Browsing by Subject "Graph Fourier transform (GFT)"

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Open Access
Graph fractional Fourier transform
(2024-09) Alikaşifoğlu, Tuna
The fractional Fourier transform (FRFT) parametrically generalizes the Fourier transform (FT) by a transform order, representing signals in intermediate time-frequency domains. The FRFT has multiple but equivalent definitions each offering benefits like derivational ease and computational efficiency. Concurrently, graph signal processing (GSP) extends traditional signal processing to irregular graph structures, enabling concepts like sampling, filtering, and Fourier transform for graph signals. The graph fractional Fourier transform (GFRFT) is recently extended to the GSP domain. However, this extension only generalizes one definition of FRFT based on specific graph structure with limited transform order range. Ideally, the GFRFT extension should be consistent with as many alternative definitions as possible. This work introduces a unified framework for GFRFT that supports multiple definitions with any graph structure and transform order. The proposed approach also allows faster transform matrix computations on large graphs and learnable transform order. Additionally, data sources on each vertex can also continually provide time-series signals such that graph signals are time varying. Joint time-vertex Fourier transform (JFT), with the associated framework of joint time-vertex (JTV) processing, provides spectral analysis tools for such signals. Just as the FRFT generalizes FT, we propose the joint time-vertex fractional Fourier transform (JFRFT) to generalize JFT. JFRFT provides an additional fractional analysis tool for JTV processing by extending temporal and vertex domains to fractional orders. Recently, the classical optimal Wiener filtering problem is introduced to JTV domain. However, the ordinary Fourier domain is not always optimal for separating the signal and noise; one can reach the smaller error in a fractional domain. We generalize the optimal Wiener filtering to the proposed JFRFT domains and provide a theoretical analysis and solution to the problem in the fractional JTV domains. We numerically verify our claims by presenting computational cost analysis and experiments with comprehensive comparisons to state-of-the-art approaches.
Open Access
Graph fractional Fourier transform: a unified theory
(IEEE, 2024) Alikaşifoğlu, Tuna; Kartal, Bünyamin; Koç, Aykut
The fractional Fourier transform (FRFT) parametrically generalizes the Fourier transform (FT) by a transform order, representing signals in intermediate time-frequency domains. The FRFT has multiple but equivalent definitions, including the fractional power of FT, time-frequency plane rotation, hyper-differential operator, and many others, each offering benefits like derivational ease and computational efficiency. Concurrently, graph signal processing (GSP) extends traditional signal processing to data on irregular graph structures, enabling concepts like sampling, filtering, and Fourier transform for graph signals. The graph fractional Fourier transform (GFRFT) is recently extended to the GSP domain. However, this extension only generalizes the fractional power definition of FRFT based on specific graph structures with limited transform order range. Ideally, the GFRFT extension should be consistent with as many alternative definitions as possible. This paper first provides a rigorous fractional power-based GFRFT definition that supports any graph structure and transform order. Then, we introduce the novel hyper-differential operator-based GFRFT definition, allowing faster forward and inverse transform matrix computations on large graphs. Through the proposed definition, we derive a novel approach to select the transform order by learning the optimal value from data. Furthermore, we provide treatments of the core GSP concepts, such as bandlimitedness, filters, and relations to the other transforms in the context of GFRFT. Finally, with comprehensive experiments, including denoising, classification, and sampling tasks, we demonstrate the equivalence of parallel definitions of GFRFT, learnability of the transform order, and the benefits of GFRFT over GFT and other GSP methods.¹¹ The codebase is available at https://github.com/koc-lab/gfrft-unified.
Open Access
Optimal fractional fourier filtering for graph signals
(IEEE, 2021-05-19) Öztürk, Cüneyd; Özaktaş, Haldun M.; Gezici, Sinan; Koç, Aykut
Graph signal processing has recently received considerable attention. Several concepts, tools, and applications in signal processing such as filtering, transforming, and sampling have been extended to graph signal processing. One such extension is the optimal filtering problem. The minimum mean-squared error estimate of an original graph signal can be obtained from its distorted and noisy version. However, the best separation of signal and noise, and thus the least error, is not always achieved in the ordinary Fourier domain, but rather a fractional Fourier domain. In this work, the optimal filtering problem for graph signals is extended to fractional Fourier domains, and theoretical analysis and solution of the proposed problem are provided along with computational cost considerations. Numerical results are presented to illustrate the benefits of filtering in fractional Fourier domains.
Open Access
Wiener filtering in joint time-vertex fractional Fourier domains
(IEEE, 2024) Alikaşifoğlu, Tuna; Kartal, Bünyamin; Koç, Aykut
Graph signal processing (GSP) uses network structures to analyze and manipulate interconnected signals. These graph signals can also be time-varying. The established joint time-vertex processing framework and corresponding joint time-vertex Fourier transform provide a basis to endeavor such time-varying graph signals. The optimal Wiener filtering problem has been deliberated within the joint time-vertex framework. However, the ordinary Fourier domain is only sometimes optimal for separating the signal and noise; one can achieve lower error rates in a fractional Fourier domain. In this paper, we solve the optimal Wiener filtering problem in the joint time-vertex fractional Fourier domains. We provide a theoretical analysis and numerical experiments with comprehensive comparisons to existing filtering approaches for time-varying graph signals to demonstrate the advantages of our approach.