Hypercomplex spectral transformations
Hypercomplex spectral transformations
Fast algorithms of hypercomplex Fourier transforms
Geometric computing with Clifford algebras
Multi-Dimensional Signal Processin Using an Algebraically Extended Signal Representation
AFPAC '97 Proceedings of the International Workshop on Algebraic Frames for the Perception-Action Cycle
The trinion Fourier transform of color images
Signal Processing
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In this article, we deal with fast algorithms for the quaternionic Fourier transform (QFT). Our aim is to give a guideline for choosing algorithms in practical cases. Hence, we are not only interested in the theoretic complexity but in the real execution time of the implementation of an algorithm. This includes floating point multiplications, additions, index computations and the memory accesses. We mainly consider two cases: the QFT of a real signal and the QFT of a quaternionic signal. For both cases it follows that the row-column method yields very fast algorithms. Additionally, these algorithms are easy to implement since one can fall back on standard algorithms for the fast Fourier transform and the fast Hartley transform. The latter is the optimal choice for real signals since there is no redundancy in the transform. We take advantage of the fact that each complete transform can be converted into another complete transform. In the case of the complex Fourier transform, the Hartley transform, and the QFT, the conversions are of low complexity. Hence, the QFT of a real signal is optimally calculated using the Hartley transform.