Discrete isotropic total variation ($\Vert Du{\Vert }_{2,1}$)

With zero forward differences at the grid boundary, discrete isotropic total variation is the sum of the Euclidean lengths of the two-component forward gradients. Its dual constraint is the product of unit Euclidean balls, and minimizing $\frac{1}{2}\Vert g-D^{\ast }p{\Vert }_2^2$ over that product yields $u_{\ast }=g-D^{\ast }{p}_{\ast }$. The exact difference adjoint operator is essential. A dual proximal gradient method projects $p+\tau D(g-D^{\ast }p)$ pointwise onto unit balls; $0<\tau \le 1/8$ is safe on an unscaled square grid.