Averaged Deep Denoisers for Image Regularization

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.