  The linear canonical transformations of geometric optics on two-dimensional
screens form the group $Sp(4,R)$, whose maximal compact subgroup is the Fourier
group $U(2)_F$; this includes isotropic and anisotropic Fourier transforms,
screen rotations and gyrations in the phase space of ray positions and optical
momenta. Deforming classical optics into a Hamiltonian system whose positions
and momenta range over a finite set of values, leads us to the finite
oscillator model, which is ruled by the Lie algebra $so(4)$. Two distinct
subalgebra chains are used to model arrays of $N^2$ points placed along
Cartesian or polar (radius and angle) coordinates, thus realizing one case of
separation in two discrete coordinates. The $N^2$-vectors in this space are
digital (pixellated) images on either of these two grids, related by a unitary
transformation. Here we examine the unitary action of the analogue Fourier
group on such images, whose rotations are particularly visible.
