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We introduce a novel regulariser based on the natural vector field operationsgradient, divergence, curl and shear. For suitable choices of the weightingparameters contained in our model it generalises well-known first- andsecond-order TV-type regularisation methods including TV, ICTV and TGV$^2$ andenables interpolation between them. To better understand the influence of eachparameter, we characterise the nullspaces of the respective regularisationfunctionals. Analysing the continuous model, we conclude that it is notsufficient to combine penalisation of the divergence and the curl to achievehigh-quality results, but interestingly it seems crucial that the penaltyfunctional includes at least one component of the shear or suitable boundaryconditions. We investigate which requirements regarding the choice of weightingparameters yield a rotational invariant approach. To guarantee physicallymeaningful reconstructions, implying that conservation laws for vectorialdifferential operators remain valid, we need a careful discretisation that wetherefore discuss in detail.

Unified Models for Second-Order TV-Type Regularisation in Imaging - A New Perspective Based on Vector Operators