Jump-amplitude inequality for total variation denoising ($[u{]}_{{\nu }_u}([f{]}_{{\nu }_u}-[u{]}_{{\nu }_u})\ge 0{\mathcal{H}}^{n-1}\text{-a.e. on }{J}_u$)

For scalar total variation denoising with $f\in BV(\Omega )\cap L^2(\Omega )$ on a bounded Lipschitz domain in arbitrary dimension, the inequality holds with common oriented BV traces on a hypersurface. It says that every surviving jump has the same direction as its data jump and no larger amplitude. Reversing the normal changes both differences and preserves the inequality.

Use residual-preserving clipping of an ROF minimizer and the jump-amplitude inequality for a bounded ROF minimizer. At each finite jump, choose an integer clipping level above both output traces. The clipped function has those same traces, and its residual-adjusted data have the same traces as $f$. Taking a countable union over the integer clipping levels proves the result. This gives the no-new-jumps property of total variation denoising under the full $L^2\cap BV$ assumptions, without asserting regularity of unbounded-forcing perimeter minimizers.

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 $[u]([f]-[u])\ge 0$ 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.

Assume $w\in BV\cap L^{\infty }$ minimizes scalar total variation denoising for $g\in BV\cap L^2$. The data $g$ may be unbounded. For the local flow of $\phi e_j$, use $(1-\theta )w+\theta w\circ {\Phi }_{\pm t}$ as two competitors. The total variation under opposite smooth flows and convexity of the total variation seminorm bound the sum of regularizer changes by $O(t^2)$. Minimality and the opposite-flow fidelity identity for quadratic data, evaluated using the BV jump-product limit with one bounded factor, implyLet $\theta \downarrow 0$. The integrands define finite signed Radon measures, since $w$ is bounded and the jump variation of $g$ is finite. Arbitrary nonnegative smooth $\phi$ make their densities nonnegative. Testing every coordinate direction removes the factor $|{\nu }_w\cdot e_j|$ and proves the stated inequality.