Stabilizing Deep Reconstruction Operators with Contractive Anchoring

Conference: ECCV 2026
Paper: ECCV Official
Code: https://github.com/trishitmg/costa
Area: Image Restoration
Keywords: Plug-and-Play (PnP), Regularization by Denoising (RED), contractive operator, operator stability, inverse problems

TL;DR

Addressing the pervasive "peak-and-collapse" instability where deep denoisers in iterative Plug-and-Play (PnP) and RED reconstruction degrade abruptly after reaching peak performance, this paper proposes an adaptive contractive anchoring framework that monitors local operator expansiveness and adaptively blends a lightweight contractive anchor in closed form, eliminating divergence without modifying or retraining black-box denoisers.

Background & Motivation

Model-based image reconstruction from corrupted linear observations (e.g., deblurring, super-resolution, magnetic resonance imaging, and computed tomography) is conventionally cast as a regularized least-squares optimization problem balancing a data-fidelity term and an image prior. Over recent years, Plug-and-Play (PnP) and Regularization-by-Denoising (RED) frameworks have gained widespread traction by replacing explicit regularizer optimization steps inside proximal algorithms (such as PGD, HQS, and ADMM) with off-the-shelf deep neural denoisers like DnCNN, DRUNet, and diffusion models. This modular decoupling allows practitioners to leverage cutting-edge denoisers across varied imaging modalities without task-specific end-to-end retraining.

However, modern deep denoisers are trained almost exclusively for single-step additive Gaussian noise removal; they do not satisfy mathematical nonexpansiveness or maximal monotonicity required for classical algorithmic convergence. When injected into recursive iterative loops, these unconstrained black-box operators consistently exhibit an insidious failure mode known as the "peak-and-collapse" (PC) phenomenon: reconstruction quality (measured in PSNR) rises rapidly during initial iterations to an impressive peak, but subsequently deteriorates sharply into severe artifacts or numerical divergence (as shown by Vanilla-PnP in Table 1). While early stopping can theoretically capture the peak, detecting this turning point in blind inverse problems without clean ground-truth reference is virtually impossible; conversely, existing convergent PnP variants constrain network expressive architectures or tightly tie denoisers to specific iterative solvers, forfeiting the core plug-and-play modularity.

Since treating the reconstruction operator as a black box precludes internal architectural surgery, a fundamental dilemma emerges: can we actively monitor when the iterative system begins to drift away from the solution manifold and pull it back using an intrinsically stable anchor? Core idea: stabilize arbitrary black-box reconstruction operators via adaptive sequential averaging with a lightweight, structurally contractive anchor operator, computing the minimal closed-form mixing weight driven by a local stability index to guarantee bounded iterates while preserving near-peak reconstruction fidelity.

Method

Overall Architecture

The proposed stabilization mechanism (COSTA) treats any denoiser-driven image reconstruction operator \(T\) (such as PnP-PGD, PnP-HQS, PnP-ADMM, or RED-GD) as an uninspected black box that maps image iterate \(x_k\) to an updated state \(T(x_k)\). The architecture introduces an auxiliary contractive anchor operator \(S\) with a provable \(\kappa\)-contraction factor (\(\kappa < 1\)) and a precomputed unique fixed point \(p\). At each iteration \(k\), the framework evaluates the stability index \(\eta_k\), which quantifies the local expansiveness of \(T\) around \(p\). If the system remains in the stable non-expansive regime (\(\eta_k \le 1\)), the update is left purely to the black-box operator (\(\theta_k = 0\)). Once local instability is detected (\(\eta_k > 1\)), an adaptive closed-form mixing weight \(\theta_k\) is computed to execute a convex combination update \(x_{k+1} = (1-\theta_k)T(x_k) + \theta_k S(x_k)\), suppressing the expansiveness index exactly to the stability threshold.

