Design of Time-Vertex Node-Variant Graph Filters
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Design of Time-Vertex Node-Variant Graph Filters Hairong Feng1 · Junzheng Jiang1 · Haitao Wang1 · Fang Zhou2 Received: 2 June 2020 / Revised: 9 September 2020 / Accepted: 10 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract This paper addresses the node-variant graph filter on the joint time-vertex domain. An efficient algorithm is proposed to design the filter that can provide a good approximation to desirable linear operators for the time-varying signals on graphs. As a case study, the application of the designed filter on the signal denoise is addressed, where the filter is designed to approximate an inverse filtering operator. Numerical experiments conducted on the synthetic and real-world datasets demonstrate that the proposed filter is superior to the polynomial counterpart. Keywords Node-variant graph filter · Time-vertex node-variant graph filter · Joint time-vertex domain · Inverse filtering
1 Introduction Graph signal processing (GSP) is a prominent framework for dealing with the data on irregular domain, e.g., sensor, transportation, social, and temperature networks [25,27,28,31]. In the framework, several important concepts in the traditional signal processing have been extended to the graph setting, including the graph Fourier transform (GFT) [26], graph filter (GF) [24,25,31], graph filter bank [10,15], etc. Among these concepts, GF is a powerful tool to process the signal on graph and it has been
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Haitao Wang [email protected] Hairong Feng [email protected] Junzheng Jiang [email protected] Fang Zhou [email protected]
1
School of Information and Communication, Guilin University of Electronic Technology, Guilin 541004, China
2
School of Life and Environmental Sciences, Guilin University of Electronic Technology, Guilin 541004, China
Circuits, Systems, and Signal Processing
applied to various fields, for instance, classification [24], denoising [17], interpolation [29], clustering [34], to name a few. The most common form of GF is the polynomial filter [24,25], whose spectral response is polynomial with respect to the spectrum of the graph. The polynomial GFs can be categorized into the finite impulse response (FIR) filter [23,32] and the infinite impulse response (IIR) GF whose spectral response is a rational function with respect to the spectrum of the graph [8,14]. The GFs with polynomial or rational kernels suffer from limited degrees of design freedom, which makes it difficult to achieve filters with good vertical and spectral characteristics concurrently. In order to design localized filters with the good spectral feature, more generalized GFs have been constructed, and one typical example is the node-variant graph filter (NVGF) [30]. The NVGF has wide applications, for example network coding [30], clustering [5], topology analysis of wireless sensor network [22], and denoising and recovery [20]. In many real-world applications, plenty of signals are time-varying, e.g., temperature data recorded at the consecutiv
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