Design of orthogonal graph filter bank with known eigenvalues of Laplacian matrix

IET Signal ProcessingVolume 13, Issue 5 p. 551-561 Research ArticleFree Access Design of orthogonal graph filter bank with known eigenvalues of Laplacian matrix Chien-Cheng Tseng, Corresponding Author Chien-Cheng Tseng tcc@nkust.edu.tw Department of Computer and Communication Engineering, National Kaohsiung University of Science and Technology, Kaohsiung, TaiwanSearch for more papers by this authorSu-Ling Lee, Su-Ling Lee Department of Computer Science and Information Engineering, Chang Jung Christian University, Tainan, TaiwanSearch for more papers by this author Chien-Cheng Tseng, Corresponding Author Chien-Cheng Tseng tcc@nkust.edu.tw Department of Computer and Communication Engineering, National Kaohsiung University of Science and Technology, Kaohsiung, TaiwanSearch for more papers by this authorSu-Ling Lee, Su-Ling Lee Department of Computer Science and Information Engineering, Chang Jung Christian University, Tainan, TaiwanSearch for more papers by this author First published: 01 July 2019 https://doi.org/10.1049/iet-spr.2018.5291Citations: 10AboutSectionsPDF ToolsRequest permissionExport citationAdd to favoritesTrack citation ShareShare Give accessShare full text accessShare full-text accessPlease review our Terms and Conditions of Use and check box below to share full-text version of article.I have read and accept the Wiley Online Library Terms and Conditions of UseShareable LinkUse the link below to share a full-text version of this article with your friends and colleagues. Learn more.Copy URL Share a linkShare onFacebookTwitterLinked InRedditWechat Abstract In the conventional designs of orthogonal graph filter bank, both eigenvalues and eigenvectors of Laplacian matrix are assumed to be unknown such that graph filter bank can be efficiently implemented by using polynomial graph filters; that is, the output signal at each vertex is computed only from the signal at that vertex and its K -hop neighbourhoods. However, the disadvantage of these methods is that the designed orthogonal filter bank does not have the exact perfect reconstruction (PR). In this study, the authors first assume that the eigenvalues of Laplacian matrix are known but eigenvectors are still unknown. Then, the orthogonal graph filter bank is designed by solving the simultaneous equation or Lagrange multiplier method such that the PR condition at all eigenvalues can be achieved for a given graph structure. Although the proposed filter bank is a graph-structure-dependent design, it can be efficiently implemented by using polynomial graph filters. Finally, experimental results are shown to demonstrate the effectiveness of the proposed design method. 1 Introduction In recent years, there is a massive progress in the researches of sensor network, traffic transportation network, economic network, social network and biological brain network [1-4]. This leads to our great attentions in the researches of processing irregular signals from these networks. Generally speaking, the irregular signals are represented by the graphs composed of several vertices, edges and weights. To process irregular graph signals, the graph signal processing (GSP) method has been proposed [5-18]. There are two frameworks to develop GSP methods. One is the undirected graph where the graph Laplacian matrix is used as the fundamental building block [5], the other is directed graph where the adjacency matrix of graph is used as the basic building block [6, 7]. In these two frameworks, the graph Fourier transforms (GFTs) have been defined. The eigenvalues and eigenvectors of graph Laplacian matrix or adjacency matrix correspond to the frequencies and spectral Fourier basis. In [5-9], the generalisation of the filtering operation in the vertex domain over graphs has been investigated. The frequency selective graph filters (low-pass, band-pass and high-pass filters) have been defined and designed. However, the function of single graph filter is limited, so it is an interesting research topic to combine many graph filters into a filter bank for improving its signal processing capability. Thus, graph filter bank has been developed in the GSP research area. In the literature, the graph filter-bank design and its applications have been studied in [10-18]. In [10], the perfect reconstruction (PR) two-channel wavelet filter bank for graph-structured data has been constructed by Narang and Ortega. The method to design quadrature mirror filter (QMF) bank for bipartite graph has been investigated. In [11], the orthogonal QMF bank is extended to the compact support biorthogonal wavelet filter banks for arbitrary undirected graphs. In [12, 13], the M -channel oversampled graph filter banks are designed and graph signal denoising experiments are