Bounded-variation contraction under clipping

For $T(t)=\min (b,\max (a,t))$ and $u\in BV(\Omega )$, $T\circ u\in BV(\Omega )$ and $|D(T\circ u)|(\Omega )\le |Du|(\Omega )$. This is the contraction property of a one-Lipschitz scalar composition, obtainable by smooth approximation and sequential lower semicontinuity. It proves range preservation for total variation denoising when the data lie in $[a,b]$.

For a scalar BV space function, clipping to $[-M,M]$ partitions its coarea formula for BV functions into middle and tail levels. The clipped function has the original superlevel sets at heights in $(-M,M)$. The positive tail levels of $u-T_{M}u$ correspond to original heights above $M$, and its negative tail levels to heights below $-M$. Thus their total variation seminorms add exactly. This is stronger than the triangle inequality and is specific to scalar monotone clipping. It supports residual-preserving clipping of an ROF minimizer.