The jump set under geometric regularisation. Part 2: Higher-order approaches
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In Part 1, we developed a new technique based on Lipschitz pushforwards for proving the jump set containment property H^m-1(J_u ∖ J_f)=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 ∈BV_loc(Ω; R^m), instead of just bounded deformation, w ∈BD(Ω). Third, w does not jump on J_u 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 L^1) norm of the symmetrised gradient Ew into an L^p 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 TGV^2 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).
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