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

Tuomo Valkonen

doi:10.1016/j.jmaa.2017.04.037

paper

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

arXiv:1407.2334 · doi:10.1016/j.jmaa.2017.04.037

Abstract

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, $w \in \mbox{BV}_\mbox{loc}(Ω; \mathbb{R}^m)$, instead of just bounded deformation, $w \in \mbox{BD}(Ω)$. 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 $E w$ 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 $\mbox{TGV}^2$ 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 $\mbox{BV}(Ω)$.

Part 1: arXiv 1407.1531 [math.FA]

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

Abstract

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