  Plug-and-Play (PnP) and Regularization-by-Denoising (RED) are recent
paradigms for image reconstruction that leverage the power of modern denoisers
for image regularization. In particular, they have been shown to deliver
state-of-the-art reconstructions with CNN denoisers. Since the regularization
is performed in an ad-hoc manner, understanding the convergence of PnP and RED
has been an active research area. It was shown in recent works that iterate
convergence can be guaranteed if the denoiser is averaged or nonexpansive.
However, integrating nonexpansivity with gradient-based learning is
challenging, the core issue being that testing nonexpansivity is intractable.
Using numerical examples, we show that existing CNN denoisers tend to violate
the nonexpansive property, which can cause PnP or RED to diverge. In fact,
algorithms for training nonexpansive denoisers either cannot guarantee
nonexpansivity or are computationally intensive. In this work, we construct
contractive and averaged image denoisers by unfolding splitting-based
optimization algorithms applied to wavelet denoising and demonstrate that their
regularization capacity for PnP and RED can be matched with CNN denoisers. To
our knowledge, this is the first work to propose a simple framework for
training contractive denoisers using network unfolding.
