Quaternion Linear Canonical Wavelet Transform and The Corresponding Uncertainty Inequalities

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.