  A common approach to solve inverse imaging problems relies on finding a
maximum a posteriori (MAP) estimate of the original unknown image, by solving a
minimization problem. In thiscontext, iterative proximal algorithms are widely
used, enabling to handle non-smooth functions and linear operators. Recently,
these algorithms have been paired with deep learning strategies, to further
improve the estimate quality. In particular, proximal neural networks (PNNs)
have been introduced, obtained by unrolling a proximal algorithm as for finding
a MAP estimate, but over a fixed number of iterations, with learned linear
operators and parameters. As PNNs are based on optimization theory, they are
very flexible, and can be adapted to any image restoration task, as soon as a
proximal algorithm can solve it. They further have much lighter architectures
than traditional networks. In this article we propose a unified framework to
build PNNs for the Gaussian denoising task, based on both the dual-FB and the
primal-dual Chambolle-Pock algorithms. We further show that accelerated
inertial versions of these algorithms enable skip connections in the associated
NN layers. We propose different learning strategies for our PNN framework, and
investigate their robustness (Lipschitz property) and denoising efficiency.
Finally, we assess the robustness of our PNNs when plugged in a
forward-backward algorithm for an image deblurring problem.
