A Modified Spline Graph Filter Bank
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A Modified Spline Graph Filter Bank Amir Miraki1 · Hamid Saeedi-Sourck1 Received: 19 May 2020 / Revised: 29 August 2020 / Accepted: 4 September 2020 © Springer Science+Business Media, LLC, part of Springer Nature 2020
Abstract Two-channel critically sampled spline graph filter bank (SGFB) is one of the existing solutions for signal processing on arbitrary graphs. This paper addresses an approach to modify the filter design for SGFB. Our method improves the shape of analysis filters in the spectral domain and also reduces the computational complexity for the synthesis section. Additionally, the proposed method exploits polynomial filters and does not need to compute eigendecomposition for the Laplacian matrix of the underlying graph. Numerical results show the efficacy of our approach by comparing its performance in graph signal decomposition and denoising with the existing solutions. Keywords Graph signal processing · Filter bank · Spectral kernel · Optimization · Denoising
1 Introduction Graphs provide a natural representation for data in many networks including energy, sensor, and neuronal networks [17]. A large amount of data being produced from nodes have inspired analyzing data supported on graphs, namely graph signal processing (GSP)[17]. Generally, GSP is considered as a functional tool for analyzing data on the irregular complex structures. Graph filter banks (GFBs) are one of the significant techniques for the processing of graph signals [1,4–6,8–11,13,15,16,20–24]. For GFB structures, the type of graph is important to satisfy the desired properties such as perfect reconstruction (PR). Authors in [13] proposed a two-channel GFB especially designed for bipartite graphs. Additionally, some efforts have been performed to improve the design of filters, i.e.,
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Hamid Saeedi-Sourck [email protected] Amir Miraki [email protected]
1
Electrical Engineering Department, Yazd University, Yazd 89158-18411, Iran
Circuits, Systems, and Signal Processing
[8,20,21]. There are some other GFBs proposed for especial graphs such as tree graphs [6], Ω-structure graphs [22], and circulant graphs [4]. The filter banks in [4,6,8,13,20–22] force to work with some specific types of graphs. However, graph signals typically live on general graphs that are not with especial connectivities. To extend the filter bank approach to arbitrary graphs, authors in [13] suggested decomposing the input graph into multiple bipartite subgraphs. Additionally, [15,24] have expanded the input graph to create a bipartite graph. However, this leads to an oversampled filter bank, which may not be desirable for some applications such as compression. Some GFBs on arbitrary graphs have been studied in [1,5,9–11,16,23]. Authors in [5] extended GFB in [4] to arbitrary graphs. Notably, the proposed structure in [5] is a critically sampled spline graph filter bank (SGFB) with PR property. Different from other GFBs, the synthesis section of SGFB works with only one inverse filter. As a result, this modification increases degrees of freedom to d
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