diffusion paper shifts the emphasis from injecting gradients to managing their stability. Under the model $\tau>0$6, the likelihood gradient is

$\tau>0$7

while the diffusion prior contributes a score-based gradient. The paper reports two instabilities: conflict between prior and likelihood gradient directions, and temporal fluctuation in the likelihood gradient itself (Wu et al., 9 Jul 2025). Its Stabilized Progressive Gradient Diffusion introduces a same-timestep warm-up with $\tau>0$8 inner updates,

$\tau>0$9

and Adaptive Directional Momentum,

$$G'_x(\boldsymbol{x})=f_h(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),\quad G'_y(\boldsymbol{x})=f_v(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),$$0

to stabilize the likelihood guidance before the denoising step (Wu et al., 9 Jul 2025). This is a distinctly modern meaning of gradient restoration: not only estimating gradients, but shaping their temporal behavior so that the restoration trajectory remains stable.

5. Graph Laplacians on gradients and unrolled interpretable networks

Gradient Graph Laplacian Regularizer methods provide one of the most mathematically explicit bridges between model-based restoration and trainable networks. For an $$G'_x(\boldsymbol{x})=f_h(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),\quad G'_y(\boldsymbol{x})=f_v(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),$$1 image patch vectorized as $$G'_x(\boldsymbol{x})=f_h(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),\quad G'_y(\boldsymbol{x})=f_v(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),$$2, the restoration objective is

$$G'_x(\boldsymbol{x})=f_h(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),\quad G'_y(\boldsymbol{x})=f_v(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),$$3

where $$G'_x(\boldsymbol{x})=f_h(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),\quad G'_y(\boldsymbol{x})=f_v(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),$$4 and $$G'_x(\boldsymbol{x})=f_h(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),\quad G'_y(\boldsymbol{x})=f_v(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),$$5 extract rows and columns, and $$G'_x(\boldsymbol{x})=f_h(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),\quad G'_y(\boldsymbol{x})=f_v(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),$$6 are gradient-induced nodal graph Laplacians (Cai et al., 2024). The regularizer is built by applying a graph Laplacian to line gradients rather than to pixel intensities. The paper states that for a 1D line graph, the smallest-eigenvalue eigenspace of the induced Laplacian spans all linear signals, so the prior promotes piecewise planar reconstruction rather than merely piecewise constant reconstruction (Cai et al., 2024).

To solve the resulting convex quadratic program, the paper introduces ADMM variants. In the single-auxiliary-variable case, the key linear systems are

$$G'_x(\boldsymbol{x})=f_h(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),\quad G'_y(\boldsymbol{x})=f_v(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),$$7

and

$$G'_x(\boldsymbol{x})=f_h(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),\quad G'_y(\boldsymbol{x})=f_v(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),$$8

with $$G'_x(\boldsymbol{x})=f_h(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),\quad G'_y(\boldsymbol{x})=f_v(\boldsymbol{x})\cdot L\!\left(f(\boldsymbol{x});\beta,\tau\right),$$9 (Cai et al., 2024). These are solved by conjugate gradient descent, and the entire iterative procedure is unrolled into a network with learned ADMM parameters, learned CGD coefficients, and periodic graph-learning modules akin to self-attention (Cai et al., 2024). The resulting UPnPGGLR network uses approximately $$G'(\boldsymbol{x})=w(\boldsymbol{x})\,\nabla I(\boldsymbol{x})$$0M parameters and is reported to be robust under covariance shift while remaining competitive with generic DL architectures in denoising, interpolation, and non-blind deblurring (Cai et al., 2024).

