= No-new-jumps property of total variation denoising
{title2=$\mathcal H^{n-1}(J_u\setminus J_f)=0$}
For scalar <total variation denoising> with $f\in BV(\Omega)\cap L^2(\Omega)$ on a bounded <Lipschitz domain> in arbitrary dimension, the reconstructed <jump set of a bounded-variation function> lies in the data <jump set of a bounded-variation function> up to a surface-null set. The <jump-amplitude inequality for total variation denoising> proves this: outside $J_f$, the two <BV traces on a hypersurface> of $f$ agree, so $[u]([f]-[u])\ge0$ forces $[u]=0$. Surviving jumps cannot be stronger than the corresponding data jumps and must have the same orientation. Existing jumps may disappear.
The proof uses <scalar total variation splitting under clipping>, <residual-preserving clipping of an ROF minimizer>, <total variation under opposite smooth flows> and the <BV jump-product limit with one bounded factor>. It requires neither bounded data nor bounded output. The bounded-forcing geometric proof through <noncontact of ROF level boundaries> remains an alternative for bounded images; it is not used to infer regularity under unbounded forcing.
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