Stabilizing Deep Reconstruction Operators with Contractive Anchoring

Technical Summary: Stabilizing Deep Reconstruction Operators with Contractive Anchoring

Problem Statement
Pretrained deep denoisers are widely used in model-based image reconstruction (IR) tasks, such as deblurring and super-resolution, via Plug-and-Play (PnP) and Regularization-by-Denoising (RED) algorithms. These methods allow the use of powerful, task-agnostic denoisers without retraining for specific inverse problems. However, a critical limitation arises when these denoisers, trained only for single-step inference, are employed as regularizers in iterative reconstruction schemes.

The authors identify a prevalent failure mode termed "peak-and-collapse" (PC). In this scenario, reconstruction quality (measured by PSNR) improves during early iterations, reaches a peak, and then abruptly deteriorates. This instability renders standard PnP and RED algorithms unreliable in practice, as identifying the optimal stopping point is difficult. Existing solutions often require retraining denoisers with specific architectural constraints (e.g., non-expansivity) or optimizing the denoiser jointly with the reconstruction model, which limits the reuse of existing, high-performance pretrained networks.

Methodology
The paper proposes a data-driven stabilization framework that acts as a "drop-in" mechanism, requiring no modification or retraining of the underlying pretrained denoiser or the proximal solver. The core approach involves controlling the instability of the IR operator using a contractive anchor.

1. Stability Index ($\eta$): Since the IR operator $\mathcal{T}$ is treated as a black box, the authors define a local stability index $\eta_p(\mathbf{x}, \mathcal{T})$ relative to a reference point $\mathbf{p}$. This index measures the relative expansion or contraction induced by the operator. Instability is characterized by $\eta > 1$.
2. Adaptive Contractive Anchoring: The method introduces a contractive operator $\mathcal{S}$ (the anchor) with a unique fixed point $\mathbf{p}$. The unstable operator $\mathcal{T}$ is blended with $\mathcal{S}$ to form an averaged operator $\mathcal{T}_\theta = (1-\theta)\mathcal{T} + \theta\mathcal{S}$.
   • Theoretical Guarantee: Lemma 1 and Theorem 1 demonstrate that averaging with a contraction strictly reduces the stability index. The authors derive an explicit threshold $\tilde{\theta}$ that guarantees the stability index is reduced to a target level $\xi$ (specifically $\xi=1$).
   • Algorithm 1: The proposed algorithm adaptively computes the mixing weight $\theta_k$ at each iteration. If $\eta > 1$, $\theta_k$ is set to the minimal value required to bring $\eta$ down to 1; otherwise, $\theta_k = 0$. This ensures stability is enforced only when necessary, preserving the reconstruction quality driven by the original operator $\mathcal{T}$.
3. Construction of the Contractive Anchor: To serve as the anchor $\mathcal{S}$, the authors design a lightweight, noise-aware denoiser $\mathcal{D}_\sigma$ that is contractive by construction.
   • The denoiser is modeled as a gradient-step operator $\mathcal{D}(\mathbf{x}) = \mathbf{x} - \gamma \nabla \Phi(\mathbf{x})$.
   • The potential function $\Phi$ is parameterized using trainable convolutional filters and spline-based nonlinearities with slopes constrained to $[0, 1]$.
   • Through specific reparameterization of the convolution weights and constraints on the step size $\gamma$ and regularization parameter $\tau$, the authors prove (Proposition 2 and 3) that the resulting operator is a $\kappa$-contraction ($\kappa < 1$) for any noise level $\sigma$. This anchor is trained as a standard Gaussian denoiser but retains provable contractivity.

Key Contributions

1. Stability Index Formalization: The introduction of a local stability index $\eta$ that quantifies the expansiveness of black-box IR operators, enabling the detection and control of instability without internal knowledge of the network.
2. Adaptive Stabilization Algorithm: A provably convergent algorithm (Algorithm 1) that adaptively adjusts the influence of a contractive anchor to prevent divergence. It operates as a complete drop-in wrapper, requiring no hyperparameter tuning beyond the anchor's training.
3. Trainable Contractive Anchor: The design of a lightweight, noise-aware denoiser that is contractive by construction. This allows the use of unconstrained training procedures while guaranteeing the mathematical properties required for the stabilization framework.

Experimental Results
The framework was evaluated across multiple proximal solvers (PGD, HQS, ADMM, RED-GD), diverse denoiser architectures (CNNs like DnCNN/DRUNet, Diffusion models, Transformers), and various imaging tasks (motion deblurring, Gaussian deblurring, super-resolution).

• Stability: The method consistently prevents the peak-and-collapse behavior observed in vanilla PnP and Equivariant PnP. While baseline methods often diverge (indicated by $\times$ in results) or suffer significant PSNR degradation after the peak, the stabilized variants maintain final PSNR values close to their peak performance.
• Performance: The stabilized methods achieve competitive or superior results compared to state-of-the-art pipelines like DPIR, GSPnP, and DEAL. For instance, in motion deblurring, the stabilized HQS+GS-DRUNet achieved 30.70 dB, matching DPIR (30.69 dB), while standard PnP-HQS diverged.
• Efficiency: The computational overhead is reported to be approximately 20% compared to vanilla PnP. The method preserves reconstruction quality because the mixing weight $\theta$ remains small during stable phases, meaning updates are still primarily driven by the high-performance black-box operator.

Significance and Claims
The paper claims that this framework provides a robust solution to the instability of deep reconstruction operators without sacrificing the modularity of the PnP/RED paradigm. By decoupling the stabilization mechanism from the specific denoiser architecture, the authors enable the reliable use of existing, powerful pretrained models in iterative reconstruction tasks. The approach offers provable guarantees of bounded iterates and prevents the abrupt degradation that currently limits the practical deployment of PnP methods. The authors position this as a step toward inherently reliable IR systems, noting that future work could involve incorporating these principles directly into the training of IR systems.