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We propose an automatic parameter selection strategy for the problem of imagesuper-resolution for images corrupted by blur and additive white Gaussian noisewith unknown standard deviation. The proposed approach exploits the structureof both the down-sampling and the blur operators in the frequency domain andcomputes the optimal regularisation parameter as the one optimising a suitableresidual whiteness measure. Computationally, the proposed strategy relies onthe fast solution of generalised Tikhonov $\ell_2$-$\ell_2$ problems asproposed in a work from Zhao et al. These problems naturally appear as substepsof the Alternating Direction Method of Multipliers (ADMM) optimisation approachused to solve super-resolution problems with non-quadratic and oftennon-smooth, sparsity-promoting regularisers both in convex and in non-convexregimes. After detailing the theoretical properties defined in the frequencydomain which allow to express the whiteness functional in a compact way, wereport an exhaustive list of numerical experiments proving the effectiveness ofthe proposed approach for different type of problems, in comparison withwell-known parameter selection strategy such as, e.g., the discrepancyprinciple.

ADMM-based residual whiteness principle for automatic parameter selection in super-resolution problems