ABSTRACT
This chapter presents quaternion Fourier transforms (QFT). The two-sided QFT was first introduced for the analysis of two-dimensional (2D) linear time invariant partial-differential systems. In further theoretical investigations a special split of quaternions was established, then called plus-minus split. Here this split continues to be developed, interpreted geometrically as orthogonal 2D planes split (OPS), and generalized to a freely steerable split of quaternions into two completely orthogonal 2D analysis planes. The new general form of the OPS split allows to find new geometric interpretations for the action of the QFT on quaternion-valued signals. The second major result presented is a variety of new steerable forms of the QFT, their geometric interpretation, and for each form OPS split theorems, which allow fast and efficient numerical implementation with standard FFT software. Then properties of the QFT are studied, like the uncertainty principle, convolution, and a Wiener Khinchine type theorem for the QFT. It is also shown how QFTs can be further specialized to a windowed QFT, a quaternionic Fourier Mellin transform, and generalized to a quaternion domain Fourier transform, and in special relativity to a volume-time Fourier transform, and a spacetime Fourier transform. [191 words]
