  Ground-based solar image restoration is a computationally expensive procedure
that involves nonlinear optimization techniques. The presence of atmospheric
turbulence produces perturbations in individual images that make it necessary
to apply blind deconvolution techniques. These techniques rely on the
observation of many short exposure frames that are used to simultaneously infer
the instantaneous state of the atmosphere and the unperturbed object. We have
recently explored the use of machine learning to accelerate this process, with
promising results. We build upon this previous work to propose several
interesting improvements that lead to better models. As well, we propose a new
method to accelerate the restoration based on algorithm unrolling. In this
method, the image restoration problem is solved with a gradient descent method
that is unrolled and accelerated aided by a few small neural networks. The role
of the neural networks is to correct the estimation of the solution at each
iterative step. The model is trained to perform the optimization in a small
fixed number of steps with a curated dataset. Our findings demonstrate that
both methods significantly reduce the restoration time compared to the standard
optimization procedure. Furthermore, we showcase that these models can be
trained in an unsupervised manner using observed images from three different
instruments. Remarkably, they also exhibit robust generalization capabilities
when applied to new datasets. To foster further research and collaboration, we
openly provide the trained models, along with the corresponding training and
evaluation code, as well as the training dataset, to the scientific community.
