Arbitrary unitary rotation of three-dimensional pixellated images

TL;DR

This paper presents a method for performing arbitrary unitary rotations on three-dimensional pixellated images using algebraic techniques, enabling lossless transformations between Cartesian and spherical representations.

Contribution

It introduces a novel algebraic framework for exact, lossless rotation of 3D pixel arrays by leveraging su(3) and so(3) algebraic structures.

Findings

Enables precise rotation of 3D pixel images without information loss

Provides a mathematical basis for transforming between Cartesian and spherical bases

Facilitates concatenation and inversion of rotations in 3D image processing

Abstract

Using the coefficients introduced by Bargmann and Moshinsky for the reduction of the su($3$) algebra of Cartesian three-dimensional oscillator multiplet states into so($3$) angular momentum submultiplets, we implement unitary rotations of three-dimensional Cartesian arrays that form finite pixellated "volume images." Transforming between the Cartesian and spherical bases, the subgroup of rotations in the latter is converted into rotations of the former, allowing for proper concatenation and inversion of these unitary transformations, which entail no loss of information.