= Residual-preserving clipping of an ROF minimizer
{title2=$w=T_Mu,\quad g=f-u+w,\quad w\in\arg\min_v\{\alpha\operatorname{TV}(v)+\tfrac12\|v-g\|_2^2\}$}
If $u$ minimizes scalar <total variation denoising> with $f\in BV\cap L^2$, set $w=T_Mu$, $r=u-w$ and $g=f-r$. The <scalar total variation splitting under clipping> gives $\operatorname{TV}(u)=\operatorname{TV}(w)+\operatorname{TV}(r)$. Compare $u$ with $v+r$ and use $\operatorname{TV}(v+r)\le\operatorname{TV}(v)+\operatorname{TV}(r)$; cancelling the tail shows that $w$ minimizes the same model for $g$. It is bounded while $g$ may be unbounded, and $w-g=u-f$ exactly. This reduction concerns the actual <minimizer>, rather than convergence of <minimizers> for truncated data.
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