Distributed signal reconstruction method and system of m-channel oversampled graph filter bank

Abstract

The application discloses a distributed signal reconstruction method and system of an M-channel oversampling graph filter bank, the method converts a signal reconstruction problem into a set of a series of local optimization problems through a truncation operator matrix, then solves the local optimization problems by using a distributed conjugate gradient method, obtains a local step value and a local increment value, and performs data fusion to update a reconstructed graph signal vector, a residual error vector and a search direction vector; finally, the method reduces errors through an iteration process to obtain an optimal solution of the reconstructed signal; each node on the network in the system has independent computing and communication capabilities, and all the nodes participate in system analysis and processing of a signal task. The method avoids a problem of computing a matrix inverse, and distributes calculation of the step value and the increment value in the conjugate gradient iteration, so that the signal reconstruction process of the filter bank can be realized in a completely distributed manner on the distributed system, the calculation complexity is effectively reduced, the robustness of the system is enhanced, and the method is suitable for analysis and processing of a large-scale node graph signal on the network.

Description

TECHNICAL FIELD

[0001] The present application relates to the technical field of graph signal processing, in particular to a distributed signal reconstruction method and system of an M-channel oversampled graph filter bank. BACKGROUND

[0002] The graph signal processing theory models the non-regular network data and the interaction between the data of the sensor network, the traffic network, the biological and social network as a graph, and develops the technology for analyzing and processing the non-regular network data. The graph filter bank is a key technology for multi-resolution analysis of the non-regular network data in the graph signal processing theory, has good local characteristics, and has been widely applied to the classification, compression and denoising of the non-regular network data. The graph filter bank includes an analysis filter bank, a sampling operation and a synthesis filter bank, the analysis filter bank decomposes the input signal into subband signals of different frequency ranges, the sampling operation can reduce the redundancy of the subband signals, and the synthesis filter bank reconstructs the subband signals to restore the original signal.

[0003] The graph filter bank can be divided into two categories according to the number of channels, that is, two channels and M channels. Compared with the two-channel critical sampling graph filter bank, the M-channel oversampled graph filter bank can divide the frequency spectrum of the graph signal more finely, and in actual application, more refined band-limited signals can be extracted for analysis and processing. The design of the M-channel oversampled graph filter bank needs to meet the complete reconstruction characteristic. In the existing signal reconstruction method, the centralized method depends on the central processor to receive and process the data from all nodes on the network, and sends the processing result to the nodes again. When the number of graph nodes is large, the computational complexity is high, the data of the whole graph cannot be processed, and the robustness is poor. SUMMARY

[0004] Therefore, the purpose of the present application is to provide a distributed signal reconstruction method and system of an M-channel oversampled graph filter bank, which avoids the problem of high computational complexity and poor robustness caused by solving the matrix inverse in the centralized method, and simultaneously calculates the step size value and the increment value in a distributed manner in the conjugate gradient iteration, so that the signal reconstruction process of the M-channel oversampled graph filter bank can be realized in a completely distributed manner. The system has independent computing and communication capabilities on each node on the network, and all nodes participate in the task of analyzing and processing the signal, which can effectively avoid the problem that the whole network signal cannot be processed due to the failure of part of the nodes, and with the increase of the number of nodes on the network, the data that needs to be received, processed and sent on each node will not increase accordingly, which is suitable for analyzing and processing the graph signal on a large-scale network.

[0005] To achieve the above purpose, the present application provides the following technical scheme:

[0006] The application provides a distributed signal reconstruction method of an M-channel oversampled graph filter bank, which comprises the following steps:

[0007] Step 1: first, a truncation operator matrix is defined according to the related characteristics of a graph, and a signal reconstruction problem (a global optimization problem) of the M-channel oversampled graph filter bank is converted into a set of a series of local optimization problems through the truncation operator matrix;

[0008] Step 2: a local step value and a local increment value are obtained by solving the local optimization problem, and the local step value and the local increment value are fused, and the reconstructed graph signal vector, the residual vector and the search direction vector are updated by using the fused step value and the increment value;

[0009] Step 3: the error is reduced through an iterative process to obtain an optimal solution of the reconstructed signal.

