Two-dimensional quaternion wavelet transform

In this paper we introduce the continuous quaternion wavelet transform (CQWT). We express the admissibility condition in terms of the (right-sided) quaternion Fourier transform. We show that its fundamental properties, such as inner product, norm relation, and inversion formula, can be established whenever the quaternion wavelets satisfy a particular admissibility condition. We present several examples of the CQWT. As an application we derive a Heisenberg type uncertainty principle for these extended wavelets. 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. [1,2,7,16]). 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. He [21] and Zhao and Peng [23] constructed the continuous quaternion wavelet transform of quaternion-valued functions. They also demonstrated a number of properties of these extended wavelets using the classical Fourier transform (FT). In [6], using the (two-sided) QFT Traversoni proposed a discrete quaternion wavelet transform which was applied by Bayro-Corrochano [12] and Zhou et al. [13]. Recently, in [18,19], we introduced an extension of the WT to Clifford algebra by means of the kernel of the Clifford Fourier transform [8]. The purpose of this paper is to construct the 2-D continuous quaternion wavelet transform (CQWT) based on quaternion algebra. We emphasize that our approach is significantly different from previous work in the definition of the exponential kernel. Our construction uses the kernel of the (right-sided) QFT which in general does not commute with quaternions. The previous papers considered the kernel of the FT which commutes with the quaternions so that the properties of the extension of the WT to quaternion algebra is a quite similar to the classical wavelets. In the present paper we use the (right-sided) QFT to investigate some important properties of the CQWT. Special attention is devoted to inner product, norm relation, and inversion formula. We show that these fundamental properties can be established whenever the admissible quaternion wavelets satisfy a particular admissibility condition. Using the properties of the CQWT and the uncertainty principle for the (right-sided) QFT [16] we establish an uncertainty principle for the CQWT.