Wavelet Analysis of Quaternion-Valued Time-Series

Abstract

In this paper we introduce quaternion-valued multi-resolution analysis. Applying the theory of matrix-valued wavelet analysis, we give the construction of scaling functions and wavelets by identifying the quaternion-valued function with the complex duplex matrix-valued function.

Matrix-Valued and Quaternion Wavelets

“…When applied to a signal whose values are complex-structured matrices, by isomorphism the resulting matrix wavelet transform is equivalent to a complex wavelet transform; see (17). …”

“…Quaternion wavelets are investigated in [5] and [17] by using two di↵erent but equivalent representations of quaternions as structured matrices in C 2⇥2 . Their frequency-domain approach makes use of quaternion Fourier transforms which are poorly suited to the task.…”

“…Their frequency-domain approach makes use of quaternion Fourier transforms which are poorly suited to the task. We can express the Fourier transforms used in He and Yu [17] and in Bahri [5] directly in terms of the quaternion scaling function : R ! H and quaternion scaling sequence {g`}.…”

“…Consequently, the quaternion-valued wavelets in [17] are invalid. With the exception of the Haar filter, none of the scaling filters given are orthogonal to their even shifts.…”

Constructions of Vector‐Valued Filters and Vector‐Valued Wavelets

“…The construction of filter banks is very important in applied aspects. The analogous theory can be extended to the cases of vector-valued and matrix-valued function spaces see [8][9][10][11] . For example, Xia and Suter in 11 proposed vector-valued wavelets and vector filter banks and established a sufficient condition on the matrix-valued filters such that the solution of the corresponding two-scale dilation equation is a matrix-valued scaling function for a matrix-valued multiresolution analysis.…”