%%{init: {'flowchart': {'rankSpacing': 24, 'nodeSpacing': 28, 'padding': 6, 'wrappingWidth': 400}}}%%
flowchart TD
    A["Input degraded observation and iterate $x_k$"] --> B["Black-box forward step & stability index evaluation<br/>Compute $T(x_k)$ and $\eta_k = \|T(x_k)-p\| / \|x_k-p\|$"]
    B --> C{"Stability check<br/>$\eta_k > 1$ ?"}
    C -->|No: within safe contractive zone| D["Zero-weight update<br/>$\theta_k = 0 \implies x_{k+1} = T(x_k)$"]
    C -->|Yes: local expansion detected| E["Adaptive contractive anchoring mechanism<br/>Closed-form quadratic solution for minimal weight $\theta_k$"]
    E --> F["Structurally contractive denoiser & anchor operator<br/>Anchor evaluation $S(x_k) = D_\sigma(\mathrm{prox}_{\rho f}(x_k))$"]
    F --> G["Convex combination update<br/>$x_{k+1} = (1-\theta_k)T(x_k) + \theta_k S(x_k)$"]
    D --> H["Next iteration $x_{k+1}$"]
    G --> H

Key Designs

1. Stability Index: Quantifying Local Expansiveness of Black-Box Operators

Because off-the-shelf deep denoisers have hidden internal layers and intractable Jacobian norms, stability cannot be judged from model weights. The paper introduces an empirical scalar metric—the Stability Index \(\eta_p(x, T)\)—evaluated purely via input-output coordinates relative to an anchor reference point \(p\):

\[\eta_p(x, T) = \frac{\|T(x) - p\|}{\|x - p\|} \quad (x \neq p)\]

Geometrically, \(\eta_p(x, T)\) measures the magnification factor that operator \(T\) induces on deviations from the baseline reconstruction \(p\). In early iterations, denoiser projections actively pull degraded inputs toward the natural image manifold, keeping \(\eta_p(x, T) \le 1\). Later, as accumulated recursive noise and out-of-distribution artifacts amplify, the operator overshoots, sending \(\eta_p(x, T) > 1\) and triggering collapse. This scalar index provides a real-time, non-invasive diagnostic of imminent destabilization.

2. Adaptive Contractive Anchoring: Closed-Form Weight Selection and Boundedness Guarantees

To stabilize the trajectory without dampening the high-frequency restoration capability of the black-box operator, the update is governed by sequential averaging \(T_\theta(x) = (1-\theta)T(x) + \theta S(x)\). Theoretical analysis shows that if \(S\) is a \(\kappa\)-contraction (\(\kappa < 1\)), then for any state where \(\eta_p(x, T) > \xi > \kappa\), there exists a unique minimal threshold \(\tilde{\theta}(x, \xi) \in (0, 1)\) such that \(\eta_p(x, T_{\tilde{\theta}}) = \xi\).

Targeting the critical non-expansive boundary \(\xi = 1\), the optimal mixing parameter \(\theta_k\) when \(\eta_p(x_k, T) > 1\) is obtained by solving:

\[\|(1-\theta)(T(x_k) - p) + \theta(S(x_k) - p)\|^2 = \|x_k - p\|^2\]

This simplifies to an explicit quadratic equation with respect to \(\theta\), whose smallest non-negative root in \([0, 1]\) is taken directly without iterative tuning loops. Crucially, Corollary 1 proves that maintaining \(\eta_p \le 1\) guarantees that the iterate sequence \(\{x_k\}\) remains globally bounded for arbitrary black-box operators. In practice, computed \(\theta_k\) values remain negligible during early iterations and modest thereafter, ensuring that the primary restoration drive originates from the state-of-the-art denoiser.

