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Design of Nonsubsampled Graph Filter Banks Via Lifting Schemes Junzheng Jiang ,Davidb.Tay ,Qiyusun , and Shan Ouyang

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Design of Nonsubsampled Graph Filter Banks Via Lifting Schemes Junzheng Jiang ,Davidb.Tay ,Qiyusun , and Shan Ouyang IEEE SIGNAL PROCESSING LETTERS, VOL. 27, 2020 441 Design of Nonsubsampled Graph Filter Banks via Lifting Schemes Junzheng Jiang ,DavidB.Tay ,QiyuSun , and Shan Ouyang Abstract—Graph filter banks play a crucial role in the vertex general, difficult to analyse and design, except in cases with and spectral representation of graph signals. The notion of two- special graphs, such as bipartite graphs [6], [7]. channel nonsubsampled graph filter banks (NSGFBs) on an undi- In [9], the notion of two-channel nonsubsampled graph filter rected graph was introduced recently. The absence of downsam- banks (NSGFB) was introduced by Jiang, Cheng and Sun. pling/upsampling operators allows greater flexibility in the design A two-channel NSGFB consists of an analysis filter bank of NSGFBs that achieve perfect reconstruction. However the design {H H } {G G } of NSGFBs that take the spectral response into account has not been 0, 1 and a synthesis filter bank 0, 1 . The absence of adequately addressed yet. Based on the polynomial/rational lifting the DU operators leads to the following perfect reconstruction scheme, this letter presents a simple method to design NSGFBs with (PR) condition [9] good spectral response and perfect reconstruction. Experimental results will demonstrate the effectiveness of the proposed method in G0H0 + G1H1 = I. (I.1) tailoring the spectral responses of the lifted NSGFBs. Application of the NSGFB to denoising will also be considered. The analysis filter bank {H0, H1} of an NSGFB on a graph G is said to be normal if (i) the lowpass graph filter passes Index Terms—Graph signal processing, nonsubsampled graph H D1/21 D1/21 filter bank, lifting scheme, Laplacian matrix. the weighted constant signal, i.e. 0 G = G ; and (ii) 1/2 the highpass graph filter blocks that signal, i.e. H1DG 1 = 0, T where DG is the degree matrix of the graph G and 1 =[1... 1] . I. INTRODUCTION In [9], the authors proposed two methods to design normal RAPH signal processing (GSP) is a research field that is NSGFBs satisfying the PR condition (I.1) but do not explicitly gaining prominence and finds a variety of applications take the spectral response into account. The spectral approach whereG the data is defined over an irregular domain, e.g. sensor to graph signal transforms provides a representation that is networks and social networks [1], [2]. One of the main ideas similar to the Fourier transform for regular-domain signals. behind GSP is the exploitation of pairwise relationships between A spectral/frequency domain interpretation of the transforms data values defined over nodes via a graph model. Some of allows a distinction between low-frequency and high-frequency the fundamental principles from traditional signal processing components of a graph signal. Another advantage with the spec- λ λ have been extended to the graph domain, giving rise to the tral approach is that the spectral filter h( ) ( spectral variable) graph Fourier transform [3], graph filter [4], graph wavelet filter can be designed independently of the graph structure [5]–[8]. bank [5]–[9], etc. Graph filter banks (GFBs), in particular, is a When the filter is applied to a specific graph, the filter adjusts λ topic that has drawn significant attention as they play a crucial itself to the structure of the graph when is substituted with a L role in the vertex and spectral representation of graph signals. graph matrix, e.g. Laplacian , which encodes the structure of GFBs with downsampling/upsampling (DU) operators are, in the graph. In this letter, we propose a simple yet flexible method to design PR NSGFBs that is based on the lifting scheme applied to the analysis filters. It allows the tailoring of the spectral Manuscript received October 18, 2019; revised January 22, 2020; accepted February 8, 2020. Date of publication February 27, 2020; date of current version response via a lifting filter. Analysis filters with good frequency March 19, 2020. This work was supported in part by the National Natural Science characteristics can be designed by optimizing the lifting filters. Foundation of China under Grant 61761011 and Grant 61871425, in part by the Transform based on lifting has also been previously proposed Natural Science Foundation of Guangxi under Grant 2017GXNSFAA198173, in [11] for graph signals. Subsampling is however used in [11] and in part by the National Science Foundation under Grant DMS-1816313. The and some edge information are lost because of this. The filters associate editor coordinating the review of this manuscript and