The Quaternion Boostlet Transform: Definition, Properties and Uncertainty Principles

Abstract

In this article, we introduce the notion of Quaternion Boostlet Transform (QBT), a hypercomplex framework designed to unify the analysis of multi-component wavefields by merging the algebraic richness of quaternions with the relativistic, hyperbolic geometry of the boostlet system. By treating coupled physical phenomena such as acoustic pressure with particle velocity or orthogonally polarized elastic displacements as single quaternion-valued entities, the QBT preserves intrinsic geometric correlations that are typically lost in component-wise processing. We also establish a rigorous mathematical foundation for the transform, including the admissibility condition, a convolution-based representation, a Plancherel theorem for energy conservation, and an explicit inversion formula ensuring perfect signal reconstruction. Furthermore, the work derives a comprehensive set of uncertainty principles namely Heisenberg, logarithmic, and Pitt's inequalities that define the precise localization constraints of QBT coefficients in the augmented phase space. The theoretical development is substantiated with illustrative examples, wherein the QBT is applied to a quaternion-valued plane wave featuring coupled pressure-velocity components and to a Gaussian-modulated circularly polarized elastic wave packet. These examples demonstrate how the transform naturally encodes wavefront orientation and polarization state through quaternion phase, offering a physically coherent and sparse dictionary for vector-valued wavefield analysis in acoustics and seismology.