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

The two-sided quaternionic Fourier transformation (QFT) was introduced in [2] for the analysis of 2D linear time-invariant partialdifferential systems. In further theoretical investigations [4, 5] a special split of quaternions was introduced, then called ±split. In the current chapter we analyze this split further, interpret it geometrically as an 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 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.