[0010] Further, the truncation operator matrix D i,2r in the step 1 is a diagonal matrix, and the jth diagonal element D i,2r (j,j) is defined by the following formula:

[0011]

[0012] In the formula, C(i,2r) represents a set of the node i and all neighbor nodes within 2r orders of the node i; and C(i,2r) needs to satisfy the following conditions:

[0013]

[0014] In the formula, C(i,2r) represents a set of the node i and all neighbor nodes within 2r orders of the node i; and C(i,2r) needs to satisfy the following conditions: represents the Beurling dimension of the graph, represents the Beurling density of the graph, and 2r represents a neighborhood radius;

[0015] Further, the set of a series of local optimization problems in the step 1 is represented as:

[0016]

[0017] In the formula, ‖·‖2 represents the square of the two-norm of a matrix or a vector, represents the square of the two-norm of a matrix or a vector, represents the reconstructed graph signal vector, D i,2r represents the truncation operator matrix, R b represents an intermediate matrix of the M-channel, represents a subband signal vector after subband processing, and M represents the number of channels of the filter bank.

[0018] Further, the solving of the local optimization problem in the step 2 is performed according to the following steps:

[0019] Step 21: initialization operation:

[0020] Step 22: Calculate the local step size value of the ith node in the mth iteration is:

[0021]

[0022] where D i,2r denotes the truncated operator matrix, r (m-1) denotes the residual vector in the m-1th iteration, d (m-1) denotes the search direction vector in the m-1th iteration, denotes the Hessian matrix;

[0023] Step 23: Fuse the local step size value:

[0024] Step 24: Update the reconstructed image signal vector in the mth iteration the residual vector r (m) is:

[0025]

[0026] where D denotes the reconstructed image signal vector in the m-1th iteration, r (m-1) denotes the residual vector in the m-1th iteration, denotes the fused step size value in the mth iteration, denotes the Hessian matrix, d (m-1) denotes the search direction vector in the m-1th iteration;

[0027] Step 25: Calculate the local increment value of the ith node in the mth iteration is:

[0028]

[0029] where D i,2r denotes the truncated operator matrix, r (m-1) and r (m) denote the residual vectors in the m-1th and mth iterations, respectively;

[0030] Step 26: Fuse the local increment value:

[0031] Step 27: Update the search direction vector d (m) is:

[0032]

[0033] where r (m) denotes the residual vector in the mth iteration, d represents the incremental value after fusion in the m-th iteration. (m-1) This represents the search direction vector in the (m-1)th iteration;

[0034] Step 28: Determine ||r (m) || ∞ Does <ε hold true?

[0035] If the condition is met, the iteration terminates, and the reconstructed graph signal vector from the m-th iteration is reconstructed. Output the final reconstruction result; otherwise, set the iteration count m = m + 1 and return to step 22.

[0036] Furthermore, the initialization operation in step 21 is performed according to the following steps:

[0037] Step 211: Let the initial sub-band signal vector Let the initial reconstructed graph signal vector It is a zero vector;

[0038] Step 212: Calculate the intermediate matrix {R0,...,R} of the first M / 2 channels. M / 2-1} is: R p =Q L H p ,p=0,...M / 2-1; The intermediate matrix {R} of the M / 2 channels after calculation M / 2 ,...,R M-1} is: R q =Q H H q ,q=M / 2,...M-1; the coefficient matrix R is calculated as follows: Calculate the coefficient vector for: In the formula, {Q L Q H} represents the downsampling matrix, {H0,...,H M-1} represents the analysis filter matrix, This represents the subband signal vector after subband processing. [·] T Indicates transpose;

[0039] Step 213: Calculate the initial gradient vector for: Calculate the Hessian matrix for: In the formula, R represents the coefficient matrix of channel M. This represents the initial reconstructed image signal vector of the M channel. Represents the coefficient vector of the M channel;

[0040] Step 214: Let the initial residual vector be: In the formula, Let d represent the initial gradient vector; let the initial search direction vector be d. (0) =r (0) Let the initial number of iterations m = 1; and give the iteration termination condition ε.

[0041] Furthermore, the fusion local step size value in step 23 is performed according to the following steps:

[0042] Step 231: Obtain the local step size values ​​of all nodes in the m-th iteration. The maximum value in and minimum value In the formula, N is the total number of nodes in the graph;

[0043] Step 232, divide the interval Divide the intervals evenly into Ω sub-intervals and set the local step size values. The portion of the interval located in the l-th interval is included in the set S. l In the given information, l = 1, 2, ..., Ω;

[0044] Step 233: Calculate set S l The average value of the inner elements is expressed as

[0045] Step 234: Calculate the weight ω of the l-th subinterval. l ,l=1,2,...,Ω is:

[0046]

[0047] In the formula, |S l | represents set S l The number of elements in the graph, N is the total number of nodes in the graph, and ζ is a constant;

[0048] Step 235: Calculate the fusion step size value in the m-th iteration. for:

[0049]