made. In [14], the near orthogonal graph filter banks are designed by using Bernstein polynomial approximation and constrained optimisation. This method allows the trade-off between transient band sharpness, ripple magnitude and reconstruction error. In [15], the polyphase techniques for constructing biorthogonal bipartite graph filter bank are presented. In [16], the spline-like wavelet filter banks for multiresolution analysis of graph-structured data is constructed. In [17, 18], the classical multirate signal processing theory is extended to GSP cases and M -channel graph filter bank is studied. Owing to the rapid progress of graph filter bank, it is an interesting research topic to design PR filter bank for processing graph signals. This is the purpose of this paper. In the conventional designs of orthogonal graph filter bank, both eigenvalues and eigenvectors of Laplacian matrix are assumed to be unknown such that graph filter bank can be efficiently implemented by using polynomial graph filters. However, the disadvantage of these methods is that the designed orthogonal filter bank does not have the exact PR. In this paper, we assume that the eigenvalues of Laplacian matrix are known but eigenvectors are still unknown. Thus, the orthogonal graph filter bank is designed by solving the simultaneous equation or Lagrange multiplier method such that the PR condition at all eigenvalues can be achieved for a given graph structure. This paper is organised as follows. In Section 2, the background of GSP is first reviewed and some basics in graphs theory are described. In Section 3, the design problem of two-channel graph filter bank is described. The critically sampled two-channel filter bank that operates on a bipartite graph is studied. In Section 4, the design methods of graph filter banks are described for known and unknown eigenvalues of Laplacian matrix. In Section 5, some design examples of orthogonal graph filter banks on path graph, ring graph, cube graph, star graph and complete bipartite graph are presented. In Section 6, some discussion are made and application examples are illustrated. Finally, a conclusion is made. 2 Preliminaries In this section, the background of GSP is first reviewed. Then, some basics in graph theory are described. Let graph signal be defined on a finite undirected graph , where is the vertex set and is the edge set. Adjacency matrix is the symmetric matrix whose elements give the weight of the edge connecting vertices to . We assume that the graph is loopless; that is, . Let be the sum of all edge weights connected to vertex , then is the diagonal degree matrix. The normalised graph Laplacian matrix is defined as whose eigendecomposition is given by (1)where matrix is composed of the orthonormal eigenvectors and diagonal matrix is composed of the real eigenvalues ; that is (2a) (2b)The properties and are satisfied in the above decomposition. The spectrum is defined as the set of eigenvalues of matrix and it is always a subset of the closed set for any graph. Since the eigenvalues of matrix may have multiplicities, the set of all distinct eigenvalues is defined as , where M is the number of distinct eigenvalues. Clearly, is a subset of ; that is, . The GFT of signal is defined as . Moreover, its corresponding inverse GFT is given by . Now, some basics in isomorphic graph and bipartite graph are described below. Definition 1.The graph is said to be isomorphic to graph if there exist a one-to-one correspondence and a one-to-one correspondence such that two vertices and are end vertices of some edge in , then and are end vertices of the edge in . The ring graphs in Figs. 1 a and b are isomorphic because two graphs have the same ring structure with different vertex ordering. The vertices of two isomorphic graphs can be labelled in such a way that the corresponding adjacency matrices are equal. The property of Laplacian matrices of isomorphic graphs are described below. Fig. 1Open in figure viewerPowerPoint Illustrative example of three isomorphic graphs (a) Ring graph with continuous vertex order, (b) Ring graph with interlaced vertex order, (c) Bipartite graph Fact 1.Let and be the Laplacian matrices of two isomorphic graphs and , then there exists a permutation matrix such that that is, Laplacian matrices and are similar, so they have the same eigenvalues.The details of the above Fact 1 can be found in [19]. Since the ring graphs in Figs. 1 a and b are isomorphic, both graphs have the same eigenvalues; that is, the spectrum of these two graphs are the same. However, the eigenvectors of the graphs in Figs. 1 a and b are different. There exists a permutation relation between the eigenvectors of two isomorphic graphs. Definition 2.A graph is called as a bipartite graph if the vertex set can be partitioned into two non-empty