The Orthogonal 2D Planes Split of Quaternions and Steerable Quaternion Fourier Transformations

TL;DR

This paper introduces a new geometric interpretation and generalization of quaternionic Fourier transforms, enabling steerable analysis planes and efficient computation methods for 2D signal processing.

Contribution

It presents a novel orthogonal 2D planes split of quaternions, generalizes it to steerable splits, and develops new steerable quaternion Fourier transform forms with geometric insights and computational advantages.

Findings

Introduced the orthogonal 2D planes split (OPS) of quaternions.

Generalized the OPS to steerable splits with geometric interpretation.

Developed new steerable QFT forms with efficient FFT-based implementation.

Abstract

The two-sided quaternionic Fourier transformation (QFT) was introduced in \cite{Ell:1993} for the analysis of 2D linear time-invariant partial-differential systems. In further theoretical investigations \cite{10.1007/s00006-007-0037-8, EH:DirUP_QFT} a special split of quaternions was introduced, then called $\pm$split. In the current \change{chapter} we analyze this split further, interpret it geometrically as \change{an} \emph{orthogonal 2D planes split} (OPS), and generalize it to a freely steerable split of $\H$ into two orthogonal 2D analysis planes. The new general form of the OPS split allows us to find new geometric interpretations for the action of the QFT on the signal. The second major result of this work is a variety of \emph{new steerable forms} of the QFT, their geometric interpretation, and for each form\change{,} OPS split theorems, which allow fast and efficient numerical implementation with standard FFT software.