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

TL;DR

This paper explores the Fourier U(2) group and its role in separating discrete variables in digital images, linking geometric optics transformations to finite oscillator models and their applications in image processing.

Contribution

It introduces a novel framework connecting the Fourier U(2) group with finite oscillator models for discrete variable separation in digital images.

Findings

Unitary transformations relate Cartesian and polar pixel grids.

Rotations in the Fourier group are visually evident in image transformations.

Finite oscillator models effectively represent discrete optical phase space.

Abstract

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.