  A recent result by Lasica, Moll and Mucha about the $\ell^1$-anisotropic
Rudin-Osher-Fatemi model in $\mathbb{R}^2$ asserts that the solution is
piecewise constant on a rectilinear grid, if the datum is. By means of a new
proof we extend this result to $\mathbb{R}^n$. The core of our proof consists
in showing that averaging operators associated to certain rectilinear grids map
subgradients of the $\ell^1$-anisotropic total variation seminorm to
subgradients.
