In the two space dimensions of screens in optical sy stems, rotations, gyrations, and fractional Fourier transformations form the Fourier subgroup of the symplectic group of linear canonical transformations: U(2) F \(\subset\) Sp(4,R). Here we study the action of this Fourier group on pixellated images within generic rectangular \(N_x\) \(\times\) \(N_y\) screens; its elements here compose properly and act unitarily, i.e., without loss of information.