  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.
