A Unified Framework for 2D Nonseparable Fractional Fourier Transform: From Geometric Completeness to Applications

TL;DR

This paper introduces a comprehensive 2D nonseparable fractional Fourier transform framework that unifies existing transforms, maintains geometric consistency with the Wigner distribution, and demonstrates practical advantages in signal analysis and image processing.

Contribution

A novel 2D nonseparable FRFT framework derived from symplectic and orthogonal groups, unifying existing transforms and ensuring geometric consistency with the 2D Wigner distribution.

Findings

The proposed transform encompasses SFRFT, GT, and CFRFT as special cases.

Numerical simulations show improved analysis of coupled chirp signals.

Applications in filtering and image encryption demonstrate robustness and efficiency.

Abstract

The one-dimensional (1D) fractional Fourier transform (FRFT) generalizes the Fourier transform, offering significant advantages in the time-frequency analysis of non-stationary signals. While various 2D extensions exist, such as the 2D separable FRFT (SFRFT), gyrator transform (GT), coupled FRFT (CFRFT), and earlier nonseparable definitions, they suffer from fragmented theoretical frameworks and a fundamental lack of geometric consistency with the 2D Wigner distribution (WD). Addressing these limitations, we propose a unified 2D nonseparable FRFT (NSFRFT) framework. Theoretically derived from the intersection of the symplectic and special orthogonal groups (isomorphic to the unitary group $\mathrm{U}(2)$), this transform inherently possesses four degrees of freedom and mathematically incorporates the 2D SFRFT, GT, and CFRFT as special cases. Unlike prior algebraic generalizations, it strictly preserves the rigid 4D rotational geometry of the 2D WD, ensuring geometric consistency and numerical stability. We derive its essential properties and develop efficient discrete algorithms with a computational complexity of $O(N^{2}\log N)$. Numerical simulations validate the superiority of the 2D NSFRFT in analyzing coupled chirp signals and demonstrate its robustness in filtering and image encryption and decryption applications.