  We consider a variational model for single-image super-resolution based on
the assumption that the gradient of the target image is sparse. We enforce this
assumption by considering both an isotropic and an anisotropic $\ell^0$
regularisation on the image gradient combined with a quadratic data fidelity,
similarly as studied in [1] for general signal recovery problems. For the
numerical realisation of the model, we propose a novel efficient ADMM splitting
algorithm whose substeps solutions are computed efficiently by means of
hard-thresholding and standard conjugate-gradient solvers. We test our model on
highly-degraded synthetic and real-world data and quantitatively compare our
results with several variational approaches as well as with state-of-the-art
deep-learning techniques. Our experiments show that $\ell^0$
gradient-regularised super-resolved images can be effectively used to improve
the accuracy of standard segmentation algorithms when applied to QR and cell
detection, and landcover classification problems, in comparison to the results
achieved by other approaches.
