A novel wavelet transform in the quaternion quadratic-phase domain (Q6591705)

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.