  In this paper we present a bilevel optimization scheme for the solution of a
general image deblurring problem, in which a parametric variational-like
approach is encapsulated within a machine learning scheme to provide a high
quality reconstructed image with automatically learned parameters. The
ingredients of the variational lower level and the machine learning upper one
are specifically chosen for the Helsinki Deblur Challenge 2021, in which
sequences of letters are asked to be recovered from out-of-focus photographs
with increasing levels of blur. Our proposed procedure for the reconstructed
image consists in a fixed number of FISTA iterations applied to the
minimization of an edge preserving and binarization enforcing regularized
least-squares functional. The parameters defining the variational model and the
optimization steps, which, unlike most deep learning approaches, all have a
precise and interpretable meaning, are learned via either a similarity index or
a support vector machine strategy. Numerical experiments on the test images
provided by the challenge authors show significant gains with respect to a
standard variational approach and performances comparable with those of some of
the proposed deep learning based algorithms which require the optimization of
millions of parameters.
