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A Unified Fractional Spectral Framework for Spatiotemporal Graph Signals: Bi-Fractional Transform and Geodesic Coupling

Mingzhi Wang, Manjun Cui, Feiyue Zhao, Yangfan He, Zhichao Zhang

TL;DR

The two-dimensional graph bi-fractional Fourier transform is proposed, which assigns independent fractional orders to the factor graphs of a Cartesian product, enabling decoupled spectral control while preserving separability, unitarity, and invertibility.

Abstract

Graph signal processing extends spectral analysis to data supported on irregular domains. Existing fractional transforms for two-dimensional graph signals, including the two-dimensional graph fractional Fourier transform (GFRFT), typically impose a shared fractional order across dimensions, which limits adaptivity to heterogeneous spatiotemporal spectra. To address this limitation, we propose the two-dimensional graph bi-fractional Fourier transform, which assigns independent fractional orders to the factor graphs of a Cartesian product, enabling decoupled spectral control while preserving separability, unitarity, and invertibility. To further resolve the basis ambiguity in temporal fractional analysis, we develop a geodesic-coupled GFRFT by constructing a coupling path along the principal geodesic on the unitary manifold, thereby unifying graph-induced and discrete temporal bases with guaranteed unitarity and a closed-form inverse. Building on these transforms, we derive a differentiable Wiener-type filtering framework with a hybrid optimization strategy: the fractional orders are learned end-to-end from data, while the coupling parameter is fixed as a structural regularizer. Experiments on real-world time-varying graph datasets and dynamic image restoration tasks demonstrate consistent gains over state-of-the-art fractional transforms and competitive learning-based baselines.

A Unified Fractional Spectral Framework for Spatiotemporal Graph Signals: Bi-Fractional Transform and Geodesic Coupling

TL;DR

The two-dimensional graph bi-fractional Fourier transform is proposed, which assigns independent fractional orders to the factor graphs of a Cartesian product, enabling decoupled spectral control while preserving separability, unitarity, and invertibility.

Abstract

Graph signal processing extends spectral analysis to data supported on irregular domains. Existing fractional transforms for two-dimensional graph signals, including the two-dimensional graph fractional Fourier transform (GFRFT), typically impose a shared fractional order across dimensions, which limits adaptivity to heterogeneous spatiotemporal spectra. To address this limitation, we propose the two-dimensional graph bi-fractional Fourier transform, which assigns independent fractional orders to the factor graphs of a Cartesian product, enabling decoupled spectral control while preserving separability, unitarity, and invertibility. To further resolve the basis ambiguity in temporal fractional analysis, we develop a geodesic-coupled GFRFT by constructing a coupling path along the principal geodesic on the unitary manifold, thereby unifying graph-induced and discrete temporal bases with guaranteed unitarity and a closed-form inverse. Building on these transforms, we derive a differentiable Wiener-type filtering framework with a hybrid optimization strategy: the fractional orders are learned end-to-end from data, while the coupling parameter is fixed as a structural regularizer. Experiments on real-world time-varying graph datasets and dynamic image restoration tasks demonstrate consistent gains over state-of-the-art fractional transforms and competitive learning-based baselines.
Paper Structure (21 sections, 52 equations, 3 figures, 6 tables, 1 algorithm)

This paper contains 21 sections, 52 equations, 3 figures, 6 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. PRELIMINARIES
  3. Graph Signals
  4. 2D-GFRFT
  5. JFRFT
  6. Two-Dimensional Graph Bi-Fractional Fourier Transform
  7. Definition
  8. Properties
  9. Geodesic-Coupled Graph Fractional Fourier Transform
  10. Definition
  11. Properties
  12. Learnable Wiener Filtering in Fractional Spectral Domains
  13. Learnable Spectral Filtering via Gradient Descent
  14. Computational Efficiency and Optimization Analysis
  15. Applications on Real-World Data
  16. ...and 6 more sections

Figures (3)

  • Figure 1: Overview of the proposed GC-GFRFT framework and its applications.
  • Figure 2: Visual comparison on the REDS sequence for dynamic image deblurring. For each frame, the first row shows the selected region on the full, and the second row shows the corresponding magnified patch.
  • Figure 3: Visual comparison on the REDS sequence for dynamic image denoising. For each frame, the first row shows the selected region on the full image, and the second row shows the corresponding magnified patch.