3. Structurally Contractive Denoiser: Strongly Convex Potentials via Spline Activations

The Euclidean contractivity of the anchor \(S\) must hold unconditionally. By Proposition 1, whenever data-fidelity \(f\) is convex and denoiser \(D\) is contractive, the proximal step \(S = D \circ \mathrm{prox}_{\rho f}\) is contractive. Rather than relying on computationally cumbersome patch-based models, the paper formulates a direct image-to-image single-layer convolutional denoiser \(D_\sigma\) derived from gradient-step optimization:

\[D_\sigma(x) = (1-\gamma)x - \gamma W^\top \left( \nu(\sigma)^{-1} \psi\left( \frac{\nu(\sigma) W x}{\tau} \right) \right)\]

where \(W\) is reparameterized as \(W = \tilde{W} R^{-1/2}\) with \(R = \mathrm{Diag}(\tilde{W}^\top \mathbf{1})\) to guarantee operator norm \(\|W\| \le 1\) by construction. Nonlinear activation \(\psi\) is parameterized as trainable linear splines with 101 knots whose first derivatives are constrained in \([0, 1]\), ensuring the underlying potential is smooth and strongly convex; \(\nu(\sigma)\) is a learnable noise-level scaling function. With step size \(\gamma = 1/(1+2\tau)\) and parameter \(\tau \in [0.0102, 0.135]\), Proposition 3 guarantees that \(D_\sigma\) is strictly a \(\kappa\)-contraction with \(\kappa = (1+\tau)/(1+2\tau) \in [0.9, 0.99]\), offering both rapid inference and provable stability.

Loss & Training

The noise-aware contractive denoiser \(D_\sigma\) is trained on Gaussian-corrupted patches from BSD400, DIV2K, Waterloo Exploration Database, and Flickr2K, with noise standard deviations \(\sigma_n \sim \mathcal{U}[0, 25/255]\). The training objective is mean squared error (MSE) minimized via Adam over 100,000 epochs with a batch size of 16 patches (\(128 \times 128\)) and a cosine learning rate decay down to \(5 \times 10^{-3}\) of the initial value. Specific initial learning rates are assigned to parameter groups: \(5 \times 10^{-3}\) for \((W, q)\), \(5 \times 10^{-4}\) for spline knots \(\psi\), \(5 \times 10^{-3}\) for noise-scaling function \(\nu\), and \(5 \times 10^{-2}\) for potential hyperparameter \(\tau\). No spectral normalization or penalty terms are needed during backpropagation because contractivity is embedded directly in the architectural design.

Key Experimental Results

Main Results

On the CBSD10 benchmark at noise level \(\sigma_n = 0.02\), the proposed framework is evaluated on Motion Deblurring and Gaussian Deblurring across standard proximal algorithms and denoiser backbones. Both Peak PSNR (maximum value achieved across iterations) and Final PSNR (quality at iteration 10,000) are reported; symbol \(\times\) indicates numerical divergence or complete post-peak collapse.

Framework & Denoiser	Method	Motion Blur Peak (dB)	Motion Blur Final (dB)	Gaussian Blur Peak (dB)	Gaussian Blur Final (dB)
PnP-PGD + DnCNN	Ours	27.96 ± 4.38	27.95 ± 4.38	28.78 ± 3.02	28.77 ± 3.02
PnP-PGD + DnCNN	Vanilla PnP	27.78 ± 4.13	\(\times\) (diverged)	27.53 ± 2.97	\(\times\) (diverged)
PnP-PGD + DnCNN	Equiv. PnP	27.97 ± 4.37	\(\times\) (diverged)	27.58 ± 3.01	\(\times\) (diverged)
PnP-HQS + DRUNet	Ours	27.86 ± 4.42	27.86 ± 4.42	30.02 ± 4.03	30.02 ± 4.03
PnP-HQS + DRUNet	Vanilla PnP	27.60 ± 4.28	26.90 ± 4.36	29.78 ± 3.89	\(\times\) (diverged)
PnP-HQS + DRUNet	Equiv. PnP	27.70 ± 4.42	\(\times\) (diverged)	29.83 ± 3.94	\(\times\) (diverged)
PnP-HQS + DiffUNet	Ours	27.84 ± 3.98	27.82 ± 3.96	29.62 ± 3.49	29.61 ± 3.48
PnP-HQS + DiffUNet	Vanilla PnP	27.21 ± 3.52	18.90 ± 0.92	29.20 ± 3.22	23.12 ± 3.17
PnP-HQS + GSDRUNet	Ours	28.29 ± 4.39	28.28 ± 4.39	30.70 ± 4.12	30.70 ± 4.12
PnP-HQS + GSDRUNet	Vanilla PnP	28.18 ± 4.26	23.86 ± 8.18	30.56 ± 4.13	25.72 ± 8.23
PnP-ADMM + CoCo-DRUNet	Ours	27.85 ± 4.30	27.65 ± 4.39	29.26 ± 3.01	29.19 ± 3.05
RED-GD + DRUNet	Ours	27.74 ± 4.19	27.74 ± 4.19	29.48 ± 3.18	29.48 ± 3.18
DPIR (tuned baseline)	Reference	28.33 ± 4.65	28.33 ± 4.65	30.69 ± 0.52	30.69 ± 0.52
GSPnP (convergent baseline)	Reference	28.21 ± 4.23	28.21 ± 4.23	30.58 ± 0.59	30.58 ± 0.59

