Adv. Appl. Cliﬀord Algebras (2018) 28:59
2018 Springer International Publishing AG,
part of Springer Nature
Applied Cliﬀord Algebras
Discrete Wavelets with Quaternion and
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,
Keywords. Quaternion wavelets, Cliﬀord wavelets, Wavelet ﬁlter coeﬃ-
cients, Wavelet functions.
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  and .
Wavelets have been generalised to complex wavelets  and matrix val-
ued wavelets  (actually called “vector-valued wavelets”) but, as we found
in our review of the literature , 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. . 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.