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arXiv 2511.16111stat.MLcs.LGmath.SP
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旋转参数化图分数阶傅里叶变换：定义、性质和最优滤波

Rotation-Parameterized Graph Fractional Fourier Transform: Definition, Properties, and Optimal Filtering

School of Mathematics and Statistics, Nanjing University of Information Science and Technology（南京信息工程大学数学与统计学院）；School of Communication and Artificial Intelligence, Nanjing Institute of Technology（南京理工大学通信与人工智能学院）；School of Integrated Circuits, Nanjing Institute of Technology（南京理工大学集成电路学院）；Jiangsu Province Engineering Research Center of IntelliSense Technology and System（江苏省智能感知技术与系统工程研究中心）；Hubei Key Laboratory of Applied Mathematics, Hubei University（湖北省应用数学重点实验室）；Key Laboratory of System Control and Information Processing, Ministry of Education, Shanghai Jiao Tong University（教育部系统控制与信息处理重点实验室，上海交通大学）

Feiyue Zhao, Mingzhi Wang, Yangfan He, Zhichao Zhang

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AI总结：

本文提出旋转参数化图分数阶傅里叶变换（RP-GFRFT），通过统一分数阶和旋转参数化的谱分析，解决现有方法在旋转基控制和零角度退化方面的不足，提升图信号处理的去噪、重建和特征保留性能。

摘要：

图谱表示在图信号处理中是基础，为分析图结构数据提供严谨的框架。图分数阶傅里叶变换（GFRFT）通过分数阶参数扩展图傅里叶变换（GFT），实现灵活的谱分析并保持数学一致性。角图傅里叶变换（AGFT）通过旋转GFT特征向量引入角度控制；然而现有构造可能无法在零角度时精确还原为GFT，削弱理论一致性和可解释性。为解决这些互补的局限性，即GFRFT缺乏基于旋转的基控制和AGFT的零角度退化问题，本文提出旋转参数化图分数阶傅里叶变换（RP-GFRFT），统一分数阶和旋转参数化的谱分析。构造了一个保持退化的旋转矩阵族以保证在零角度时精确还原为GFT。然后提出了两种RP-GFRFT变体，I-RP-GFRFT和II-RP-GFRFT，并通过理论分析确认其幺正性、可逆性、还原行为和光滑参数依赖性。将分数阶和旋转角度联合优化用于自适应图谱滤波。在真实世界信号、图像和点云上的实验表明，RP-GFRFT在去噪精度、重建质量和特征保留方面优于GFRFT、AGFT和代表性滤波基线。

英文摘要：

Graph spectral representations are fundamental in graph signal processing, providing a rigorous frameworkforanalyzing graph-structured data. The graph fractional Fourier transform (GFRFT) extends the graph Fourier transform (GFT) through a fractional-order parameter, enabling flexible spectral analysis with mathematical consistency. The angular graph Fourier transform (AGFT) further introduces angular control by rotating GFT eigenvectors; however, existing constructions may fail to reduce exactly to the GFT at zero angle, weakening theoretical consistency and interpretability. To address these complementary limitations, namely the lack of rotation-based basis control in GFRFT and the defective zero-angle degeneracy of AGFT, this paper proposes the rotation-parameterized graph fractional Fourier transform (RP-GFRFT), which unifies fractional order and rotation-parameterized spectral analysis. A degeneracy preserving rotation matrix family is constructed to guarantee exact GFT reduction at zero angle. TwoRP-GFRFTvariants,I-RP-GFRFTandII-RP-GFRFT,arethenformulated, with theoretical analyses confirming their unitarity, invertibility, reduction behavior, and smooth parameter dependence. The fractional order and rotation angle are jointly optimized for adaptive graph spectral filtering. Experiments on real-world signals, images, and point clouds demonstrate that RP-GFRFT improves denoising accuracy, reconstruction quality, and feature preservation over GFRFT, AGFT, and representative filtering baselines.

机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。

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