Quaternionic shearlet transform

The shearlet transform has been shown to be a valuable and powerful time-frequency analyzing tool for optics and non-stationary signal processing. In this article, we propose a novel transform called quaternionic shearlet transform which is designed to represent quaternion-valued signals at different scales, locations and orientations. We investigate the fundamental properties of quaternionic shearlet transform including Parseval's formula, Moyal's principle, inversion formula, and characterization of its range using the machinery of quaternion Fourier transform and quaternion convolution. We conclude our investigation by deriving an analogue of the classical Heisenberg-Pauli-Weyl uncertainty inequality and the associated logarithmic version for the quaternionic shearlet transform.