Design And Distributed Implementation Of Graph Filters

Different from conventional signals such as discrete time series and digital images,data from real scenes are often unevenly distributed in space and time.The latest research advances in Graph Signal Processing(GSP)make it possible to analyze irregular structure data in vertex domain and spectral domain.It provides a framework for applying classical signal processing to large data sets by defining signals on graphs.Graph filter is one of the important cornerstone of graph signal processing,which has strong data analysis ability and rich practical application scenarios.In this paper,ARMA(Auto Regressive Moving Average)graph filter and Node-Variant FIR graph filter were researched,as well as FIR(Finite Impulse Reponse)The main work of graph filter in the signal recovery problem is as follows:(1)This paper studies the design of ARMA graph filter.Taking the error between the expected frequency response provided by users and the frequency response of ARMA graph filter as the optimization objective of graph filter coefficient solving,this paper proposes a variant iteration method of Steigliz-Mc Bride based on LMS(Least Mean Square)to solve the optimization problem.The resulting ARMA graph filter coefficients are independent of the underlying graph structure,and then the conjugate gradient algorithm is used to solve the ARMA graph filter output signals.Simulation results show that the proposed iterative algorithm has better approximation effect compared with Prony algorithm,and the frequency response of ARMA graph filter is more stable.(2)This paper studies the asynchronous realization of Node-Variant FIR graph filter.In order to improve the situation that all nodes have the same weight in each shift of ordinary FIR graph filter and the limitation of node synchronous communication on distributed computing,this paper proposes an asynchronous implementation method of node to FIR graph filter.In this method,the graph filter coefficient in each shift is made into a diagonal matrix with different elements,so as to achieve the purpose of different corresponding weights of all nodes in each shift.In addition,some nodes are randomly selected according to the update set to update each round of communication and all nodes are forced to communicate simultaneously.By setting the error between the output signal of the asynchronous algorithm and the desired filter signal as the optimization objective,the optimization problem is solved.The simulation results show that the asynchronous implementation of Node-Variant FIR graph filter proposed in this paper has faster convergence speed than the synchronous implementation.(3)This paper studies the signal recovery based on FIR graph filter.Firstly,the signal recovery problem is described as a convex optimization problem to minimize the total variation function of the graph,and an accurate closed-form solution(GTVR)can be obtained through derivation.Since the exact closed-form solution requires a large amount of computation and is prone to numerical instability in the process of matrix inversion,In this paper,an exact solution of graph filter and an approximate iterative solution(GF)of graph filter are proposed to replace the closed solution.Simulation results show that compared with GTVR method,GF method requires less computation and has similar signal recovery performance.