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

TL;DR

This paper extends a technique for analyzing the jump set containment property in higher-order regularization models, including TGV, and explores conditions under which the property holds, with applications to infimal convolution TV and TGV approximation.

Contribution

It generalizes the jump set containment analysis to higher-order regularizers like TGV, identifying key conditions and modifications needed for the property to hold.

Findings

Jump set containment holds under specific conditions for higher-order regularizers.

Replacing the symmetrised gradient norm with an $L^p$ norm ($p>1$) enables the property.

The technique applies to infimal convolution TV and TGV strict approximation.

Abstract

In Part 1, we developed a new technique based on Lipschitz pushforwards for proving the jump set containment property $\mathcal{H}^{m-1}(J_u \setminus J_f)=0$ 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, $w \in \mbox{BV}_\mbox{loc}(\Omega; \mathbb{R}^m)$, instead of just bounded deformation, $w \in \mbox{BD}(\Omega)$. 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 $\mbox{TGV}^2$ 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 $\mbox{BV}(\Omega)$.