Joint Time-Vertex Fractional Fourier Transform

TL;DR

The paper introduces the Joint Time-Vertex Fractional Fourier Transform (JFRT), a two-parameter extension of the Joint Time-Vertex Transform (JFT) that performs analysis in both fractional time and fractional vertex domains. It establishes key theoretical properties (index additivity, commutativity, reversibility, reduction to identity, and unitary behavior under unitary GFTs) and derives a Tikhonov-regularized denoising framework in the JFRT domain using a joint fractional Laplacian. The authors develop fractional joint filters, provide separable formulations, analyze computational costs, and validate the approach with extensive experiments on synthetic and real-world datasets, showing improved denoising and clustering performance over state-of-the-art methods. The work extends fractional Fourier analysis to graph-based joint signals, offering flexible processing options for time-varying graph data with practical impact on denoising, reconstruction, and classification tasks.

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.

Figures (9)

• Figure 1: Graph signals for the delay parameter $d=35$, and at time instances, from left-to-right, $t=0, 0.009, 0.099$ secs., which correspond to the $1^{\text{st}}$, ${10}^{\text{th}}$ and ${100}^{\text{th}}$ columns of the joint time-vertex signal $\mathbf{X}$, where $f_s = 1\,kHz$.
• Figure 2: Time series signals for the first three vertices in the sensor network. The original version is obtained with the $d=30$, the noisy version with $\sigma=0.15$, and the filtered version with Adjacency method and $\alpha = 1.34, \beta=1.01$, and $c=35$. Error reduces to $16.42\%$ from $21.36\%$ after filtering.
• Figure 3: Error surfaces for both the adjacency (left) and Laplacian (right) based joint time-vertex fractional filtering methods, for $\sigma=0.1,d=50, c=1$. Best $\alpha,\beta$ pairs are obtained according to the defined error metric and found to be ${(\alpha,\beta)}_{\text{adj}}=(0.69,1.03)$ and ${(\alpha,\beta)}_{\text{lap}}=(0.69,1.07)$.
• Figure 4: The results of the regularization-based denoising applied to the Molene dataset. (Left) varying regularization parameters are considered while fractional orders are fixed to $\alpha = 0.905$ and $\beta =1$. (Right) fractional orders are considered with fixed regularization parameters $\tau_g = 3.8, \tau_t = 4$.
• Figure 5: In the NOAA dataset, both yearly (top row) and monthly (bottom row) settings are considered. Performances of denoising with respect to (Upper left) varying $\alpha,\beta$ with fixed $\tau_t = 3.4,\tau_g = 0.4$. (Upper right) varying $\tau_t,\tau_g$ with fixed $\alpha = 0.965, \beta = 1.005$. (Lower left) varying $\alpha,\beta$ with fixed $\tau_t = 1.1, \tau_g = 0.4$ and (Lower right) varying $\tau_t,\tau_g$ with fixed $\alpha = 1.09, \beta = 1.01$.

Theorems & Definitions (23)

• Remark 1
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• Remark 3
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• Proposition 1