  Let $u \in \mbox{BV}(\Omega)$ solve the total variation denoising problem
with $L^2$-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 proof
unfortunately depends heavily on the co-area formula, as do many results in
this area, and as such is not directly extensible to higher-order,
curvature-based, and other advanced geometric regularisers, such as total
generalised variation (TGV) and Euler's elastica. These have received increased
attention in recent times due to their better practical regularisation
properties compared to conventional total variation or wavelets. We prove
analogous jump set containment properties for a general class of regularisers.
We do this with novel Lipschitz transformation techniques, and do not require
the co-area formula. In the present Part 1 we demonstrate the general technique
on first-order regularisers, while in Part 2 we will extend it to higher-order
regularisers. In particular, we concentrate in this part on TV and, as a
novelty, Huber-regularised TV. We also demonstrate that the technique would
apply to non-convex TV models as well as the Perona-Malik anisotropic
diffusion, if these approaches were well-posed to begin with.
