Unitary rotation and gyration of pixellated images on rectangular screens

TL;DR

This paper investigates how the Fourier subgroup of the symplectic group acts on pixellated images on rectangular screens, ensuring transformations like rotations and gyrations are unitary and information-preserving.

Contribution

It characterizes the unitary action of the Fourier group on pixellated images in rectangular screens, extending the understanding of linear canonical transformations in digital image processing.

Findings

The Fourier group acts unitarily on pixellated images.

Transformations preserve information without loss.

Application to optical systems and image manipulation.

Abstract

In the two space dimensions of screens in optical sy stems, rotations, gyrations, and fractional Fourier transformations form the Fourier subgroup of the symplectic group of linear canonical transformations: U(2) F $\subset$ Sp(4,R). Here we study the action of this Fourier group on pixellated images within generic rectangular $N_x$ $\times$ $N_y$ screens; its elements here compose properly and act unitarily, i.e., without loss of information.