A Comprehensive Comparative Study of Two-Dimensional and Three-Dimensional Fractional Fourier Transforms

Abstract

The Fractional Fourier Transform (FrFT) has established itself as a fundamental mathematical tool in signal processing, optics, and communications, with applications spanning filter design, signal recovery, phase retrieval, pattern recognition, and image encryption. While the one-dimensional FrFT is theoretically well-understood, modern applications increasingly involve multidimensional data, particularly images (2D) and medical/geophysical volumes (3D). This paper presents a comprehensive comparative study of separable two-dimensional and three-dimensional Fractional Fourier Transforms. We provide complete mathematical proofs of fundamental properties including linearity, index additivity, shift and modulation theorems, Parseval’s relation, eigenfunction expansions, and convolution structures for both 2D and 3D cases. Through systematic comparison, we demonstrate that while separable FrFTs preserve many desirable properties in higher dimensions, they are inherently limited to separable phase space rotations. We establish the relationship between separable multidimensional FrFTs and more general non-separable linear canonical transforms, identifying scenarios where separable transforms suffice and where genuinely coupled transforms become necessary.

Introduction

The Fractional Fourier Transform (FrFT) generalizes the classical Fourier transform by introducing a continuous order parameter, interpreted as a rotation in the time–frequency plane. While 1D FrFT is well-studied, multidimensional FrFTs are essential for applications in image processing, optics, medical imaging, geophysics, and volumetric data analysis.

This work contributes by:

1. Providing complete mathematical proofs for 2D and 3D separable FrFTs.

2. Identifying the separable phase-space rotation property and its geometric interpretation.

3. Establishing relationships between separable multidimensional FrFTs and non-separable linear canonical transforms.

Mathematical Preliminaries

• 1D FrFT:
  For a function f(x)∈L2(R)f(x) \in L^2(\mathbb{R})f(x)∈L2(R) and order α\alphaα, the FrFT is defined using a kernel Kα(x,u)K_\alpha(x,u)Kα?(x,u).
  Key properties include:

  • Index additivity (semigroup): Kα∗Kβ=Kα+βK_\alpha * K_\beta = K_{\alpha+\beta}Kα?∗Kβ?=Kα+β?

  • Conjugation: Kα‾=K−α\overline{K_\alpha} = K_{-\alpha}Kα??=K−α?

• Separable Multidimensional FrFT:

  • 2D FrFT: F(α,β)[f(x,y)]=?f(x,y)Kα(x,u)Kβ(y,v)dxdyF^{(\alpha,\beta)}[f(x,y)] = \iint f(x,y) K_\alpha(x,u) K_\beta(y,v) dx dyF(α,β)[f(x,y)]=?f(x,y)Kα?(x,u)Kβ?(y,v)dxdy

  • 3D FrFT: F(α,β,γ)[f(x,y,z)]=?f(x,y,z)Kα(x,u)Kβ(y,v)Kγ(z,w)dxdydzF^{(\alpha,\beta,\gamma)}[f(x,y,z)] = \iiint f(x,y,z) K_\alpha(x,u) K_\beta(y,v) K_\gamma(z,w) dx dy dzF(α,β,γ)[f(x,y,z)]=?f(x,y,z)Kα?(x,u)Kβ?(y,v)Kγ?(z,w)dxdydz

Key Properties

1. Linearity:

   • 2D: F(α,β)[af+bg]=aF(α,β)[f]+bF(α,β)[g]F^{(\alpha,\beta)}[af+bg] = a F^{(\alpha,\beta)}[f] + b F^{(\alpha,\beta)}[g]F(α,β)[af+bg]=aF(α,β)[f]+bF(α,β)[g]

   • 3D: Same principle extends with triple integrals.

2. Index Additivity:

   • 2D: F(α2,β2){F(α1,β1)[f]}=F(α1+α2,β1+β2)[f]F^{(\alpha_2,\beta_2)}\{F^{(\alpha_1,\beta_1)}[f]\} = F^{(\alpha_1+\alpha_2,\beta_1+\beta_2)}[f]F(α2?,β2?){F(α1?,β1?)[f]}=F(α1?+α2?,β1?+β2?)[f]

   • 3D: Extends similarly to three dimensions.

3. Shift Theorem (2D example):

   • Shifting input by (x0,y0)(x_0, y_0)(x0?,y0?) results in phase modulation and coordinate shift in FrFT output.

Applications

• 2D FrFT: Image processing, optical systems.

• 3D FrFT: Medical imaging, geophysical analysis, volumetric data analysis.

Conclusion

This paper has presented a comprehensive comparative study of 2D and 3D separable Fractional Fourier Transforms, providing complete proofs of linearity, index additivity, shift and modulation theorems, Parseval’s relation, eigenfunction expansions, and generalized convolution theorems. Our systematic comparison confirms that separable multidimensional FrFTs successfully extend unitarity, invertibility, and complete eigenfunction expansions from the 1D case, but are fundamentally limited to block-diagonal symplectic (separable) phase-space rotations. For applications involving separable systems, the separable FrFT is an efficient and theoretically complete tool. When coupling between dimensions or full phase-space rotation is required, non-separable generalizations such as the NSFrFT or angular graph fractional transforms become necessary. Future directions include: fast algorithms for non-separable multidimensional FrFTs, uncertainty principles in higher-dimensional fractional domains, adaptive fractional transforms with learnable parameters, extensions to graph-structured domains, and applications in quantum information processing.

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Copyright

Copyright © 2026 S. D. Kambale. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.