Angular Graph Fractional Fourier Transform: Theory and Application
Feiyue Zhao, Yangfan He, Zhichao Zhang
TL;DR
The paper addresses limitations of fixed graph spectral transforms by unifying fractional-order analysis with angular spectral control. It introduces AGFRFT, a degeneracy‑friendly framework with two variants (Type I and II) that preserves GFT degeneration at zero angle and unitarity while enabling joint optimization over the angle and fractional order. Theoretical results establish unitarity, invertibility, smooth parameter dependence, and degeneracy properties, and the approach is validated across real data, images, and 3D point clouds, showing improved spectral concentration and reconstruction quality over GFRFT and AGFT. The work offers a versatile, learnable tool for adaptive angular fractional spectral processing in graph signal processing with strong potential for integration into broader graph-based learning systems.
Abstract
Graph spectral representations are fundamental in graph signal processing, offering a rigorous framework for analyzing and processing graph-structured data. The graph fractional Fourier transform (GFRFT) extends the classical graph Fourier transform (GFT) with a fractional-order parameter, enabling flexible spectral analysis while preserving mathematical consistency. The angular graph Fourier transform (AGFT) introduces angular control via GFT eigenvector rotation; however, existing constructions fail to degenerate to the GFT at zero angle, which is a critical flaw that undermines theoretical consistency and interpretability. To resolve these complementary limitations - GFRFT's lack of angular regulation and AGFT's defective degeneracy - this study proposes an angular GFRFT (AGFRFT), a unified framework that integrates fractional-order and angular spectral analyses with theoretical rigor. A degeneracy-friendly rotation matrix family ensures exact GFT degeneration at zero angle, with two AGFRFT variants (I-AGFRFT and II-AGFRFT) defined accordingly. Rigorous theoretical analyses confirm their unitarity, invertibility, and smooth parameter dependence. Both support learnable joint parameterization of the angle and fractional order, enabling adaptive spectral processing for diverse graph signals. Extensive experiments on real-world data denoising, image denoising, and point cloud denoising demonstrate that AGFRFT outperforms GFRFT and AGFT in terms of spectral concentration, reconstruction quality, and controllable spectral manipulation, establishing a robust and flexible tool for integrated angular fractional spectral analysis in graph signal processing.