disjoint subsets and such that every edge in graph connects one vertex from to one vertex from . The vertex subsets satisfy the relation and , where is the empty set. Fig. 1 c shows a bipartite graph which is the same as the ring graph in Fig. 1 b, so the ring graph with even number of vertices is a special bipartite graph. A more general result to check whether a graph is bipartite is given by the fact below. Fact 2.A non-null graph is bipartite if and only if it does not contain any cycle of odd length.The proof of this fact can be found in [20]. This fact tells us that the following graphs are all bipartite: ring graph with even number of vertices, path graph, tree graph, grid graph, star graph and cube graph because they do not contain any cycle of odd length. The eigenvalue of Laplacian matrix of bipartite graph satisfies the following fact [21]. Fact 3.For each eigenvalue of the Laplacian matrix of bipartite graph G, the value is also an eigenvalue of G.On the basis of this fact, the design of PR two-channel wavelet filter banks for graph-structured data has been presented in [10]. However, the explicit eigenvalues of bipartite graph are not considered in [10]. In this paper, we assume that the eigenvalues are known to design PR graph filter bank. Some explicit eigenvalues of Laplacian matrices of special graphs are listed below [21]: (1) For the complete bipartite graph on N vertices, the eigenvalues of are 0, 1 (with multiplicity ) and 2. (2) For the path graph on N vertices, the eigenvalues of are , for . (3) For the ring graph on N vertices, the eigenvalues of are , for . (4) For the star graph on N vertices, the eigenvalues of are 0, 1 (with multiplicity ) and 2. (5) For the cube graph on vertices, the eigenvalues of are (with multiplicity ), for . Since complete bipartite graph, ring graph with even N, path graph, star graph and cube graph are all bipartite graphs with known eigenvalues of Laplacian matrix. Thus, it is interesting to design orthogonal graph filter bank with known eigenvalues. The designed filter banks have PR for these bipartite graphs. This is the purpose of this paper. 3 Two-channel graph filter bank Fig. 2Open in figure viewerPowerPoint Two-channel graph filter bank In this section, the design problem of two-channel graph filter bank is described. The critically sampled two-channel filter bank studied in [10] operates on a bipartite graph whose vertex set V includes two disjoint subsets and with . Without losing generality, the vertex in are labelled continuously as and the complement set is then . Fig. 2 depicts the block diagram of two-channel filter bank which is composed of two analysis graph filters , two synthesis graph filters and down-up-sampling operations and . In [10], it has been shown that the overall output signal of the filter bank is the sum of the output signals and of the two channels; that is (3)where diagonal matrices with (4) Thus, the distortion-free PR filter-bank condition yields the results (5a) (5b)On the basis of the eigendecomposition of Laplacian matrix , four graph filters can be written as (6a) (6b)where , , and are four spectral kernels. Substituting (6a) and (6b) into (5a) and (5b) and using the eigenproperty of bipartite graph [10], the PR condition of bipartite graph filter bank becomes (7a) (7b)for . Therefore, the two-channel graph filter-bank design problem is how to design low-pass filter kernels , and high-pass filter kernels , such that the PR condition in (7a)–(7b) is satisfied. For the orthogonal graph filter-bank design, the following spectral kernels are selected: (8)Thus, the problem becomes how to design a single spectral kernel to satisfy the ideal low-pass spectral response and the following PR condition: (9)Clearly, these are N discrete constraints in the graph spectrum rather than infinite continuous constraints on the interval . Depending on whether the eigenvalues of matrix are known or not, the design problem can be solved in the following two different cases: Case 1: The eigenvalues and eigenvectors of matrix are both unknown. In this case, the orthogonal filter bank with near PR can be designed by using polynomial graph filters. Case 2: The eigenvalues are known but eigenvectors are unknown. In this case, the orthogonal filter bank with PR can be designed by polynomial graph filters. In next section, the designs of orthogonal graph filter bank in the above two cases are investigated in details. 4 Design of graph filter bank In this section, the design methods of orthogonal graph filter bank in two different cases are presented. The details are described below. 