Quaternion and Clifford Fourier Transforms and Wavelets

Abstract

Abstract. We survey the historical development of quaternion and Clifford Fourier transforms and wavelets. Keywords. Quaternions, Clifford Algebra, Fourier Transforms, Wavelet Transforms.The development of hypercomplex Fourier transforms and wavelets has taken place in several different threads, reflected in the overview of the subject presented in this chapter. We present in Section 1 an overview of the development of quaternion Fourier transforms, then in Section 2 the development of Clifford Fourier transforms. Finally, since wavelets are a more recent development, and the distinction between their quaternion and Clifford algebra approach has been much less pronounced than in the case of Fourier transforms, Section 3 reviews the history of both quaternion and Clifford wavelets.We recognise that the history we present here may be incomplete, and that work by some authors may have been overlooked, for which we can only offer our humble apologies. [90,91] on Clifford Fourier and Laplace transforms further explained in Section 2.2. Ernst and Delsuc's quaternion transforms were two-dimensional (that is they had two independent variables) and proposed for application to nuclear magnetic resonance (NMR) imaging. Written in terms of two independent time variables 1 t 1 and t 2 , the forward transforms 1 The two independent time variables arise naturally from the formulation of twodimensional NMR spectroscopy. Quaternion Fourier Transforms (QFT)1

Discrete octonion Fourier transform and the analysis of discrete 3-D data

“…commutativity and associativity. Non-commutativity is also encountered with Fourier transforms based on quaternion algebra and (in general) Clifford algebras (Brackx et al 2013). In order to deal with these problems, we have defined the algebra of quadruplecomplex numbers (based on the double-complex numbers introduced by Ell (1993); Kurman (1958)).…”

“…As in the case of classical signal processing, so the discrete counterpart of this theory has so far mainly focused on signals with real and complex values, as well as their complex spectra. In recent years, however, more and more works have started to appear, which authors use in their research hypercomplex algebras, among others quaternions and octonions (Brackx et al 2013;Hahn and Snopek 2016;Lian 2019;Snopek 2015;Wang et al 2017). The area of applications is focused so far on the study of neural networks (Popa 2016(Popa , 2018, analysis of color and multispectral images (Ell et a.…”

On harmonic analysis of vector‐valued signals

“…This relies on the generalisation of Euler's formula exp iθ = cos θ + i sin θ, i 2 = −1, which applies for complex numbers, to a more general case exp ♦θ = cos θ + ♦ sin θ where ♦ represents a square root of −1, that is ♦ 2 = −1, and the exponential function is defined for an argument consisting of ♦ multiplied by a real scalar θ. In the case of hypercomplex algebras, ♦ would be an element of the algebra (for examples, see [11,12]), but in general it could be something else, for example a matrix as in [10]. The 'scaling' concept then includes (but often not in an easily understood manner) a change of orientation of the exponential as discussed above, in order to construct an arbitrarily-oriented ellipse in N dimensions by adding two (or more) exponentials rotating in opposite senses.…”

“…But more significantly, it is possible to define transforms with more than one hypercomplex exponential (for example, one each side of the signal, or two different exponentials on the same side, or multiple different exponentials on each side). For a discussion of many possibilities, see [13]. Interpretation of a transform in which exponentials are arranged on both sides of the signal function is not simple.…”

“…In this approach, the signal values may be represented by vectors in the linear algebra sense (that is a degenerate matrix with one row or column), and the exponential by a matrix exponential with a matrix root of −1, that is a matrix that squares to give a negated identity matrix. The formulation is as given in (13) except that i is replaced by a matrix. Unlike the case of hypercomplex algebras, where the dimension N is limited to powers of two, real matrix roots of −1 exist for other even values of N (but not for odd values of N [10, Theorem 2, p 651]).…”