[Paper Review] Joint Time-Vertex Fractional Fourier Transform

Alikaşifoğlu, Tuna, Bünyamin Kartal+2 more|arXiv (Cornell University)|Mar 15, 2022

Molecular spectroscopy and chirality4 citations

TL;DR

This paper introduces the Joint Time-Vertex Fractional Fourier Transform (JFRT), a generalization of the Joint Fourier Transform (JFT) that extends fractional Fourier analysis to joint time-vertex graph signals. By applying fractional orders to both temporal and graph spectral domains, JFRT enables flexible, intermediate-domain signal representation, achieving improved denoising and clustering performance over standard JFT in numerical experiments with synthetic and real-world mesh data.

ABSTRACT

Graph signal processing (GSP) facilitates the analysis of high-dimensional data on non-Euclidean domains by utilizing graph signals defined on graph vertices. In addition to static data, each vertex can provide continuous time-series signals, transforming graph signals into time-series signals on each vertex. The joint time-vertex Fourier transform (JFT) framework offers spectral analysis capabilities to analyze these joint time-vertex signals. Analogous to the fractional Fourier transform (FRT) extending the ordinary Fourier transform (FT), we introduce the joint time-vertex fractional Fourier transform (JFRT) as a generalization of JFT. The JFRT enables fractional analysis for joint time-vertex processing by extending Fourier analysis to fractional orders in both temporal and vertex domains. We theoretically demonstrate that JFRT generalizes JFT and maintains properties such as index additivity, reversibility, reduction to identity, and unitarity for specific graph topologies. Additionally, we derive Tikhonov regularization-based denoising in the JFRT domain, ensuring robust and well-behaved solutions. Comprehensive numerical experiments on synthetic and real-world datasets highlight the effectiveness of JFRT in denoising and clustering tasks that outperform state-of-the-art approaches.

Motivation & Objective

• Address the need for advanced spectral analysis tools in joint time-vertex signal processing, where signals evolve over time on irregular graph structures.
• Extend the classical fractional Fourier transform (FRT) to the graph signal processing (GSP) domain, enabling fractional analysis in both time and vertex (graph) domains.
• Develop a unified framework that generalizes the Joint Fourier Transform (JFT) and provides a flexible, parameterized transformation for time-varying graph signals.
• Enable improved signal processing performance in applications such as denoising and clustering by leveraging the additional degrees of freedom in fractional orders.
• Establish theoretical properties of the JFRT, including index additivity, reversibility, unitarity under specific conditions, and reduction to known transforms as special cases.

Proposed method

• Propose the Joint Time-Vertex Fractional Fourier Transform (JFRT) as a two-parameter transformation, applying fractional Fourier transforms independently to the time and graph (vertex) domains.
• Define the JFRT using the eigen-decomposition of the graph Laplacian and the Fourier transform of time signals, combining them via a tensor product structure.
• Derive the inverse JFRT and prove its reversibility, index additivity, and unitarity when the underlying graph Fourier transform is unitary.
• Formulate a Tikhonov regularization-based denoising method in the JFRT domain, using a joint fractional Laplacian to regularize signals in both time and graph domains.
• Implement the JFRT via a two-step process: apply fractional Fourier transform to time sequences per vertex, then apply graph fractional Fourier transform across vertices.
• Use nearest-neighbor graphs to model spatial connectivity and apply rectangular windowing with overlap to extract time sequences for analysis.

Experimental results

Research questions

• RQ1 Can the fractional Fourier transform be generalized to joint time-vertex graph signals to enable intermediate-domain spectral analysis?
• RQ2 What theoretical properties (e.g., additivity, unitarity, reversibility) does the proposed JFRT satisfy under different graph topologies?
• RQ3 How does the JFRT improve signal denoising and clustering performance compared to the standard JFT and other hybrid transforms?
• RQ4 Does the JFRT achieve better performance than the ordinary JFT in real-world applications such as motion tracking on 3D meshes?
• RQ5 Can the JFRT serve as a unifying framework that generalizes the JFT, 2D DFT, and 2D DFRT as special cases?

Key findings

• The JFRT achieves higher clustering accuracy than the standard JFT, with peak performance at fractional orders (e.g., 0.6–0.8) for both -10 dB and -20 dB SNR conditions.
• At -10 dB SNR, the best JFRT order achieved an average clustering accuracy of approximately 92%, significantly outperforming the ordinary JFT and other hybrid configurations.
• The distribution of clustering accuracy across 20 trials was more concentrated and higher in the fractional domain, indicating robustness and improved performance over non-fractional alternatives.
• The JFRT-based denoising method effectively suppresses sparse Gaussian noise (SNR -10 dB and -20 dB), with performance superior to standard JFT and hybrid transforms.
• The JFRT reduces to the standard JFT at order (1,1), to identity at (0,0), and to 2D discrete fractional Fourier transform for directed circular graphs at (1,1), confirming consistency with known transforms.
• The JFRT is unitary when the underlying graph Fourier transform is unitary, and it satisfies index additivity and reversibility, validating its mathematical soundness.