The Design Methods And Its Distributed Implementation Of Non-uniform And M-channel Oversampled Graph Filter Banks

Graph filter banks is one of the research hotspots in graph signal processing,which is widely used in image processing,biological networks system and wireless sensor network.Similar to traditional filter banks,graph filter banks can divide different frequency components of signal into different channels for separate processing based on good frequency division characteristics.And now,there are many methods to design graph filter banks.In the study of graph filter banks,there are two basic issues need to be taken into consideration.On the one hand,current research of M-channel graph filter banks does not involve the design of non-polynomial graph filter banks and non-uniform spectrum division.Existing design of M-channel graph filter banks is mainly based on the polynomial filter which is designed by the polynomial approximation method,and then the entire graph filter bank is designed according to the structural characteristics of the graph filter bank.But there is not only polynomial filter banks,but also non-polynomial one.On the other hand,there is currently a few of research on design methods and its distributed implementation of Mchannel oversampled graph filter banks with non-polynomial filter.With the continuous growth of irregular domain data dimensions,the demand for distribtued processing is getting higher and higher,so the distributed implementation of graph filter banks is very necessary.In M-channel graph filter banks,the realization of distributed processing is mainly realized by polynomial form filters,and does not involve the distributed realization of nonpolynomial and sample graph filter banks.Meanwhile,existing distributed method with nonpolynomial filter does not take the sampling operator into consideration.According to the problem of existing methods,this paper proposes the design methods and its distributed implementation of non-uniform and M-channel oversampled graph filter banks.Firstly,a construction and design method of M-channel non-uniform oversampled graph filter banks is proposed in this paper.According to the uneven frequency distribution in the graph spectrum,the non-polynomial analysis filter bank with low order is designed by approximating the polynomial filter with high order.Given the analysis filter banks and sampling matrix,the reconstruction problem can be formulated as an unconstrained optimization problem.To avoid the matrix inverse with high computation cost when graph is with large order,the precondition gradient method is proposed to solve this problem,which is enable the distributed manner.Numerical results show that the proposed method can satisfy the perfect reconstruction and have a better frequency selectivity and better localized characteristic in vertex domain.And it also shows that the proposed method is also suitable for existing down-sampling methods in vertex domain.Compared with existing methods,the proposed method has a faster convergence rate since the precondition operator can increase the convergence rate effectively.Secondly,a design method and its distributed implementation of M-channel oversampled graph filter banks is proposed in this paper.Given the analysis filter banks and sampling matrix,the reconstruction problem can be formulated into a quadratic optimization problem.According to the sparse pattern of graph topology,the global problem can be split into a series of local problems resides on each node.Then the solution of local problems can be patched into a value of each node.To reduce the approximation error,iteration is used to eliminate it.In this method,a matrix associated with graph topology is used to approximate the Hessian matrix inverse,and the pseudo-inverse of this matrix is equivalent to the inverse of local matrices of small sizes.It is proved that the proposed algorithm is of linear convergence.Numerical results demonstrate that the proposed method has a faster convergence rate and can achieve perfect reconstruction.Compared with existing methods,the proposed method has better performance in graph signal denoising,especially in image denoising.