The Continuous quaternion Algebra-Valued Wavelet Transform and the Associated Uncertainty Principle

TL;DR

This paper extends wavelet transforms to quaternion algebra using the quaternion Fourier transform, establishing fundamental properties and uncertainty principles, including Heisenberg-Pauli-Weyl, logarithmic, and Hardy's UPs, in the quaternion domain.

Contribution

It introduces a novel quaternion-valued wavelet transform based on the quaternion Fourier transform and derives associated uncertainty principles, expanding the theoretical framework of quaternion signal analysis.

Findings

Established fundamental properties of the quaternion wavelet transform.

Derived the Heisenberg-Pauli-Weyl uncertainty principle for the quaternion wavelet transform.

Generalized logarithmic and Hardy's uncertainty principles to the quaternion wavelet domain.

Abstract

The purpose of this article is to extend the wavelet transform to quaternion algebra using the kernel of the two-sided quaternion Fourier transform (QFT). We study some fundamental properties of this extension such as scaling, translation, rotation, Parseval's identity, inversion theorem, and a reproducing kernel, then we derive the associated Heisenberg-Pauli-Weyl uncertainty principle UP. Finally, using the quaternion Fourier representation of the CQWT we generalize the logarithmic UP and Hardy's UP to the CQWT domain.