Design Of Time-vertex Joint Graph Filter And Filter Bank

In recent years,as one of the investigation hotspots in the signal processing domain,graph signal processing(GSP)has the key superiorities of analyzing and processing irregular field data,which has fascinated researchers.Graph filter(GF)and graph filter bank are the core contents of GSP.They can provide a feasible method for irregular data processing,and have received widespread attention.However,most of the data in reality vary with time,such as sea surface temperature data,propagation data of the epidemic disease.Static GSP has some limitations on the processing of these data,without considering their time relevance.Based on this,the time-vertex joint GSP came into being.As a generalization of static GSP,it can describe the time-varying characteristics of data,and has more realistic significance.Nevertheless,in the time-vertex joint graph,the research on graph filters and filter banks is still in the development stage,which is a challenging research topic.This paper mostly focuses on the theory and design of time-vertex joint graph filter and filter bank.The specific work is summarized as follows:(1)For the node-variant(NV)GF on the static graph,it does not consider the evolution characteristics of data in the time domain.Besides,the design freedom and flexibility of the two-dimensional(2D)polynomial time-vertex joint GF are inadequate.For solving these issues,a design algorithm of NV GF on the time-vertex joint graph is proposed in this paper.In this algorithm,the design problem of filter coefficients is divided into a series of sub-optimization problems,and solved in closed form.This can make obtained filter approach the ideal linear operator commendably.Then,using the algorithm,a filter is designed to approximate the desired inverse filter operator for signal denoising.The experiments indicate that the overall performance of the designed filter is better than that of the polynomial filter.(2)The general theoretical research and design methods of time-vertex joint graph filter bank are lack.In addition,the spectrum filtering theory and distributed implementation of the filter bank are not considered.To solve these problems,the generalized product graph model is primarily proposed in this paper.And it is used to prove the relationship between the 2D polynomial time-vertex joint GF and the spectrum function of joint graph.Then,based on the proposed generalized product graph spectrum theory,the theory and design method of M-channel time-vertex joint graph nonsubsampled filter bank(JGNSFB)are developed.A prominent feature of the filter bank is that it can be fully implemented in a distributed manner.In the construction of the filter bank,the design problem of analysis filter banks is reduced to an unconstrained optimization problem.The optimization problem is with respect to the function of passband flatness and stopband energy.By solving this problem,filters with various frequency domain shapes can be designed.Then,two different types of synthesis filter banks are designed.One is polynomial synthesis filter banks,whose design problem is a quadratic programming problem with constraint.The constraint condition is the perfect reconstruction(PR)condition in the frequency domain of joint graph.The other is non-polynomial synthesis filter bank,which is constructed by the Lagrange multiplier method.The constraint of this problem is the PR condition on the time-vertex joint graph.Next,the large-scale matrix inverse problems that may occur in solving non-polynomial filters are addressed.Accordingly,this paper proposes a distributed algorithm with exponential convergence to realize the filter,and gives novel theoretical proofs.The experiments show that the designed filter banks have good frequency selectivity and PR characteristics.It is also indicated that the proposed distributed algorithm has obvious advantages in handling large-scale data.Moreover,in the denoising experiments of time-varying data sets,it is verified that the proposed filter bank has obvious superiority over other methods.