A Study Of Complex Filter Banks For Graph Signals

Graphs provide a flexible model for irregular data such as biological networks,social networks,and sensor networks.The graph is widely used and essential for the research of graph signal processing.The graph wavelet filterbank is one of the most important processing tools for graph signals.The low channel coefficients can represent the approximation of the original graph,and the high channel coefficients can represent the detail of the original graph.The corresponding graph wavelet transform of the real graph filterbank is transformed into a real transform,which cannot provide phase information.As an extension of the real wavelet filterbank on the graph,this paper proposes three structures of the complex wavelet filterbank on the graph.Firstly,a complex wavelet filter bank based on the complex Laplace matrix is proposed.The eigenvalues and eigenvectors of the real Laplacian matrix of the undirected graph are real,and it can only construct the real filter bank,this paper proposes to use the complex Laplacian matrix to construct the graph complex wavelet filterbank.This method constructs the complex weight matrix of the graph,where the amplitude of the complex weight is determined by the similarity of the signal values of adjacent vertices.Imitating the phase of the complex sinusoid,the relative position relationship between adjacent vertices is used to define the phase of the complex weight,the complex Laplacian matrix thus determined as a positive semi-definite Hermitian matrix.The paper proves the perfect reconstruction of the graph complex wavelet filterbank.The experiments show that the fast algorithm based on Chebyshev polynomial approximation can approximate perfect reconstruction,and the complex wavelet coefficients can provide important structural information.And the angle of the complex Laplace matrix can determine the direction selectivity of the complex filter bank.Secondly,we propose the orthogonal and biorthogonal separable complex filterbanks on the graph.The complex filterbank uses the two-channel structure of the real filterbank,and the filters in it are designed as the separable complex filters composed of existing real filters.This paper derives the conditions that the orthogonal separable complex filterbank satisfies antialiasing,perfect reconstruction and orthogonality,and the conditions that the biorthogonal separable complex filterbank satisfies anti-aliasing and perfect reconstruction,then further derive the conditions to be satisfied by the real filter constituting the complex filter.Simulation experiments show that the complex wavelet coefficients can provide the structural information of the graph.Moreover,the complex filter is not designed from the complex number domain to avoid complex operations in the complex number domain.Finally,a dual-tree complex wavelet filterbank on directed graph is proposed.The dualtree complex filterbank takes the unitary shift operator matrix S as the base matrix and is composed of real filter banks that do not interfere with each other.One tree obtains the real part of the complex transform,and one tree obtains the imaginary part of the complex transform.And the overall complex transformation of the complex filter bank as ?=?h+j?g.By defining the mapping relationship between the S domain and the classical domain,the linearphase biorthogonal filter and the q-shift filter in the classical domain can be mapped into graph filters and applied to the dual-tree complex filterbank.In addition,this paper uses the discrete wavelet framework to implement graph multi-scale analysis,which avoids the down-sampling operation on the directed graph.The experiment shows that the dual-tree complex filter bank can realize the perfect reconstruction of the graph signal,and the complex wavelet coefficients can provide the structure information of the graph.The complex filterbank also has better performance in denoising.The wavelet of multi-dimensional dual-tree complex filter bank has directivity.