4.1 Design method in case 1 In this case, the eigenvalues and eigenvectors of Laplacian matrix are both unknown in order to make the kernel design independent of the edge weight and vertex order of the bipartite graph, so the kernel is designed as continuous function of rather than the function of the discrete eigenvalues . Therefore, the discrete PR condition in (9) is changed to a more strict continuous condition below: (10)Moreover, the low-pass graph filter is chosen as a polynomial graph filter with order K(11)Since the Laplacian signal of a graph signal can be computed by (12)where the neighbourhood is the set of the vertices connected to , the output signal of graph filter at each vertex can be computed only from the signal at that vertex and its K -hop neighbourhoods. Substituting (1) into (11), it yields the result (13)where kernel is given by the polynomial form (14)Thus, the design problem becomes how to determine filter coefficients such that the PR condition in (10) is satisfied as well as possible. To achieve this purpose, the filter coefficients are found by minimising the following cost function: (15)Moreover, to let graph filter be a low-pass filter with some vanishing moments, the following equality constraints are usually incorporated into the optimisation process: The positive integers and are chosen to control the flatness of spectral response at . In [14, 15], some algorithms have been presented to solve this constrained optimisation problem. After is designed, the other three graph filters can be obtained from by using (8); that is, and (16)Although the implementation of graph filter bank does not require the explicit computation of the entire set of eigenvalues and eigenvectors of the Laplacian matrix in case 1, the designed results do not satisfy the PR condition exactly. Thus, this method is suitable for filter-bank processing on a large-sized graph, in which eigendecomposition of is difficult to be computed. 4.2 Design method in case 2 In this case, we assume that the eigenvalues of matrix are known but eigenvectors are unknown. Since complete bipartite graph, ring graph with even N vertices, path graph, star graph and cube graph are all bipartite graphs that have explicit expressions of the eigenvalues, the graph filter bank on these graphs can be designed by using the method in case 2. To accomplish this design, the ideal spectral kernel is defined as follows: (17)This leads to the following spectral response: (18)If we choose kernel , then it is easy to check the PR condition in (9) is satisfied for any graph spectrum . Here, the low-pass graph filter is also chosen as a polynomial graph filter in (11) for efficient implementation. The proposed method is based on the following fact. Fact 4.If the filter coefficients are determined by letting spectral response in (14) be equal to the ideal response in (17) for all distinct eigenvalues ; that is, the following M equalities are imposed on the design: (19)then the PR condition in (9) is satisfied. Proof.Since the are the distinct eigenvalues set of the spectrum , (19) yields the result (20)Using (17) and (20), we have (21)So, the PR condition in (9) is satisfied. The proof is completed.Now, Fact 4 is used to design graph filter bank with PR. The equation in (19) can be rewritten as the form (22)where (23)The solution of linear equation (22) depends on the choices of K and M. Now, two choices are studied. In the first choice, is selected. Thus, the solution of (22) is unique and given by (24)Since the matrix and vector only depend on the eigenvalue set , it is easy to show the following fact is valid. □ Fact 5.If and graphs and have the same eigenvalue sets and , then the filter coefficient vectors of both graphs are the same even though the topologies of these two graphs are different. Although the filter coefficient vectors are the same, the filters of both graphs are different because the Laplacian matrices are different.Now, one example is illustrated to show this fact holds. Let be a star graph and be a complete bipartite graph, then the eigenvalue sets and are the same and given by . This leads that the filter coefficient vectors of both graphs are the same though the topologies are different.In the second choice, we assume so that there are extra degree of freedoms to improve the spectral response . Thus, the filter coefficients are determined by minimising the following cost function with equality constraint : (25)where passband region and stopband region . The is a prescribed small positive number. Moreover, the spectral response in (14) can be written into the inner product form (26)where the coefficient vector and vector . Substituting (26) into (25), we have (27)where , and . Thus, the design problem can be formulated as the optimisation problem (28)This problem can be easily solved by using the Lagrange multiplier method and its optimal solution is given by (29)Since the constraint is satisfied in the optimal solution, the response satisfies the PR condition in (9) exactly. After is designed, the graph filter can be obtained from by using (16) and the filters at synthesis bank are given by and . Finally, it is worth mentioning that the method in case 2 is suitable for signal processing on a graph, in which eigenvalues of are computable or they have explicit expressions. In the next section, several design examples are presented to demonstrate the effectiveness of the above methods. 