Convolution, Correlation, and Uncertainty Principles for the Quaternion Offset Linear Canonical Transform

Abstract

Quaternion Fourier transform (QFT) has gained significant attention in recent years due to its effectiveness in analyzing multi-dimensional signals and images. This article introduces two-dimensional (2D) right-sided quaternion offset linear canonical transform (QOLCT), which is the most general form of QFT with additional free parameters. We explore the properties of 2D right-sided QOLCT, including inversion and Parseval formulas, besides its relationship with other transforms. We also examine the convolution and correlation theorems of 2D right-sided QOLCT, followed by several uncertainty principles. Additionally, we present an illustrative example of the proposed transform, demonstrating its graphical representation of a given signal and its transformed signal. Finally, we demonstrate an application of QOLCT, where it can be utilized to generalize the treatment of swept-frequency filters.

Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications

“…The quaternion number system was first described by Hamilton as a generalization of complex numbers [13,14]. In recent years, researchers have extended integral transforms into the quaternion algebra domain, leading to the development of theoretical frameworks such as quaternion Fourier transform (QFT) [15][16][17][18], quaternion fractional Fourier transform (QFRFT) [19], quaternion windowed fractional Fourier transform (QWFRFT) [20][21][22][23][24], quaternion linear canonical transform (QLCT) [25,26], and quaternion offset linear canonical transform [27]. Several important properties of QLCT have been investigated, including linearity, time shift, modulation, reconstruction formula, boundedness , and uncertainty principles in [28][29][30].…”

Solution of the Optimization Problem of Magnetotelluric Sounding in Quaternions by the Differential Evolution Method

“…In recent years, scientists have been increasingly interested in using quaternion Fourier transforms (QFTs) to solve partial differential equations. In [4,5], the basic properties of direct and inverse partial differential transformations using QFT were studied and proved, and the applications of QFT for solving problems of thermal conductivity and the wave equation were considered. It is shown that the use of the QFT makes it possible to exclude one of the spatial or temporal variables.…”

Weighted Convolution for Quaternion Linear Canonical Cosine Transform and Its Application

“…It exhibits unique advantages, strong flexibility, and processing ability when dealing with non-stationary and non-Gaussian signals. Significant progress has been made in signal sampling and filtering [3,4], discrete algorithms [5,6], convolution theory [7][8][9], time-frequency analysis [10][11][12], parameter estimation [13,14], uncertainty principles [11,15,16], image encryption [17,18], etc. The linear canonical cosine transform (LCcT) is widely recognized for its excellent orthogonal properties that enable efficient compression of energy distribution.…”

“…In recent years, with the continuous progress of mathematical theory, researchers have begun to extend the concept of integral transforms to the field of quaternion algebra. This extension has led to new theoretical frameworks, such as the quaternion Fourier transform (QFT) [27,28], the quaternion fractional Fourier transform (QFRFT) [29,30], the quaternion linear canonical transform (QLCT) [31][32][33][34], and the quaternion offset linear canonical transform (QOLCT) [7,35,36]. These theoretical frameworks provide new methods and tools for processing and analyzing quaternion signals.…”