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

TL;DR

This paper introduces a novel bi-fractional Fourier transform for spatiotemporal graph signals, allowing independent spectral control across dimensions, and a geodesic-coupled transform to unify temporal bases, enhancing analysis flexibility.

Contribution

It proposes the two-dimensional graph bi-fractional Fourier transform and a geodesic-coupled GFRFT, enabling decoupled spectral control and unification of graph and temporal bases with guaranteed unitarity.

Findings

Demonstrates improved performance on real-world datasets.

Achieves consistent gains over state-of-the-art fractional transforms.

Provides a differentiable Wiener-type filtering framework with learned fractional orders.

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.