Reduced Biquaternion Windowed Linear Canonical Transform: Properties and Applications

Abstract

The quaternion windowed linear canonical transform is a tool for processing multidimensional data and enhancing the quality and efficiency of signal and image processing; however, it has disadvantages due to the noncommutativity of quaternion multiplication. In contrast, reduced biquaternions, as a special case of four-dimensional algebra, possess unique advantages in computation because they satisfy the multiplicative exchange rule. This paper proposes the reduced biquaternion windowed linear canonical transform (RBWLCT) by combining the reduced biquaternion signal and the windowed linear canonical transform that has computational efficiency thanks to the commutative property. Firstly, we introduce the concept of a RBWLCT, which can extract the time local features of an image and has the advantages of both time-frequency analysis and feature extraction; moreover, we also provide some fundamental properties. Secondly, we propose convolution and correlation operations for RBWLCT along with their corresponding generalized convolution, correlation, and product theorems. Thirdly, we present a fast algorithm for RBWLCT and analyze its computational complexity based on two dimensional Fourier transform (2D FTs). Finally, simulations and examples are provided to demonstrate that the proposed transform effectively captures the local RBWLCT-frequency components with enhanced degrees of freedom and exhibits significant concentrations.

Weighted Convolution for Quaternion Linear Canonical Cosine Transform and Its Application

“…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.…”