Design of Hermite Polynomial Graph Filter and Its Application to Sensor Network Data Denoising

In this paper, the design and application of Hermite polynomial graph filter is studied. First, the basics of graph signal processing (GSP) is briefly reviewed and the design problem of graph filter is described. Second, the orthogonal Hermite polynomial is applied to design graph filter by minimizing the integral weighted squares errors between ideal and actual spectral responses of filter in the spectral domain. The mathematical formula of graph filter coefficients is derived and the design examples of lowpass filter and heat kernel smoothing (HKS) filter are illustrated. Third, the order recursive relation of Hermite polynomial is employed to derive a distributed implementation structure of graph filter by using graph Laplacian matrix as a graph shift matrix. Finally, the application examples of sensor network data denoising at USA and Taiwan are demonstrated to show the efficacy of the proposed polynomial graph filter method by using improvement of signal to noise ratio (SNR).