Total variation denoising in l1 anisotropy

We aim at constructing solutions to the minimizing problem for the variant of the Rudin-Osher-Fatemi denoising model with rectilinear anisotropy and to the gradient flow of its underlying aniso-tropic 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