Bounded-variation contraction under clipping
= 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 $[a,b]$.