Equivariant Denoisers for Image Restoration
Marien Renaud, Arthur Leclaire, Nicolas Papadakis
TL;DR
ERED tackles the challenge of encoding transformation invariances in image restoration priors by introducing a $\pi$-equivariant regularization of denoising within a Plug-and-Play framework. The approach defines $r_{\sigma}^{\pi}$ and $\,s_{\sigma}^{\pi}$ via averaging over transformation groups and constructs an equivariant denoiser $\tilde{D}_{\sigma}$ to implement the prior, with convergence guarantees under unbiased and biased settings. The authors prove critical-point convergence to the ideal equivariant objective as $\sigma \to 0$ and provide biased convergence results for practical denoisers, supported by experiments on deblurring and despeckling that show modest PSNR gains when using equivariant transforms such as flips and rotations. Overall, the work offers a principled, theoretically grounded route to enforce geometric priors in learned denoisers, with tangible benefits for certain restoration tasks and stochastic PnP schemes.
Abstract
One key ingredient of image restoration is to define a realistic prior on clean images to complete the missing information in the observation. State-of-the-art restoration methods rely on a neural network to encode this prior. Moreover, typical image distributions are invariant to some set of transformations, such as rotations or flips. However, most deep architectures are not designed to represent an invariant image distribution. Recent works have proposed to overcome this difficulty by including equivariance properties within a Plug-and-Play paradigm. In this work, we propose a unified framework named Equivariant Regularization by Denoising (ERED) based on equivariant denoisers and stochastic optimization. We analyze the convergence of this algorithm and discuss its practical benefit.