  Image restoration remains a challenging task in image processing. Numerous
methods tackle this problem, often solved by minimizing a non-smooth penalized
co-log-likelihood function. Although the solution is easily interpretable with
theoretic guarantees, its estimation relies on an optimization process that can
take time. Considering the research effort in deep learning for image
classification and segmentation, this class of methods offers a serious
alternative to perform image restoration but stays challenging to solve inverse
problems. In this work, we design a deep network, named DeepPDNet, built from
primal-dual proximal iterations associated with the minimization of a standard
penalized likelihood with an analysis prior, allowing us to take advantage of
both worlds.
  We reformulate a specific instance of the Condat-Vu primal-dual hybrid
gradient (PDHG) algorithm as a deep network with fixed layers. The learned
parameters are both the PDHG algorithm step-sizes and the analysis linear
operator involved in the penalization (including the regularization parameter).
These parameters are allowed to vary from a layer to another one. Two different
learning strategies: "Full learning" and "Partial learning" are proposed, the
first one is the most efficient numerically while the second one relies on
standard constraints ensuring convergence in the standard PDHG iterations.
Moreover, global and local sparse analysis prior are studied to seek a better
feature representation. We apply the proposed methods to image restoration on
the MNIST and BSD68 datasets and to single image super-resolution on the BSD100
and SET14 datasets. Extensive results show that the proposed DeepPDNet
demonstrates excellent performance on the MNIST and the more complex BSD68,
BSD100, and SET14 datasets for image restoration and single image
super-resolution task.