5 Design examples In this section, some design examples of orthogonal graph filter banks are presented. To evaluate performance, the reconstruction error is defined as (30)and the maximum error is . Clearly, if the error is satisfied, the PR condition in (9) holds. Now, the design examples on path graph, ring graph, cube graph, star graph and complete bipartite graph are illustrated. Example 1.In this example, the orthogonal graph filter-bank design on path graph is studied. Fig. 3 shows a path graph with . If the vertex order is properly labelled, path graph is a special bipartite graph. The eigenvalues of Laplacian matrix of path graph are given by for . It is clear that and . Since all eigenvalues are distinct, we have and (31) Now, the designed results with and are reported below.First, the results with are studied. The parameters are chosen as and . If the design method in [14] is employed, the filter coefficients of the method in case 1 can be obtained from Table 2 in [14]. Figs. 4 a and b depict the designed spectral responses , and reconstruction errors . The is equal to , so the PR condition is not satisfied by the method in [14]. Moreover, solving the simultaneous equation in (24), the filter coefficients of the method in case 2 can be computed. Figs. 4 c and d show the designed spectral responses , and reconstruction errors . It is clear that the is equal to zero, so the PR condition is satisfied by the proposed method.Next, the results with are studied. The parameters are chosen as and . If the design method in [14] is employed, the filter coefficients of the method in case 1 can be obtained from Table 2 in [14]. Figs. 5 a and b depict the designed spectral responses , and reconstruction errors . The is equal to , so the PR condition is not satisfied by the method in [14]. Moreover, using the Lagrange multiplier method in (29) and , the filter coefficients of the method in case 2 can be computed. Figs. 5 c and d show the designed spectral responses , and reconstruction errors . It is clear that the is equal to zero, so the PR condition is exactly satisfied by the proposed method.Finally, the graph signal reconstruction error in vertex domain is studied. The parameters are chosen as and . Fig. 6 a shows the adjacency matrix with size . Fig. 6 b depicts the input graph signal . To evaluate the performance, the reconstruction error in vertex domain is defined by (32)where , and are the graph signals at the synthesis bank in Fig. 2. If the designed graph filter in Fig. 4 a is used, Figs. 6 c and d show the graph signals and . The reconstruction error in this design is equal to , so the graph signal is not perfectly reconstructed by the method in [14]. Moreover, if the designed graph filter in Fig. 4 c is used, Figs. 6 e and f show the output graph signals and . The reconstruction error is equal to zero, so the graph signal is perfectly reconstructed by the proposed method. Fig. 3Open in figure viewerPowerPoint Illustrative example of path graph with N = 7 (a) Path graph with continuous vertex order, (b) Path graph with interlaced vertex order, (c) Isomorphic bipartite graph Fig. 4Open in figure viewerPowerPoint Designed results with K + 1 = M = 7 for path graph (a) Designed spectral responses |h0 (λ)| (solid line) and |h1 (λ)| (dashed line) by the method in [14], (b) Reconstruction errors E (μn) by the method in [14], (c) Designed spectral responses |h0 (λ)| (solid line) and |h1 (λ)| (dashed line) by the method in case 2, (d) Reconstruction errors E (μn) by the method in case 2 Fig. 5Open in figure viewerPowerPoint Designed results with K + 1>M for path graph. The parameters are chosen as K = 10 and M = 7 (a) Designed spectral responses |h0 (λ)| (solid line) and |h1 (λ)| (dashed line) by the method in [14]. (b) Reconstruction errors E (μn) by the method in [14], (c) Designed spectral responses |h0 (λ)| (solid line) and |h1 (λ)| (dashed line) by the method in case 2, (d) Reconstruction errors E (μn) by the method in case 2 Fig. 6Open in figure viewerPowerPoint Graph signals at the input and output of graph filter bank on path graph (a) Adjacency matrix A of the path graph in Fig. 3 c, (b) Input graph signal f, (c) Graph signal fL by the method in [14], (d) Graph signal fH by the method in [14], (e) Graph signal fL by the method in case 2, (f) Graph signal fH by the method in case 2 Example 2.In this example, the orthogonal graph filter-bank