  The Plug-and-Play (PnP) framework makes it possible to integrate advanced
image denoising priors into optimization algorithms, to efficiently solve a
variety of image restoration tasks generally formulated as Maximum A Posteriori
(MAP) estimation problems. The Plug-and-Play alternating direction method of
multipliers (ADMM) and the Regularization by Denoising (RED) algorithms are two
examples of such methods that made a breakthrough in image restoration.
However, while the former method only applies to proximal algorithms, it has
recently been shown that there exists no regularization that explains the RED
algorithm when the denoisers lack Jacobian symmetry, which happen to be the
case of most practical denoisers. To the best of our knowledge, there exists no
method for training a network that directly represents the gradient of a
regularizer, which can be directly used in Plug-and-Play gradient-based
algorithms. We show that it is possible to train a network directly modeling
the gradient of a MAP regularizer while jointly training the corresponding MAP
denoiser. We use this network in gradient-based optimization methods and obtain
better results comparing to other generic Plug-and-Play approaches. We also
show that the regularizer can be used as a pre-trained network for unrolled
gradient descent. Lastly, we show that the resulting denoiser allows for a
better convergence of the Plug-and-Play ADMM.
