Adv. Appl. Clifford Algebras (2018) 28:59
c
2018 Springer International Publishing AG,
part of Springer Nature
https://doi.org/10.1007/s00006-018-0876-5
Advances in
Applied Clifford Algebras
Discrete Wavelets with Quaternion and
Clifford Coefficients
Peter Fletcher
∗
Abstract. In this paper we use matrix representations of quaternions and
Clifford algebras and solve the same matrix equations in each case to find
Daubechies quaternion and Clifford scaling filters. We use paraunitary
completion of the polyphase matrix to find corresponding quaternion
and Clifford wavelet filters. We then use the cascade algorithm on our
filters to find quaternion and Clifford scaling and wavelet functions,
which we illustrate using all possible projections onto two and three
dimensions: to our knowledge, this is the first time that this has been
done. We discuss the shapes of these functions and conclude with a
consideration of what we could actually do with our filters.
Mathematics Subject Classification. Primary 42C40; Secondary 11R52,
15A66, 65T60.
Keywords. Quaternion wavelets, Clifford wavelets, Wavelet filter coeffi-
cients, Wavelet functions.
1. Introduction
A wavelet transform gives the frequencies present in a signal and when or
where those frequencies occurred; wavelets are the basis functions onto which
the signal is projected. The literature on real wavelets and wavelet transforms
is extensive, but two good introductory sources are [22] and [12].
Wavelets have been generalised to complex wavelets [13] and matrix val-
ued wavelets [24] (actually called “vector-valued wavelets”) but, as we found
in our review of the literature [6], the vast majority of practical quaternion
wavelet transforms make very little use of the properties of quaternions and
their use is limited to the analysis of monochrome images, e.g. [2]. However,
we did find one author, Ginzberg, who in his PhD thesis [8, ch. 5] used a
matrix representation of quaternions and found the coefficients of a length
10 discrete scaling filter and a related discrete wavelet filter with quaternion
coefficients. We explain these terms in the next section.
∗
Corresponding author.
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