Design Of Time-vertex Joint Graph Filter Bank With Sampling

With the advent of the era of big data,the cost of collecting data is decreasing.The data collected from the real world usually have high-dimensional and irregular characteristics,such as brain neural network data,sensor network data,etc.Classical digital signal processing tools have achieved great success in processing discrete signals in the regular domain,but cannot adequately capture the correlations of irregular data.Therefore,researchers proposed a more universal graph signal processing theory based on graph model.Graph signal processing extends the classical signal processing theory from the regular time domain to the irregular graph domain,so that the correlation between data can be effectively utilized.Among them,graph filter banks have attracted much attention because of their multi-scale characteristics and multi-resolution analysis ability.However,the actual collected data usually also has time-varying characteristics,such as temperature network data,sea level pressure data,etc.In order to make better use of the temporal and spatial correlation between data,researchers proposed the time-vertex graph signal processing theory,and graph filter bank theory is also extended from static graphs to time-vertex joint graphs.Considering the complex aliasing effect caused by the sampling operation,the existing time-vertex joint graph filter banks are mainly non-subsampling structures,while the joint time-vertex graph filter bank with sampling has not been proposed yet.To fill the gap in this research field,this paper proposes two joint time-vertex graph filter banks with different sampling structures.The specific work is showed as follows:(1)For the problem that the existing joint time-vertex graph filter banks does not consider the sampling operation,this paper proposes a multi-channel joint time-vertex critical sampling graph filter bank(TVCSGFB)for processing time-varying graph signals,which satisfies the properties of critical sampling and perfect reconstruction.Different from conventional approaches based on the vertex or spectral domain,the filtering and sampling operations of our proposed framework are both carried out in the joint time-vertex domain,which effectively exploits the joint graph-temporal correlation of the data.The analysis filter bank consists of a series of 2-dimensional graph filters with different joint frequency supports,leading to the decomposition of the input graph signal into subband signals with different joint frequency supporting regions.The downsampling and upsampling(DU)operations related to critical sampling are defined in the time vertex domain and expressed in the form of Kronecker product of matrix.The design of the synthesis filter bank can be formulated as an optimization problem with minimizing the energy of the synthetic filter bank as the objective function and the perfect reconstruction condition as the constraint.Experiments show that the filter bank has excellent perfect reconstruction performance.In addition,its nonlinear approximation performance and denoising performance are significantly better than static graph filter banks.(2)For the joint time-vertex critically sampled graph filter bank,the sampling set the leads to perfect reconstruction is not unique,and the overall performance of the graph filter bank will heavily depend on the different sampling sets.For this problem,this paper proposes a joint time-vertex spectral domain spline graph filter bank,which satisfies the desirable properties of perfect reconstruction and critical sampling.The filtering and sampling of this graph filter bank are performed on the joint spectral domain,and the sampling set in the joint spectral domain is unique.Its analysis filter consists of a series of joint spline filters with different rectangular support domains,where the joint spline filter is composed of the Kronecker product of the time domain spline filter and the vertex domain spline filter.The synthesis section of this filter bank can be viewed as an inverse filtering problem.The experimental results show that the nonlinear approximation performance and denoising performance of the joint time-vertex spectral domain spline graph filter bank is significantly better than that of the spectral domain sampled graph filter bank on static graphs when processing time-varying graph signals.