3 research outputs found
The Fourier U(2) Group and Separation of Discrete Variables
The linear canonical transformations of geometric optics on two-dimensional
screens form the group , whose maximal compact subgroup is the Fourier
group ; 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 . Two distinct
subalgebra chains are used to model arrays of points placed along
Cartesian or polar (radius and angle) coordinates, thus realizing one case of
separation in two discrete coordinates. The -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
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,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 N² points placed along Cartesian or polar (radius and angle) coordinates, thus realizing one case of separation in two discrete coordinates. The N2-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