Fast Algorithms for the Hypercomplex Fourier Transforms

In multi-dimensional signal processing the Cliiord Fourier transform (CFT or in the 2-D case: quater-nionic Fourier transform/QFT) is a consequent extension of the complex valued Fourier transform. Hence, we need a fast algorithm in order to compute the transform in practical applications. Since the CFT is based on a corresponding Cliiord algebra (CA) and CAs are not commutative in general, we cannot simply apply the n-dimensional decima-tion method as in the complex case. We propose a solution for this problem for real signals. The idea is to embed the CFT in a diierent algebra which is isomorphic to the m-fold Cartesian product of the complex numbers. Since this approach only works for real signals, we have to develop a diier-ent method for Cliiord valued signals. We present two ways of calculating the CFT: one using the just mentioned transform for real signals and one approach using 1-D FFTs similar to row-column transforms in the 2-D case. All described algorithms are explicitly formulated for the 3-D case (8-D CA) and the asymptotic complexities are calculated. On modern computers all oating point operations except for divisions and square-roots are performed in the same time. Therefore, we always consider the whole number of oating point operations .