Two-Dimensional Quaternion Linear Canonical Transform A Novel Framework for Probability Modeling

TL;DR

This paper introduces the 2D Quaternion Linear Canonical Transform as a new mathematical framework to enhance probability modeling for multidimensional and complex-valued signals, broadening theoretical and practical applications.

Contribution

It extends the linear canonical transform to quaternion algebra, creating a novel framework for probability modeling in multidimensional complex-valued signal analysis.

Findings

Provides a new approach for probability distribution analysis

Enhances understanding of multidimensional complex signals

Lays groundwork for future research in signal processing and statistics

Abstract

The linear canonical transform (LCT) serves as a powerful generalization of the Fourier transform (FT), encapsulating various integral transforms within a unified framework. This versatility has made it a cornerstone in fields such as signal processing, optics, and quantum mechanics. Extending this concept to quaternion algebra, the Quaternion Fourier Transform (QFT) emerged, enriching the analysis of multidimensional and complex-valued signals. The Quaternion Linear Canonical Transform (QLCT), a further generalization, has now positioned itself as a central tool across various disciplines, including applied mathematics, engineering, computer science, and statistics. In this paper, we introduce the Two Dimensional Quaternion Linear Canonical Transform (2DQLCT) as a novel framework for probability modeling. By leveraging the 2DQLCT, we aim to provide a more comprehensive understanding of probability distributions, particularly in the context of multi-dimensional and complex-valued signals. This framework not only broadens the theoretical underpinnings of probability theory but also opens new avenues for researchers