Equivariant plug-and-play image reconstruction

TL;DR

This work targets instability in plug-and-play and regularization-by-denoising methods for linear inverse imaging problems by enforcing approximate equivariance on the denoiser. It introduces a simple Monte Carlo group-averaging scheme to transform any denoiser into a G-equivariant one, improving Jacobian symmetry and reducing the Lipschitz constant, thereby stabilizing PnP/RED/ULA iterations. The authors provide theoretical results on optimality, prior existence, and interplay with the forward operator, and validate the approach across image deblurring, super-resolution, and MRI with multiple backbone denoisers, achieving better or comparable reconstruction quality with greater stability. The practical impact lies in enabling robust, task-agnostic use of denoisers in inverse-imaging pipelines without retraining, while highlighting limitations and directions for future refinement of equivariant priors.

Abstract

Plug-and-play algorithms constitute a popular framework for solving inverse imaging problems that rely on the implicit definition of an image prior via a denoiser. These algorithms can leverage powerful pre-trained denoisers to solve a wide range of imaging tasks, circumventing the necessity to train models on a per-task basis. Unfortunately, plug-and-play methods often show unstable behaviors, hampering their promise of versatility and leading to suboptimal quality of reconstructed images. In this work, we show that enforcing equivariance to certain groups of transformations (rotations, reflections, and/or translations) on the denoiser strongly improves the stability of the algorithm as well as its reconstruction quality. We provide a theoretical analysis that illustrates the role of equivariance on better performance and stability. We present a simple algorithm that enforces equivariance on any existing denoiser by simply applying a random transformation to the input of the denoiser and the inverse transformation to the output at each iteration of the algorithm. Experiments on multiple imaging modalities and denoising networks show that the equivariant plug-and-play algorithm improves both the reconstruction performance and the stability compared to their non-equivariant counterparts.

Figures (10)

• Figure 1: Instability of algorithms relying on implicit denoising priors can be solved by incorporating equivariance. Enforcing approximate equivariance of the denoiser at test time allows to both stabilize the algorithm and to improve the reconstruction quality without needing to retrain the implicit prior. Left: PnP algorithm applied to an accelerated MRI problem. Middle: Unadjusted Langevin sampling algorithm for a motion blur problem; estimated mean and variance of the associated Markov chain are displayed. Right: RED algorithm on a $4\times$ super-resolution problem.
• Figure 2: Behaviour of the \ref{['eq:pnp_fb']} algorithm with an approximated proximity operator (blue curve) and its equivariant counterpart (red curve). Contour lines show the loss in \ref{['eq:min_pb']} with $r(x)=\|B_1x\|_1$. Stars denote the limit point of each sequence (when it exists) and green dots show the initialization points.
• Figure 3: Motion deblurring on a Set3C sample with \ref{['eq:pnp_fb']} relying on a DRUNet backbone denoiser.
• Figure 4: Average PSNR (left) and convergence criterion $\|x_{k+1}-x_{k}\|/\|x_k\|$ (right) for 3 different imaging problems as a function of the iteration number with different backbone denoisers plugged in the \ref{['eq:pnp_fb']} algorithm. Top row: DnCNN, middle row: DRUNet, bottom row: 1-Lipschitz DnCNN.
• Figure 5: Gaussian deblurring with standard deviation $\sigma = 0.02$ on a BSD10 sample (detail) for different denoising backbone plugged in the (PnP) algorithm.

Theorems & Definitions (6)

• Proposition 1
• proof
• Proposition 2
• proof
• Proposition 3
• proof