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

Abstract

In Part 1, we developed a new technique based on Lipschitz pushforwards for proving the jump set containment property Hm−1(Ju∖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∈BVloc(Ω;Rm), instead of just bounded deformation, w∈BD(Ω). 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 Lπ norm, p>1, in which case Korn's inequality holds. On the positive side, we verify the jump set containment property for second-order infimal convolution TV (ICTV) in dimension m=2. We also study the limiting behaviour of the singular part of Du, as the second parameter of TGV2 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 BV(Ω).