design on ring graph is studied. Fig. 1 shows a ring graph with . Ring graph with even N is a special bipartite graph with known eigenvalues for . It is clear that , and symmetric condition holds. So, the distinct eigenvalues are given by (33)where . Now, the designed results with and are reported below.First, the results with are studied for ring graph. The parameters are chosen as , and . The filter coefficients in case 1 method are obtained from Table 2 in [14]. Figs. 7 a and b show the spectral responses , and reconstruction errors . The maximum error is , so the PR condition is not satisfied by the method in [14]. Moreover, solving the simultaneous equation in (24), the filter coefficients in case 2 method can be computed. Figs. 7 c and d depict the spectral responses , and reconstruction errors . Since the maximum error is , the PR condition is satisfied in our designed result. Moreover, it can be found that the results in Fig. 7 are the same as the results in Fig. 4. The explanations are provided by using Fact 5 below. Since the eigenvalue set of ring graph with is the same as the set of path graph with , the filter coefficient vectors of these two graphs are the same. This leads to the same spectral responses and reconstruction errors in Figs. 4 and 7. However, the filters of these two graphs are different because the Laplacian matrices are different.Next, the results with are studied for ring graph. The parameters are chosen as , and . The filter coefficients in case 1 method are obtained from Table 2 in [14]. Figs. 8 a and b depict the spectral responses , and reconstruction errors . The is equal to , so the PR condition is not satisfied by the method in [14]. Moreover, using the Lagrange multiplier method in (29) and , the filter coefficients in case 2 method can be computed. Figs. 8 c and d show the designed spectral responses , and reconstruction errors . It is clear that the is equal to zero, so the PR condition is satisfied in our designed result. Fig. 7Open in figure viewerPowerPoint Designed results with K + 1 = M = 7 for ring graph (a) Designed spectral responses |h0 (λ)| (solid line) and |h1 (λ)| (dashed line) by the method in [14], (b) Reconstruction errors E (μn) by the method in [14], (c) Designed spectral responses |h0 (λ)| (solid line) and |h1 (λ)| (dashed line) by the method in case 2, (d) Reconstruction errors E (μn) by the method in case 2 Fig. 8Open in figure viewerPowerPoint Designed results with K + 1 > M for ring graph. The parameters are chosen as K = 10 and M = 8 (a) Designed spectral responses |h0 (λ)| (solid line) and |h1 (λ)| (dashed line) by the method in [14], (b) Reconstruction errors E (μn) by the method in [14], (c) Designed spectral responses |h0 (λ)| (solid line) and |h1 (λ)| (dashed line) by the method in case 2, (d) Reconstruction errors E (μn) by the method in case 2 Example 3.In this example, the orthogonal graph filter-bank design on cube graph is investigated. Fig. 9 shows a cube graph with . Cube graph with vertices is a special bipartite graph with known eigenvalues (with multiplicity ) for . So, the distinct eigenvalues are given by (34)where . Now, the designed results with and are presented. If the parameters are chosen as , and , Figs. 10 a and b show the spectral responses , and reconstruction errors of the filter bank designed by (24). The is equal to zero, so the proposed filter bank is PR. Moreover, if the parameters are chosen as , and , Figs. 10 c and d show the spectral responses , and reconstruction errors of the filter bank designed by (29). The is equal to zero, so the proposed filter bank is PR. Fig. 9Open in figure viewerPowerPoint Illustrative example of cube graph with N = 8 (a) Cube graph, (b) Isomorphic bipartite graph Fig. 10Open in figure viewerPowerPoint Designed results with N = 16 and M = 5 for cube graph (a) Designed spectral responses |h0 (λ)| (solid line) and |h1 (λ)| (dashed line) by (24) with K = 4, (b) Reconstruction errors E (μn) by (24) with K = 4, (c) Designed spectral responses |h0 (λ)| (solid line) and |h1 (λ)| (dashed line) by (29) with K = 10, (d) Reconstruction errors E (μn) by (29) with K = 10 Example 4.In this example, the orthogonal graph filter-bank designs on star graph and complete bipartite graph are investigated. The eigenvalues of star graph and complete bipartite graph with N vertices are given by 0, 1 (with multiplicity ) and 2. That is, spectrum is equal to . This yields the results , , and . So, (23) reduces to (35)Solving (35), the filter coefficients are given by , and . Thus, the designed spectral kernel is given by (36)The above satisfies the PR condition in (9) exactly for star graph and complete bipartite graph with N vertices. Moreover, for conventional design method in [14], the equality constraints , and are incorporated into the optimisati