一种统一的分数谱框架用于时空图信号：双分数变换与测地耦合
A Unified Fractional Spectral Framework for Spatiotemporal Graph Signals: Bi-Fractional Transform and Geodesic Coupling
AI总结：
本文提出双分数变换与测地耦合方法，用于统一时空图信号处理，通过独立分配分数阶次和构造耦合路径，提升对异构谱的适应性，并在动态图像恢复任务中取得显著性能提升。
摘要：
图信号处理将谱分析扩展到支持不规则域的数据上。现有的二维图信号分数变换，包括二维图分数傅里叶变换（GFRFT），通常在两个维度上施加共享的分数阶次，这限制了对异构时空谱的适应性。为了解决这一限制，我们提出了二维图双分数傅里叶变换，该变换为Cartesian积的因子图分配独立的分数阶次，从而在保持可分离性、单位性和可逆性的同时实现解耦的谱控制。为进一步解决时间分数分析中的基底模糊问题，我们开发了一种测地耦合的GFRFT，通过在单位元流形上构造耦合路径，从而将图诱导的和离散时间基底统一起来，并保证了单位性和闭式反变换。基于这些变换，我们推导出一个可微的维纳型滤波框架，采用混合优化策略：分数阶次从数据中端到端学习，而耦合参数则固定为结构正则化器。在真实世界的时间变化图数据集和动态图像恢复任务上的实验表明，与最先进的分数变换和基于竞争学习的基线相比，取得了持续的提升。
英文摘要：
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.
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