  We aim at constructing solutions to the minimizing problem for the variant of
Rudin-Osher-Fatemi denoising model with rectilinear anisotropy and to the
gradient flow of its underlying anisotropic total variation functional. We
consider a naturally defined class of functions piecewise constant on
rectangles (PCR). This class forms a strictly dense subset of the space of
functions of bounded variation with an anisotropic norm. The main result shows
that if the given noisy image is a PCR function, then solutions to both
considered problems also have this property. For PCR data the problem of
finding the solution is reduced to a finite algorithm. We discuss some
implications of this result, for instance we use it to prove that continuity is
preserved by both considered problems.
