  In this paper we study a variant to Chan-Vese image segmentation model with
rectilinear anisotropy. We show existence of minimizers in the $2$-phases case
and how they are related to the (anisotropic) Rudin-Osher-Fatemi denoising
model (ROF). Our analysis shows that in the natural case of a piecewise
constant on rectangles image (PCR function in short), there exists a minimizer
of the Chan-Vese functional which is also piecewise constant on rectangles over
the same grid that the one defined by the original image. In the multiphase
case, we show that minimizers of the Chan-Vese multiphase functional also share
this property in the case that the initial image is a PCR function. We also
investigate a multiphase and anisotropic version of the Truncated ROF
algorithm, and we compare the solutions given by this algorithm with minimizers
of the multiphase anisotropic Chan-Vese functional.
