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

The linear canonical transformations of geometric optics on two-dimensional screens form the group Sp(4,<), 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 points placed along Cartesian or polar (radius and angle) coordinates, thus realizing one case of separation in two discrete coordinates. The N-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.