Research On The Construction Method Of M-Band Wavelet Filter Banks In Directed Graphs

The directed graph M-band wavelet filter bank is a multi-channel signal processing method on directed graphs,which can be used for multi-scale analysis of signals on directed graphs and has high research value and important theoretical significance.Compared with undirected graphs,directed graphs are analyzed based on adjacency matrix and cannot obtain a graph representation using Laplace decomposition,which makes the existing multi-channel filter bank method on undirected graphs cannot be directly applied to directed graphs.However,most of the existing filter banks for directed graphs are only for the two-channel case,which cannot effectively meet the demand for multi-scale analysis of signals on directed graphs and restrict the development of signal processing on directed graphs.Aiming at the demand of multi-scale analysis of signals on directed graphs for multichannel filter banks on directed graphs,this paper carries out research on the construction method of M-band wavelet filter banks on directed graphs based on the traditional M-band wavelet transform and dual tree complex wavelet transform theory,combined with the characteristics of signal representation on directed graphs,and the main research contents include:1.A construction method of M-channel oversampling graph filter banks based on directed graph Hermite matrix is proposed.In order to solve the problem that a directed graph cannot directly use an asymmetric Laplacian matrix as the base matrix,the direction information of the directed edge in the directed graph is expressed as a function related to the rotation parameter,and the undirected graph symmetric adjacency matrix that uniquely corresponds to the directed graph is obtained from the asymmetric adjacency matrix of the directed graph,and then the Hadamard product of the two is taken as the adjacency matrix of the directed graph,then the directed graph Hermite Laplacian matrix related to the rotation parameter is obtained and used as the base matrix.Then,when the multi-channel oversampling graph filter bank meets the perfect reconstruction condition,the conditions that each filter in the filter bank should meet are derived and analyzed.For different scale digraphs,the selection of rotation parameters and matrix weights is discussed.Finally,design simulation experiments to verify the reconstruction characteristics and denoising performance of the designed filter banks.Simulation experiments have shown that for different graph signals,using the Chebyshev polynomial approximation algorithm,this method can achieve perfect reconstruction of the original signal,and compared to existing methods,this method can achieve higher peak signal-to-noise ratio.2.An M-channel graph filter bank construction method based on the directed graph unitary shift matrix is proposed.To address the energy non-conservation problem faced when using the adjacency matrix of the directed graph for signal shifting,a singular value decomposition of the adjacency matrix is performed to obtain the closest unitary matrix to the adjacency matrix in Hilbert space and use it as the basis matrix on the directed graph.Then,under the premise that the filter bank can be completely reconstructed on the directed graph,the relationship between the known single low-pass filter and the remaining high-pass filters in the filter bank is investigated,and the perfect reconstruction property of the multi-channel filter bank based on the unitary shift matrix is proved.Combined with the M-band dual tree complex wavelet transform,the M-band dual tree complex filter bank on the directed graph was constructed,and the reconstruction characteristics and denoising performance of the designed filter bank are simulated and verified.The simulations show that the multi-channel filter bank can obtain higher peak signal-to-noise ratio and lower mean square error than the two-channel filter bank,and has better reconstruction performance and more directional selectivity than the two-channel filter bank.Moreover,dual tree complex filter bank achieves higher peak signal-to-noise ratio and better denoising performance than the real filter bank on the directed graph.