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The Optimal Joint Time-Vertex Graph Filter Design: From Ordinary Graph Fourier Domains to Fractional Graph Fourier Domains

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Abstract

In this paper, we investigate the optimal joint time-vertex graph filter design, which aims to denoise time-varying graph signals. We first propose to design the optimal joint time-vertex ordinary Fourier (O-JTV-OF) graph filter. Specifically, we formulate the O-JTV-OF graph filter design problem based on the Cartesian product and further obtain the closed-form expression of the O-JTV-OF graph filter coefficients by solving the Wiener–Hopf equation on the time-vertex graph. Then, we extend the O-JTV-OF graph filter design to the optimal joint time-vertex fractional Fourier (O-JTV-FF) graph filter design by leveraging the graph fractional shift operator and the graph fractional Fourier transform, which can provide a more suitable space to separate the signal and noise. Our numerical results on three real-world datasets show that the proposed O-JTV-OF and O-JTV-FF graph filter design methods lead to higher signal-to-noise ratio (SNR) results than the traditional optimal static ordinary Fourier (OSOF) and the optimal static fractional Fourier (OSFF) graph filter design methods. Moreover, the proposed O-JTV-FF graph filter design method performs better than the proposed O-JTV-OF graph filter design method in terms of SNR of reconstructed signals. Compared to a linear filter, i.e., the joint Tikhonov regularization-based method, our methods also provide better performance overall, especially performing significantly better in a low-input SNR.

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Data Availability

The datasets generated during and/or analyzed during the current study are available from the corresponding author on reasonable request.

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Acknowledgements

This work was supported in part by the National Natural Science Foundation of China under Grant 62071242.

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Authors and Affiliations

  1. School of Communication and Information Engineering, Nanjing University of Posts and Telecommunications, Nanjing, 210023, China

    Zirui Ge, Haiyan Guo, Tingting Wang & Zhen Yang

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  1. Zirui Ge
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  2. Haiyan Guo
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  3. Tingting Wang
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  4. Zhen Yang
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Correspondence to Zirui Ge or Zhen Yang.

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Appendices

Appendix A

The objective function of (10) can be rewritten as:

$$\begin{aligned} \mathbb {E}\Vert \varvec{B} \varvec{c}-\varvec{x}\Vert _2^2 =\mathbb {E}\left\{ \varvec{c}^H \varvec{B}^H \varvec{B} \varvec{c}-2 \varvec{x}^H \varvec{B} \varvec{c}+\varvec{x}^H \varvec{x}\right\} . \end{aligned}$$
(25)

To obtain \(\varvec{c}^\textrm{opt}\) that minimizes \(\mathbb {E}\Vert \varvec{B} \varvec{c}-\varvec{x}\Vert _2^2\), we take the derivative of \(\mathbb {E}\Vert \varvec{B} \varvec{c}-\varvec{x}\Vert _2^2\) with respect to \(\varvec{c}\), and set it to zero. Then, we have

$$\begin{aligned} 2\mathbb {E} \left\{ \varvec{B}^H\varvec{B} \right\} \varvec{c}-2\mathbb {E} \left\{ \varvec{B^Hx} \right\} =0, \end{aligned}$$
(26)

which leads to (11).

Appendix B

Since all the eigenvalues of \(\varvec{L}_{\mathcal {G}}\) are distinct, the Vandermonde matrices \(\varvec{\Psi }_{\mathcal {T} \lambda }\) and \(\varvec{\Psi }_{\mathcal {G} \lambda }\) are full column rank. As \(\mathbb {E} \left\{ |\varvec{y}_{\mathcal {F}}\left( n \right) |^2 \right\} \) is the graph power spectral density at graph frequency point n, it is generally not strictly equal to zero.

By letting , we can rewrite \(\varvec{R}_{y,y}^{Tv}=\varvec{\Psi }^H\mathbb {E} \left\{ \varvec{Y}_{\mathcal {F}} \right\} \varvec{\Psi }\) as

(27)

where \(\varvec{\eta }_{\mathcal {F}}\) is the square root of \(\mathbb {E} \left\{ |\varvec{y}_{\mathcal {F}}\left( n \right) |^2 \right\} \). For a given matrix \(\varvec{A}\), we have \(\mathrm{{Rank}}\left( \varvec{A} \right) =\mathrm{{Rank}}\left( \varvec{AA}^T \right) \). Then, we can obtain \(\mathrm{{Rank}}\left( \varvec{R}_{y,y}^{Tv} \right) =\mathrm{{Rank}}\left( \textrm{diag}\left( |\varvec{\eta }_{\mathcal {F}} |\right) \varvec{\Psi } \right) = PQ \). That is, \(\varvec{R}_{y,y}^{Tv}\) is invertible.

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Ge, Z., Guo, H., Wang, T. et al. The Optimal Joint Time-Vertex Graph Filter Design: From Ordinary Graph Fourier Domains to Fractional Graph Fourier Domains. Circuits Syst Signal Process 42, 4002–4018 (2023). https://doi.org/10.1007/s00034-023-02298-2

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