Design and Application of Graph Filter Using Orthogonal Jacobi Polynomial

Graph filters are essential tools in graph signal processing (GSP), which has been widely employed to process irregular data from sensor, social, brain, and transportation networks. This paper explores the design and application of graph filters constructed using orthogonal Jacobi polynomials (OJP), whose weighting functions depend on two real-valued free parameters. The main tasks are as follows: First, we briefly review the fundamentals of GSP and OJP. Then, we design the graph filter’s transfer function matrix as a linear combination of Jacobi polynomials. The optimal filter coefficients are determined by minimizing the integral of the weighted squared error between the ideal and actual spectral responses within the target frequency band. An efficient implementation structure is also developed using the recursive properties of the polynomials. Next, we present design example of lowpass filter, demonstrating that approximation errors can be adjusted by tuning the two OJP parameters. Finally, a sensor network denoising example illustrates that the proposed method is better than traditional total variation regularization method when the OJP parameters are properly chosen.