A novel wavelet transform in the quaternion quadratic-phase domain
type: Item
instance of: scholarly article
title: A novel wavelet transform in the quaternion quadratic-phase domain
author: Mohammad Younus Bhat
author: Osama Abdulaziz Alamri
author: Q6179837
published in: International Journal of Wavelets, Multiresolution and Information Processing
publication date: 2024-08-22T00:00:00Z
review text: The article presents a novel mathematical tool, the quaternion quadratic-phase wavelet transform (QQPWLT), building on the foundations of quaternion algebra and the quadratic-phase Fourier transform. The QQPWLT is rigorously defined as an integral transform that combines the quaternion quadratic-phase Fourier transform's global kernel with the wavelet transform's localization capabilities. Mathematically, the QQPWLT for a quaternion-valued function \(f \in L^2(\mathbb{R}^2, \mathbb{H})\) with respect to a quaternion wavelet \(\psi \in L^2(\mathbb{R}^2, \mathbb{H})\) is given by:\N\[\N\mathbb{T}_{\psi}^\mathbb{H}[f](a,\mathfrak{y}, \theta) = \langle f, \Psi_{a, \mathfrak{y}, \theta}^{i,j} \rangle = \int_{\mathbb{R}^2} f(\mathfrak{x}) \overline{\Psi_{a, \mathfrak{y}, \theta}^{i,j}(\mathfrak{x})} \, d\mathfrak{x},\N\]\Nwhere \(\Psi_{a, \mathfrak{y}, \theta}^{i,j}(\mathfrak{x})\) is a quaternion quadratic-phase wavelet with \(a\in\mathbb{R}^{+},\,\mathfrak{y}\in\mathbb{R}^2\) and \(0\le\theta\le 2\pi\). The paper establishes essential properties of the QQPWLT, such as the orthogonality relation and the inversion formula. Additionally, it introduces Heisenberg-Pauli-Weyl inequality and the logarithmic uncertainty principles pertinent to the QQPWLT, which establish constraints on signal localization in both the time and frequency domains. Through illustrative examples, the authors substantiate the proposed transform and its theoretical implications, demonstrating its relevance in the field of signal processing.
reviewed by: Nikhil Khanna
Mathematics Subject Classification ID: 42A05
Mathematics Subject Classification ID: 42C20
Mathematics Subject Classification ID: 42B10
Mathematics Subject Classification ID: 42C40
Mathematics Subject Classification ID: 46S10
zbMATH DE Number: 7900489
zbMATH Keywords: quaternion quadratic-phase Fourier transform
zbMATH Keywords: convolution
zbMATH Keywords: orthogonality
zbMATH Keywords: inversion
zbMATH Keywords: uncertainty principle
MaRDI profile type: Publication
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DOI: 10.1142/S0219691324500024
description: scientific article; zbMATH DE number 7900489