  We propose an automatic parameter selection strategy for the problem of image
super-resolution for images corrupted by blur and additive white Gaussian noise
with unknown standard deviation. The proposed approach exploits the structure
of both the down-sampling and the blur operators in the frequency domain and
computes the optimal regularisation parameter as the one optimising a suitable
residual whiteness measure. Computationally, the proposed strategy relies on
the fast solution of generalised Tikhonov $\ell_2$-$\ell_2$ problems as
proposed in a work from Zhao et al. These problems naturally appear as substeps
of the Alternating Direction Method of Multipliers (ADMM) optimisation approach
used to solve super-resolution problems with non-quadratic and often
non-smooth, sparsity-promoting regularisers both in convex and in non-convex
regimes. After detailing the theoretical properties defined in the frequency
domain which allow to express the whiteness functional in a compact way, we
report an exhaustive list of numerical experiments proving the effectiveness of
the proposed approach for different type of problems, in comparison with
well-known parameter selection strategy such as, e.g., the discrepancy
principle.
