Quaternion Algebra-Valued Wavelet Transform

The quaternion Fourier transform (QFT), which is a nontrivial generalization of the real and complex Fourier transform (FT) using quaternion algebra has been of interest to researchers for some years (see e.g. [3, 5]). It was found that many FT properties still hold but others have to be modified. Based on the (right-sided) QFT, one can extend the classical wavelet transform (WT) to quaternion algebra while enjoying the same properties as in the classical case. In [10], using the (two-sided) QFT Traversoni proposed a discrete quaternion wavelet transform which was applied by Bayro-Corrochano [2] and Zhou et al. [11]. In [6, 8], we introduced an extension of the WT to Clifford algebra by means of the kernel of the Clifford Fourier transform [4]. The purpose of this paper is to provide an alternative proof for inner product relation property of the two-dimensional continuous quaternion wavelet transform (CQWT), which was recently studied in [7]. This property is very important to derive the inversion formula for the CQWT.