Enhanced Denoising and Convergent Regularisation Using Tweedie Scaling

TL;DR

This work addresses ill-posed image reconstruction by integrating a tunable regularisation strength into Plug-and-Play denoisers through Tweedie scaling, defining $\mathbf{D}_{\delta} = \operatorname{Id} + \frac{\mathbf{D} - \operatorname{Id}}{\delta^2}$ to control regularisation. Building on Tweedie’s identity and small-$\sigma$ expansions, the authors justify the scaling without requiring the denoiser to be exact MMSE, and they prove that the resulting PnP iterations are convergent (under mild assumptions) and constitute a convergent regularisation in the sense of existing theory. They provide theoretical links between the scaling parameter $\delta$ and the denoiser’s training quality via $\delta_{\text{opt}}$, and validate the framework experimentally with DRUNet on CBSD68, showing stability and convergence in denoising and inpainting tasks. The results offer a principled, interpretable mechanism to modulate regularisation strength in PnP methods and ensure convergence when using deep denoisers, with potential extensions to other PnP algorithms and more general denoisers.

Abstract

The inherent ill-posed nature of image reconstruction problems, due to limitations in the physical acquisition process, is typically addressed by introducing a regularisation term that incorporates prior knowledge about the underlying image. The iterative framework of Plug-and-Play methods, specifically designed for tackling such inverse problems, achieves state-of-the-art performance by replacing the regularisation with a generic denoiser, which may be parametrised by a neural network architecture. However, these deep learning approaches suffer from a critical limitation: the absence of a control parameter to modulate the regularisation strength, which complicates the design of a convergent regularisation. To address this issue, this work introduces a novel scaling method that explicitly integrates and adjusts the strength of regularisation. The scaling parameter enhances interpretability by reflecting the quality of the denoiser's learning process, and also systematically improves its optimisation. Furthermore, the proposed approach ensures that the resulting family of regularisations is provably stable and convergent.

Figures (3)

• Figure 1: $\delta \mapsto \mathbb{E}_{X, \xi}[\|\mathbf{D}_\delta(x + \sigma\xi) - x\|^2]$ for $\sigma \in \{0.01, 0.07, 0.10, 0.12\}$
• Figure 2: Convergence of $(\operatorname{PnP}(\delta, y_k))_k$ towards $\operatorname{PnP}(\delta, y)$.
• Figure 3: Data consistency and convergence (or lack thereof) for the Tweedie scaling method (blue) and homogeneous scaling (green) in Inpainting and Denoising.

Theorems & Definitions (4)

• proposition thmcounterproposition
• proof
• proposition thmcounterproposition: $\mathbf{T}_\delta$ defines convergent iterations
• proof