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 timeconsuming 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.

In this article, we introduce a new proximal interior point algorithm (PIPA). This algorithm is able to handle convex optimization problems involving various constraints where the objective function is the sum of a Lipschitz differentiable term and a possibly nonsmooth one. Each iteration of PIPA involves the minimization of a merit function evaluated for decaying values of a logarithmic barrier parameter. This inner minimization is performed thanks to a finite number of subiterations of a variable metric forward-backward method employing a line search strategy. The convergence of this latter step as well as the convergence the global method itself are analyzed. The numerical efficiency of the proposed approach is demonstrated in two image processing applications.

No abstract

Image restoration is frequently addressed by resorting to variational methods which account for some prior knowledge about the solution. The success of these methods, however, heavily depends on the estimation of a set of hyperparameters. Deep learning architectures are, on the contrary, very generic and efficient, but they offer limited control over their output. In this paper, we present iRestNet, a neural network architecture which combines the benefits of both approaches. iRestNet is obtained by unfolding a proximal interior point algorithm. This enables enforcing hard constraints on the pixel range of the restored image thanks to a logarithmic barrier strategy, without requiring any parameter setting. Explicit expressions for the involved proximity operator, and its differential, are derived, which allows training iRestNet with gradient descent and backpropagation. Numerical experiments on image deblurring show that the proposed approach provides good image quality results compared to state-of-theart variational and machine learning methods.