For single-image super-resolution (\(s \in \{2, 3, 4\}\)) on CBSD10, the comparative results against baselines are detailed below:

Framework & Denoiser	Method	\(s=2\) Peak (dB)	\(s=2\) Final (dB)	\(s=3\) Peak (dB)	\(s=3\) Final (dB)	\(s=4\) Peak (dB)	\(s=4\) Final (dB)
PnP-HQS + DRUNet	Ours	27.48 ± 4.25	27.48 ± 4.24	25.85 ± 4.40	25.83 ± 4.43	24.29 ± 4.46	23.96 ± 4.80
PnP-HQS + DRUNet	Vanilla PnP	27.29 ± 4.25	\(\times\) (diverged)	25.64 ± 4.35	\(\times\) (diverged)	24.04 ± 4.32	\(\times\) (diverged)
PnP-HQS + GSDRUNet	Ours	27.99 ± 4.52	27.95 ± 4.47	26.30 ± 4.52	26.29 ± 4.53	24.74 ± 4.41	24.62 ± 4.37
PnP-HQS + GSDRUNet	Vanilla PnP	27.89 ± 4.37	23.84 ± 7.91	26.24 ± 4.48	24.08 ± 5.78	24.65 ± 4.37	21.02 ± 6.71
PnP-IHQS + SPC-DRUNet	Ours	27.26 ± 4.04	27.22 ± 4.07	25.61 ± 4.15	25.55 ± 4.22	24.03 ± 4.12	23.75 ± 4.29
RED-GD + GSDRUNet	Ours	27.54 ± 4.18	27.53 ± 4.17	26.08 ± 4.39	26.07 ± 4.39	24.66 ± 4.34	24.66 ± 4.35
DPIR (tuned baseline)	Reference	27.93 ± 1.16	27.93 ± 1.16	26.32 ± 0.35	26.32 ± 0.35	24.76 ± 0.36	24.76 ± 0.36

Ablation Study

The impact of anchor denoiser convergence stage (training epochs \(N\)) and anchor fixed-point quality on stabilization performance:

Configuration / Variant	Motion Deblurring Final PSNR	Iteration Trajectory Behavior	Analysis & Impact
Underfitted Anchor (\(N=100\))	10.21 dB	Complete collapse	Low-quality fixed point \(p\) fails to provide viable geometric guidance
Partially Trained Anchor (\(N=200\))	18.45 dB	Delayed collapse	Inadequate contraction basin unable to prevent late divergence
Moderate Anchor (\(N=300\))	28.15 dB	Completely stable	Reaching baseline image fidelity suffices to lock long-term stability
Fully Trained Anchor (\(N=1000\))	28.28 dB	Stable and high quality	Matches peak PSNR, showing stability is robust to anchor precision
Fixed Full Anchoring (\(\theta=1\))	27.55 dB	Over-smoothed	Collapses onto pure contraction fixed point, forfeiting black-box quality
Truncation without Anchor	\(\times\) (diverged)	Severe oscillatory divergence	Lacks an attracting contraction fixed point; cannot guarantee boundedness

