The jump set under geometric regularisation. Part 2: Higher-order approaches

In Part 1, we developed a new technique based on Lipschitz pushforwards for proving the jump set containment property Hm-1(Ju setminus Jf)=0 of solutions u to total variation denoising. We demonstrated that the technique also applies to Huber-regularised TV. Now, in this Part 2, we extend the technique to higher-order regularisers. We are not quite able to prove the property for total generalised variation (TGV) based on the symmetrised gradient for the second-order term. We show that the property holds under three conditions: First, the solution u is locally bounded. Second, the second-order variable is of locally bounded variation, w ∈ mboxBV_mboxloc(Omega; ℝm), instead of just bounded deformation, w ∈ mboxBD(Omega). Third, w does not jump on Ju parallel to it. The second condition can be achieved for non-symmetric TGV. Both the second and third condition can be achieved if we change the Radon (or L1) norm of the symmetrised gradient Ew into an Lp norm, p>1, in which case Korn's inequality holds. We also consider the application of the technique to infimal convolution TV, and study the limiting behaviour of the singular part of D u, as the second parameter of mboxTGV2 goes to zero. Unsurprisingly, it vanishes, but in numerical discretisations the situation looks quite different. Finally, our work additionally includes a result on TGV-strict approximation in mboxBV(Omega).