Two-Dimensional Quaternion Fractional Fourier Transform: Definition and Probabilistic Analysis

This article presents a detailed study of the two-dimensional quaternion fractional Fourier transform (2D QFRFT) and investigates its role in the probabilistic analysis of quaternion-valued signals. The 2D formulation is constructed by applying fractional Fourier transforms independently along each spatial dimension, thereby extending classical 2D Fourier and fractional Fourier frameworks to the quaternion domain. Key analytical properties of the 2D QFRFT, including linearity, shift behavior, differentiation, convolution, and energy relations, are summarized based on existing results in the literature. Furthermore, the transform is employed to define and analyze fundamental probabilistic quantities, such as expected value and normalized probability distributions, within the 2D quaternion fractional transform domain. These results provide a systematic 2D extension of existing quaternion transform-based probabilistic models and offer a clear theoretical foundation for the representation and analysis of 2D quaternion-valued signals in non-commutative settings.