Quaternion Linear Canonical Wavelet Transform and The Corresponding Uncertainty Inequalities

TL;DR

This paper introduces a quaternion linear canonical wavelet transform for two-dimensional quaternion signals, extending wavelet analysis with new properties and uncertainty inequalities, useful in optics and signal processing.

Contribution

It proposes a novel quaternion wavelet transform that incorporates linear canonical domain features and establishes its fundamental properties and uncertainty inequalities.

Findings

Derived Parseval's formula and energy conservation for the transform

Established inversion formula and range characterization

Proved uncertainty inequalities analogous to classical results

Abstract

The linear canonical wavelet transform has been shown to be a valuable and powerful time-frequency analyzing tool for optics and signal processing. In this article, we propose a novel transform called quaternion linear canonical wavelet transform which is designed to represent two dimensional quaternion-valued signals at different scales, locations and orientations. The proposed transform not only inherits the features of quaternion wavelet transform but also has the capability of signal representation in quaternion linear canonical domain. We investigate the fundamental properties of quaternion linear canonical wavelet transform including Parseval's formula, energy conservation, inversion formula, and characterization of its range using the machinery of quaternion linear canonical transform and its convolution. We conclude our investigation by deriving an analogue of the classical Heisenberg-Pauli-Weyl uncertainty inequality and the associated logarithmic and local versions for the quaternion linear canonical wavelet transform.