= Total variation denoising of a disk
{title2=$u=(1-2\alpha/R)_+\chi_{B(0,R)}$}
For data $g=\chi_{B(0,R)}$ on $\mathbb R^2$, the minimizer of $\alpha\operatorname{TV}(u)+\tfrac12\|u-g\|_2^2$ is the displayed amplitude shrinkage, including extinction at $\alpha=R/2$. The <total variation calibration> uses $z(x)=x/R$ inside the disk and $z(x)=Rx/|x|^2$ outside. Its <continuous> normal component creates no boundary measure, and $\operatorname{div}z=(2/R)\chi_{B(0,R)}$. This <subgradient> certifies the positive branch; scaling it down certifies the zero branch. The quadratic fidelity is <strictly convex>, ensuring uniqueness.
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