  While variational methods have been among the most powerful tools for solving
linear inverse problems in imaging, deep (convolutional) neural networks have
recently taken the lead in many challenging benchmarks. A remaining drawback of
deep learning approaches is their requirement for an expensive retraining
whenever the specific problem, the noise level, noise type, or desired measure
of fidelity changes. On the contrary, variational methods have a plug-and-play
nature as they usually consist of separate data fidelity and regularization
terms.
  In this paper we study the possibility of replacing the proximal operator of
the regularization used in many convex energy minimization algorithms by a
denoising neural network. The latter therefore serves as an implicit natural
image prior, while the data term can still be chosen independently. Using a
fixed denoising neural network in exemplary problems of image deconvolution
with different blur kernels and image demosaicking, we obtain state-of-the-art
reconstruction results. These indicate the high generalizability of our
approach and a reduction of the need for problem-specific training.
Additionally, we discuss novel results on the analysis of possible optimization
algorithms to incorporate the network into, as well as the choices of algorithm
parameters and their relation to the noise level the neural network is trained
on.
