Design Of Non-Time-Varying And Two-Dimensional Separable Time-Varying Nonsubsampled Graph Filter Banks

With the development of the era of big data,the data recorded in daily life are characterized by quantification,diversification and irregularization.Among them,complex irregular distributed network structure exists widely in reality,such as epidemic transmission network,social network,global sea level temperature network,sensor temperature network and so on.It is difficult to use traditional digital signal processing to analyze and process the complex irregular network data,so graph signal processing is developed.In traditional digital signal processing,multi-scale transform can reveal the structural information of signals at different resolution levels.At the same time,they can provide a rough-to-fine analysis method,thus reducing the dimension of the signal.Similarly,algorithms dealing with irregular large-scale network data must have the ability to reduce the data dimension significantly.Graph wavelet and graph filter banks have the characteristics of multiresolution analysis,which is an important way to achieve this goal.Therefore,as an important graph signal analysis tool in graph signal processing,graph filter banks have attracked more and more scholars' attention and research.At present,the main design of graph filter banks concentrates on critically sampled and oversampled.However,it is difficult to accurately define the downsampling operation for a generalized graph signal in graph filter banks.Most of the studies of graph filter banks are based on non-time-varying graph signals,and the temporal correlation of graph signals is not considered.Therefore,the design method of non-time-varying and time-varying nonsubsampled graph filter banks is proposed to overcome the shortcomings of the existing design methods.1.In order to overcome the problem that it is difficult to accurately define the downsampling operation for a generalized graph signal in graph filter banks,this paper focus on the design algorithms of nonsubsampled graph filter banks.First,the spline filters are taken as the analysis filter banks.Then,two different methods are proposed to construct the synthesis filter banks.In the design method of vertex domain,the synthesis filter banks can be constructed with the perfect reconstruction conditions in vertex domain.By taking into account the frequency of the subband filters in the design method of frequency domain,the synthesis filter banks are designed by solving a constrained optimization problem involving the spectrum characteristics of the filters.The design methods can lead to two channel nonsubsampled graph filter banks with perfect reconstruction.Finally,taking the two channel nonsubsampled graph filter banks as a basicbuilding block,we can construct multilevel nonsubsampled graph filter banks used to realize multiresolution analysis of graph signal through cascading.Simulation results show that the designed nonsubsampled graph filter banks have perfect reconstruction property.Furthermore,the designed multilevel nonsubsampled graph filter banks have better denoising performance than the existing graph filter banks.2.In order to solve the problem that existing graph filter banks are difficult to process time-varying graph signals,a design method of two-dimensional separable time-varying nonsubsampled graph filter banks is proposed.Firstly,based on the properties of two-dimensional separable graph filters,the analysis graph filter banks of two-dimensional separable time-varying nonsubsampled graph filter banks are designed.Then,according to the perfect reconstruction conditions of joint frequency domain,using the properties of Bezouts identity and two-dimensional separable graph filters,synthesis graph filter banks are constructed in joint frequency domain.Finally,in the design method of joint time-vertex domain,the design problem of synthesis graph filter banks is formulated into an optimization problem with constraints,and the synthesis graph filter banks are solved with the perfect reconstruction conditions as the constraint function.The inverse operation of large-scale matrix exists in the solution of synthetic graph filter banks,which is not conducive to the processing of time-varying graph signals.In order to avoid large-scale matrix inversion,an iterative reconstruction algorithm is proposed to reconstruct time-varying graph signals.Simulation results show that the designed time-varying nonsubsampled graph filter banks have perfect reconstruction property,and the proposed iterative reconstruction algorithm has a lower number of iterations.Furthermore,the time-varying nonsubsampled graph filter banks have better denoising performance than the existing graph filter banks.