Total variation denoising of a disk ($u=(1-2\alpha /R{)}_{+}{\chi }_{B(0,R)}$)

For data $g={\chi }_{B(0,R)}$ on ${\mathbb{R}}^2$, the minimizer of $\alpha \mathrm{TV}(u)+\frac{1}{2}\Vert u-g{\Vert }_2^2$ 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 $\mathrm{div}z=(2/R){\chi }_{B(0,R)}$. This subgradient certifies the positive branch; scaling it down certifies the zero branch. The quadratic fidelity is strictly convex, ensuring uniqueness.