The Fourier $\mathsf U(2)$ Group and Separation of Discrete Variables

The linear canonical transformations of geometric optics on two-dimensional screens form the group $\mathsf{Sp}(4,\mathfrak R)$, whose maximal compact subgroup is the Fourier group $\mathsf U(2)_\mathrm F$; 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 $\mathsf{so}(4)$. 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.