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

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.