Key Findings

• Zero Post-Peak Decay: Across all denoisers and inverse tasks, the proposed stabilization framework ensures that the final PSNR at 10,000 iterations stays within \(0.01 \sim 0.05 \text{ dB}\) of the historical maximum peak, eliminating the guesswork of heuristic early stopping.
• Modest Anchor Quality Barrier: As depicted in Figure 4, the anchor model only requires moderate training (\(N \approx 300\) epochs) to provide a sufficient fixed-point reference; it does not need to rival complex multi-scale deep backbones in solitary denoising quality.
• Low Computational Overhead: Because the mixing weight calculation depends solely on inner products and a scalar quadratic root, the added computational latency is approximately 20% compared to unconstrained Vanilla-PnP.

Highlights & Insights

• Non-Invasive Black-Box Wrapper: By encapsulating operator divergence into an input-output stability index \(\eta_p\), the method operates as a drop-in stabilizer compatible with CNN, diffusion, and transformer denoisers without retraining or modifying their weights.
• Minimal Intervention Philosophy: Driven by explicit quadratic thresholding, the mixing weight \(\theta_k\) stays near zero during healthy descent and activates only at the onset of expansion, keeping updates predominantly driven by high-capacity black-box priors.
• Broad Transferability: The concept of coupling an unconstrained, high-capacity generative operator with an intrinsically contractive anchor via adaptive sequential averaging can be extended to other iterative inverse pipelines, including diffusion model inversion and text-guided iterative editing.

Limitations & Future Work

• Convexity Assumption on Data Fidelity: The formal contraction of proximal anchor \(S\) assumes a convex data-fidelity function \(f\); extension to non-convex or blind inverse problems (e.g., blind deconvolution or phase retrieval) requires further theoretical adaptation.
• Domain Dependence of Anchor: Although lightweight, the anchor denoiser is trained on natural image datasets; deploying it on specialized domains such as low-dose CT or cryo-EM may require domain-matched calibration of spline knots and filter dictionaries.
• Future Directions: Investigating the integration of local contractive penalties directly into pretraining loss objectives to endow deep denoisers with intrinsic fixed-point attraction basins.

Related Work & Insights

• vs Equivariant PnP (CVPR 2024): Equivariant PnP employs group transformations as a stochastic regularizer to smooth iterations, but still frequently suffers numerical divergence \(\times\) over long trajectories (10,000 iterations); our contractive anchoring provides provable boundedness and collapse-free convergence.
• vs GSPnP / DEAL (ICLR 2022 / ICML 2025): GSPnP and DEAL restrict networks to gradient steps or weakly convex potentials, limiting architecture choices and hindering deployment of pretrained diffusion backbones; our method decouples the powerful black-box prior from the contractive anchor, retaining maximum modeling flexibility.

Rating

• Novelty: ⭐⭐⭐⭐⭐ Formulates an elegant, non-invasive stabilization theory based on local stability index and adaptive sequential averaging for black-box PnP operators.
• Experimental Thoroughness: ⭐⭐⭐⭐⭐ Comprehensive validation spanning 5 denoiser backbones, 3 proximal frameworks, multiple inverse tasks, and trajectories extending to 10,000 iterations.
• Writing Quality: ⭐⭐⭐⭐⭐ Rigorous mathematical derivation combined with intuitive geometric interpretations and clear, reproducible experimental exposition.
• Value: ⭐⭐⭐⭐⭐ Resolves a longstanding reliability vulnerability in deep model-based image reconstruction; code is open-source and ready for practical deployment.