  Variational methods are widely applied to ill-posed inverse problems for they
have the ability to embed prior knowledge about the solution. However, the
level of performance of these methods significantly depends on a set of
parameters, which can be estimated through computationally expensive and
time-consuming methods. In contrast, deep learning offers very generic and
efficient architectures, at the expense of explainability, since it is often
used as a black-box, without any fine control over its output. Deep unfolding
provides a convenient approach to combine variational-based and deep learning
approaches. Starting from a variational formulation for image restoration, we
develop iRestNet, a neural network architecture obtained by unfolding a
proximal interior point algorithm. Hard constraints, encoding desirable
properties for the restored image, are incorporated into the network thanks to
a logarithmic barrier, while the barrier parameter, the stepsize, and the
penalization weight are learned by the network. We derive explicit expressions
for the gradient of the proximity operator for various choices of constraints,
which allows training iRestNet with gradient descent and backpropagation. In
addition, we provide theoretical results regarding the stability of the network
for a common inverse problem example. Numerical experiments on image deblurring
problems show that the proposed approach compares favorably with both
state-of-the-art variational and machine learning methods in terms of image
quality.
