  In recent years, large convolutional neural networks have been widely used as
tools for image deblurring, because of their ability in restoring images very
precisely. It is well known that image deblurring is mathematically modeled as
an ill-posed inverse problem and its solution is difficult to approximate when
noise affects the data. Really, one limitation of neural networks for
deblurring is their sensitivity to noise and other perturbations, which can
lead to instability and produce poor reconstructions. In addition, networks do
not necessarily take into account the numerical formulation of the underlying
imaging problem, when trained end-to-end. In this paper, we propose some
strategies to improve stability without losing to much accuracy to deblur
images with deep-learning based methods. First, we suggest a very small neural
architecture, which reduces the execution time for training, satisfying a green
AI need, and does not extremely amplify noise in the computed image. Second, we
introduce a unified framework where a pre-processing step balances the lack of
stability of the following, neural network-based, step. Two different
pre-processors are presented: the former implements a strong parameter-free
denoiser, and the latter is a variational model-based regularized formulation
of the latent imaging problem. This framework is also formally characterized by
mathematical analysis. Numerical experiments are performed to verify the
accuracy and stability of the proposed approaches for image deblurring when
unknown or not-quantified noise is present; the results confirm that they
improve the network stability with respect to noise. In particular, the
model-based framework represents the most reliable trade-off between visual
precision and robustness.
