Existing deep learning-based computer vision methods usually operate in the spatial and frequency domains, which are two orthogonal \textbf{individual} perspectives for image processing.In this paper, we introduce a new spatial-frequency analysis tool, Fractional Fourier Transform (FRFT), to provide comprehensive \textbf{unified} spatial-frequency perspectives.The FRFT is a unified continuous spatial-frequency transform that simultaneously reflects an image's spatial and frequency representations, making it optimal for processing non-stationary image signals.We explore the properties of the FRFT for image processing and present a fast implementation of the 2D FRFT, which facilitates its widespread use.Based on these explorations, we introduce a simple yet effective operator, Multi-order FRactional Fourier Convolution (MFRFC), which exhibits the remarkable merits of processing images from more perspectives in the spatial-frequency plane. Our proposed MFRFC is a general and basic operator that can be easily integrated into various tasks for performance improvement.We experimentally evaluate the MFRFC on various computer vision tasks, including object detection, image classification, guided super-resolution, denoising, dehazing, deraining, and low-light enhancement. Our proposed MFRFC consistently outperforms baseline methods by significant margins across all tasks.

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.

Our code is released publicly at https://github.com/yuhuUSTC/FRFT.

--Spectral graph embedding plays a critical role in graph representation learning by generating low-dimensional vector representations from graph spectral information. However, the embedding space of traditional spectral embedding methods often exhibit limited expressiveness, failing to exhaustively capture latent structural features across alternative transform domains. T o address this issue, we use the graph fractional Fourier transform to extend the existing state-of-the-art generalized frequency filtering embedding (GEFFE) into fractional domains, giving birth to the generalized fractional filtering embedding (GEFRFE), which enhances embedding informative-ness via the graph fractional domain. The GEFRFE leverages graph fractional domain filtering and a nonlinear composition of eigenvector components derived from a fractionalized graph Laplacian. T o dynamically determine the fractional order, two parallel strategies are introduced: search-based optimization and a ResNet18-based adaptive learning. Extensive experiments on six benchmark datasets demonstrate that the GEFRFE captures richer structural features and significantly enhance classification performance. Notably, the proposed method retains computational complexity comparable to GEFFE approaches.

Existing deep learning-based computer vision methods usually operate in the spatial and frequency domains, which are two orthogonal \textbf{individual} perspectives for image processing.In this paper, we introduce a new spatial-frequency analysis tool, Fractional Fourier Transform (FRFT), to provide comprehensive \textbf{unified} spatial-frequency perspectives.The FRFT is a unified continuous spatial-frequency transform that simultaneously reflects an image's spatial and frequency representations, making it optimal for processing non-stationary image signals.We explore the properties of the FRFT for image processing and present a fast implementation of the 2D FRFT, which facilitates its widespread use.Based on these explorations, we introduce a simple yet effective operator, Multi-order FRactional Fourier Convolution (MFRFC), which exhibits the remarkable merits of processing images from more perspectives in the spatial-frequency plane. Our proposed MFRFC is a general and basic operator that can be easily integrated into various tasks for performance improvement.We experimentally evaluate the MFRFC on various computer vision tasks, including object detection, image classification, guided super-resolution, denoising, dehazing, deraining, and low-light enhancement. Our proposed MFRFC consistently outperforms baseline methods by significant margins across all tasks.