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Let $u \in \mbox{BV}(\Omega)$ solve the total variation denoising problemwith $L^2$-squared fidelity and data $f$. Caselles et al. [Multiscale Model.Simul. 6 (2008), 879--894] have shown the containment $\mathcal{H}^{m-1}(J_u\setminus J_f)=0$ of the jump set $J_u$ of $u$ in that of $f$. Their proofunfortunately depends heavily on the co-area formula, as do many results inthis area, and as such is not directly extensible to higher-order,curvature-based, and other advanced geometric regularisers, such as totalgeneralised variation (TGV) and Euler's elastica. These have received increasedattention in recent times due to their better practical regularisationproperties compared to conventional total variation or wavelets. We proveanalogous jump set containment properties for a general class of regularisers.We do this with novel Lipschitz transformation techniques, and do not requirethe co-area formula. In the present Part 1 we demonstrate the general techniqueon first-order regularisers, while in Part 2 we will extend it to higher-orderregularisers. In particular, we concentrate in this part on TV and, as anovelty, Huber-regularised TV. We also demonstrate that the technique wouldapply to non-convex TV models as well as the Perona-Malik anisotropicdiffusion, if these approaches were well-posed to begin with.

The jump set under geometric regularisation. Part 1: Basic technique and first-order denoising