  We introduce a novel regulariser based on the natural vector field operations
gradient, divergence, curl and shear. For suitable choices of the weighting
parameters contained in our model it generalises well-known first- and
second-order TV-type regularisation methods including TV, ICTV and TGV$^2$ and
enables interpolation between them. To better understand the influence of each
parameter, we characterise the nullspaces of the respective regularisation
functionals. Analysing the continuous model, we conclude that it is not
sufficient to combine penalisation of the divergence and the curl to achieve
high-quality results, but interestingly it seems crucial that the penalty
functional includes at least one component of the shear or suitable boundary
conditions. We investigate which requirements regarding the choice of weighting
parameters yield a rotational invariant approach. To guarantee physically
meaningful reconstructions, implying that conservation laws for vectorial
differential operators remain valid, we need a careful discretisation that we
therefore discuss in detail.
