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 TGV2 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.
ASJC Scopus subject areas
- Statistik und Wahrscheinlichkeit
- Modellierung und Simulation
- Physik der kondensierten Materie
- Maschinelles Sehen und Mustererkennung
- Geometrie und Topologie
- Angewandte Mathematik