approving it for publication was Prof. Ran Tao. (Corresponding author: David Tay.) in [11] are specified in the vertex domain and shaping the spectral Junzheng Jiang is with the School of Information and Communication, response is not readily achievable. In this work there is no loss Guilin University of Electronic Technology, Guilin 541004, China, and also of edge information (as there is no subsampling) and the control with the National and Local Joint Engineering Research Center of Satellite the spectral characteristics of the filters, e.g. low-pass, is easily Navigation Positioning and Location Service, Guilin 541004, China (e-mail: achieved. [email protected]). David B. Tay is with the School of Information Technology, Deakin Univer- sity, Waurn Ponds, VIC 3216, Australia (e-mail: [email protected]). RELIMINARIES AND IFTING CHEME Qiyu Sun is with the Department of Mathematics, University of Central II. P L S Florida, Orlando, FL 32816 USA (e-mail: [email protected]). Let G =(V,E,W) be an undirected weighted graph with no Shan Ouyang is with the School of Information and Communication, { } Guilin University of Electronic Technology, Guilin 541004, China (e-mail: self-loops and multiple edges, where V = 1, 2,...,N is the [email protected]). set of nodes, E is the set of edges, and W =[wij ]1≤i,j≤N is the Digital Object Identifier 10.1109/LSP.2020.2976550 weighted adjacency matrix. The degree matrix DG is a diagonal 1070-9908 © 2020 IEEE. Personal use is permitted, but republication/redistribution requires IEEE permission. See https://www.ieee.org/publications/rights/index.html for more information. Authorized licensed use limited to: University of Central Florida. Downloaded on September 06,2020 at 21:12:07 UTC from IEEE Xplore. Restrictions apply. 442 IEEE SIGNAL PROCESSING LETTERS, VOL. 27, 2020 matrix whose i-th diagonal entry is given by dii = j∈V wij . analysis filters in (II.3) satisfies (I.1). Conversely, if the prototype P P The symmetrically normalized Laplacian matrix is defined as analysis filters {H0 , H1 } are commutative, then the lifted Lsym ≡ I − D−1/2WD−1/2 Lsym NSGFB with the synthesis filters in (II.2) satisfies (I.1). G G G . The eigendecomposition of G P sym T The commutativity between prototype synthesis filters {G0 , is given by LG = UΛU , where U is the orthogonal eigen- P G1 } follows if they both can be diagonalized by a common Λ =diag(λ1,...,λN ) vector matrix and is a diagonal eigen- nonsingular matrix. In [9], a representative class of analysis filter value matrix with eigenvalues 0 ≤ λ1 ≤···≤λN ≤ 2.The spln spln sym bank is the spline filter bank {H0,n , H1,n } of order n ≥ 1 eigenvalues of LG can be interpreted as spectral frequen- spln sym n spln sym n cies [5]–[7]. given by H0,n ≡ (I − LG /2) , H1,n ≡ (LG /2) .The G x spln A signal on the graph is represented by a vector = corresponding minimum degree synthesis filter bank {G , ··· T 0,n [x1 xN ] , where each element xi represents a numerical Gspln} Gspln ≡ quantity associated with node i. The graph Fourier transform 1,n is designed via the Bezout identity, where 0,n T sym spln sym x of a graph signal x is given by x = U x. A graph filter is Pn(LG /2), G1,n ≡ Pn(I − LG /2), and Pn is the unique represented by a matrix H that acts on an input signal x to polynomial solution of degree n − 1 to the equation (1 − n n give an output signal y = Hx. A common class of filters are n n − sym t) P (t)+t P (1 t)=1. Now the synthesis filters are poly- functions of the normalized Laplacian matrix LG , nomials of the symmetric Laplacian matrix. Therefore, they share the same orthogonal eigenvector space, and this makes H Lsym U Λ UT = h( G ):= h( ) , (II.1) them commutative. A similar argument can be made for the commutativity of the prototype analysis filters. where h is a function defined on [0, 2] and h(Λ) is a diagonal matrix with diagonal entries h(λ1),...,h(λN ). For a filter H of the form (II.1) and an input x, the graph Fourier transform of the III. DESIGN METHODOLOGY corresponding output Hx is given by Hx = h(Λ)x. Hence the The role of the analysis filter bank is to decompose the filter H has frequency response h(λi) at the spectral frequencies input signal into different spectral frequency bands, while the λi, 1 ≤ i ≤ N [1], [3], [5]–[7]. Due to the above observation, role of the synthesis filter bank is to reconstruct the signal. the function h is known as the spectral response function of the The frequency characteristic of the former is therefore more filter H in (II.1). important than the latter in many applications. With the aim to The lifting scheme, pioneered by Sweldens [10], can improve achieve good frequency characteristics for the analysis filters, the frequency response of the subband filters by using lifting we concentrate on the lifting scheme (II.3) so that the spectral filters.
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