The broader GGLR literature on manifold graphs provides the underlying theory. Given node coordinates on a manifold, local gradients are estimated by weighted least squares, stacked as $$G'(\boldsymbol{x})=w(\boldsymbol{x})\,\nabla I(\boldsymbol{x})$$1, and coupled by a gradient graph with Laplacian $$G'(\boldsymbol{x})=w(\boldsymbol{x})\,\nabla I(\boldsymbol{x})$$2, yielding

$$G'(\boldsymbol{x})=w(\boldsymbol{x})\,\nabla I(\boldsymbol{x})$$3

For denoising, the estimator is

$$G'(\boldsymbol{x})=w(\boldsymbol{x})\,\nabla I(\boldsymbol{x})$$4

and for inpainting,

$$G'(\boldsymbol{x})=w(\boldsymbol{x})\,\nabla I(\boldsymbol{x})$$5

The paper derives an MSE decomposition in the eigenbasis of $$G'(\boldsymbol{x})=w(\boldsymbol{x})\,\nabla I(\boldsymbol{x})$$6 and uses it for efficient selection of the regularization weight $$G'(\boldsymbol{x})=w(\boldsymbol{x})\,\nabla I(\boldsymbol{x})$$7 (Chen et al., 2022). Compared with GLR and GTV, the stated advantage is the promotion of planar or piecewise planar signals and the reduction of staircase artifacts (Chen et al., 2022).

6. Domain-specific interpretations, applications, and limits

The gradient-restoration viewpoint appears in domain-specific forms whose shared structure is clearer than their surface differences.

Domain	Gradient object	Restoration mechanism
Low-light enhancement	Image spatial gradients	Dark-region gradient amplification plus constrained integration
Retinex restoration	Illumination gradients	GGLR regularization for PWP illumination
Natural-image priors	$$G'(\boldsymbol{x})=w(\boldsymbol{x})\,\nabla I(\boldsymbol{x})$$8	DAE-estimated mean-shift/score field
Plug-and-play MAP	$$G'(\boldsymbol{x})=w(\boldsymbol{x})\,\nabla I(\boldsymbol{x})$$9	Learned regularizing gradient in gradient descent
Diffusion restoration	Prior and likelihood gradients	Managed guidance, purification, or primal-dual updates
Multi-task weather restoration	Parameter gradients	Top-10% gradient-magnitude masks for task-specific updates

In multi-scenario adverse-weather restoration, the gradient object is no longer spatial. Model parameters are partitioned as

$w(\boldsymbol{x})=L(I(\boldsymbol{x});\beta,\tau)$0

and the task-specific mask is

$w(\boldsymbol{x})=L(I(\boldsymbol{x});\beta,\tau)$1

with $w(\boldsymbol{x})=L(I(\boldsymbol{x});\beta,\tau)$2 set to the $w(\boldsymbol{x})=L(I(\boldsymbol{x});\beta,\tau)$3th percentile of gradient magnitudes, so that the top $w(\boldsymbol{x})=L(I(\boldsymbol{x});\beta,\tau)$4 of parameters are selected for each task (Guo et al., 2024). The method reports PSNR scores of $w(\boldsymbol{x})=L(I(\boldsymbol{x});\beta,\tau)$5 on the Raindrop dataset, $w(\boldsymbol{x})=L(I(\boldsymbol{x});\beta,\tau)$6 on the Rain dataset, and $w(\boldsymbol{x})=L(I(\boldsymbol{x});\beta,\tau)$7 on the Snow100K dataset (Guo et al., 2024). Here, gradient restoration means restoring the appropriate adaptation subspace of the network via gradient-guided masking.

In astronomy, Scaled Gradient Projection and $w(\boldsymbol{x})=L(I(\boldsymbol{x});\beta,\tau)$8-SGP frame restoration as constrained optimization of a divergence under nonnegativity and flux conservation:

$w(\boldsymbol{x})=L(I(\boldsymbol{x});\beta,\tau)$9

The iterative update is

$$\min_{J}\; \sum_{p\in\Omega}\Big(\,\big(J(p+\hat{\boldsymbol{x}})-J(p)-G'_x(p)\big)^2+\big(J(p+\hat{\boldsymbol{y}})-J(p)-G'_y(p)\big)^2\,\Big) \quad\text{s.t.}\quad J_{\min}\le J(p)\le J_{\max},$$0

with diagonal scaling, Barzilai-Borwein step sizes, and projection onto the feasible set (Gondhalekar et al., 2022). The paper reports consistent enhancement in flux conservation and often improved FWHM and ellipticity relative to SGP, with $$\min_{J}\; \sum_{p\in\Omega}\Big(\,\big(J(p+\hat{\boldsymbol{x}})-J(p)-G'_x(p)\big)^2+\big(J(p+\hat{\boldsymbol{y}})-J(p)-G'_y(p)\big)^2\,\Big) \quad\text{s.t.}\quad J_{\min}\le J(p)\le J_{\max},$$1 adaptation allowing the loss to better match mixed observational noise (Gondhalekar et al., 2022). Although the paper does not manipulate spatial gradients directly, restoration is still gradient-driven in the optimization sense.

In point-cloud restoration, the score field

$$\min_{J}\; \sum_{p\in\Omega}\Big(\,\big(J(p+\hat{\boldsymbol{x}})-J(p)-G'_x(p)\big)^2+\big(J(p+\hat{\boldsymbol{y}})-J(p)-G'_y(p)\big)^2\,\Big) \quad\text{s.t.}\quad J_{\min}\le J(p)\le J_{\max},$$2

or, for degraded data, $$\min_{J}\; \sum_{p\in\Omega}\Big(\,\big(J(p+\hat{\boldsymbol{x}})-J(p)-G'_x(p)\big)^2+\big(J(p+\hat{\boldsymbol{y}})-J(p)-G'_y(p)\big)^2\,\Big) \quad\text{s.t.}\quad J_{\min}\le J(p)\le J_{\max},$$3, is estimated as a continuous field, and restoration proceeds by deterministic gradient ascent

$$\min_{J}\; \sum_{p\in\Omega}\Big(\,\big(J(p+\hat{\boldsymbol{x}})-J(p)-G'_x(p)\big)^2+\big(J(p+\hat{\boldsymbol{y}})-J(p)-G'_y(p)\big)^2\,\Big) \quad\text{s.t.}\quad J_{\min}\le J(p)\le J_{\max},$$4

optionally followed by GLR or RGLR proximal refinement (Chen et al., 2021). The continuity of the field is enforced by cosine-annealed neighborhood aggregation, precisely to make the optimization stable as points move (Chen et al., 2021). This suggests that, beyond images, gradient restoration can be understood as restoring a continuous geometric vector field that encodes the target manifold.

The literature also makes recurring limitations explicit. Gradient amplification in dark regions can amplify noise and risk haloing or ringing; the low-light paper notes optional gradient smoothing and careful parameter choice as remedies (Tanaka et al., 2018). Graph-based PWP priors can oversmooth when their assumptions fail, such as highly textured reflectance or complex illumination (Gharedaghi et al., 2023). Learned gradient priors depend on the fidelity of the estimated score or regularizing gradient and may lack classical convergence guarantees when implemented by learned denoisers or time-dependent operators (Fermanian et al., 2022, Li et al., 10 Nov 2025). Diffusion guidance can destabilize sampling when prior and likelihood gradients conflict, motivating explicit gradient management (Wu et al., 9 Jul 2025).

Taken together, these formulations show that gradient restoration is not reducible to edge sharpening. It encompasses explicit recovery of gradient fields, graph-based smoothing of gradients, estimation of prior or regularizer gradients, stabilization of competing gradients in generative sampling, and even gradient-based selection of task-relevant parameters. The unifying principle is that reconstruction quality depends crucially on the geometry encoded by gradients—whether those gradients live in image space, manifold space, latent space, or parameter space—and that restoring that geometry is often more effective than operating on intensities alone (Tanaka et al., 2018, Bigdeli et al., 2017, Cai et al., 2024, Wu et al., 9 Jul 2025).