[0050] In the formula, ω l This represents the weight of the l-th subinterval. Describe set S l The average value of the elements in the middle;

[0051] Furthermore, the fusion of local incremental values ​​in step 26 is performed according to the following steps:

[0052] Step 261: Obtain the local increment values ​​of all nodes in the m-th iteration. The maximum value in and minimum value In the formula, N is the total number of nodes in the graph;

[0053] Step 262, divide the interval Divide the intervals evenly into Ω sub-intervals and set the local increment values. The portion of the interval located in the l-th interval is included in the set S. l In ', l = 1, 2, ..., Ω;

[0054] Step 263: Calculate set S′ l The average value of the inner elements is expressed as

[0055] Step 264: Calculate the weight ω′ of the l-th subinterval. l ,l=1,2,...,Ω is:

[0056]

[0057] In the formula, |S′ l | represents set S′ l The number of elements in the graph, N is the total number of nodes in the graph, and ζ' is a constant;

[0058] Step 265: Calculate the fusion increment value in the m-th iteration. for:

[0059]

[0060] In the formula, ω′ l This represents the weight of the l-th subinterval. Describe the set S′ l The average value of the elements in the middle;

[0061] The distributed signal reconstruction system of the M-channel oversampled graph filter bank provided by this invention consists of a memory module, a processor module, a communication module, a time synchronization module, and a data verification module. The processor module includes analysis filter banks, sampling operations (upsampling and downsampling operations), and synthesis filter banks. The system's workflow is as follows:

[0062] The graph signals obtained from the current network node are stored in the storage module; signals from neighboring nodes are received through the communication module, and after data verification by the data verification module, they are stored in the storage module; signals from the storage module are read, processed by the processor module, stored in the storage module, and then sent to neighboring nodes on the network through the communication module under the guidance of the time synchronization module.

[0063] The signal must meet the complete reconstruction condition during signal processing in the processor module. The signal is input into the analysis filter bank, and after downsampling, a sub-band signal is obtained. This sub-band signal is then upsampling and input into the synthesis filter bank, where it is processed to obtain the reconstructed signal. The input signal and the output reconstructed signal satisfy the following formula:

[0064]

[0065]

[0066] In the formula, x and Let {G0,...,G} represent the input signal vector and the output reconstructed signal vector, respectively. Let T represent the system's transfer matrix. M-1} represents the synthesis filter matrix, {H0,...,H M-1} represents the analysis filter matrix, {J0,J1} represents the sampling diagonal matrix, I represents the identity matrix, p represents the channel number of the first M / 2 channels, q represents the channel number of the last M / 2 channels, and M represents the number of channels in the filter bank.

[0067] Furthermore, the sampling operation includes upsampling and downsampling operations, and a downsampling matrix {Q} is defined. L Q H} and upsampling matrix The downsampling matrix {Q L Q H} is defined according to the following formula:

[0068]

[0069]

[0070] Among them, bipartite graph The graph nodes are divided into two disjoint sets L = {i1, i2, ..., i...} |L|} and H={i1,i2…,i |H|}, and satisfying |L|+|H|=N, where N is the total number of nodes in the graph, and E represents the set of edges in the graph; the upsampling matrix It is the downsampling matrix {Q} L Q H The transpose of}; the sampling operation is performed by first downsampling and then upsampling, as follows:

[0071]

[0072] Furthermore, the sub-band signal vector {z0,...,z} M-1}for:

[0073] zp =Q L H p x,p=0,...,M / 2-1

[0074] z q =Q H H q x,q=M / 2,...,M-1

[0075] In the formula, x represents the input signal vector, {Q L Q H} represents the downsampling matrix, {H0,...,H M-1} represents the analysis filter bank matrix, {z0,...,z M-1} represents the M-channel subband signal vector;

[0076] Furthermore, the complete reconstruction condition is:

[0077]

[0078] In the formula, {G0,...,G M-1} represents the synthesis filter matrix, {H0,...,H M-1} represents the analysis filter matrix, {Q L Q H} represents the downsampling matrix. Let I represent the upsampling matrix, p represent the channel number of the first M / 2 channels, q represent the channel number of the last M / 2 channels, and M represent the number of channels in the filter bank.

[0079] The beneficial effects of this invention are as follows:

[0080] The distributed signal reconstruction method for M-channel oversampled graph filter banks provided by this invention avoids the problems of high computational complexity and poor robustness caused by solving matrix inverses, compared with existing centralized methods. Moreover, this method adopts a distributed conjugate gradient method, in which the step size and increment values ​​are calculated in a distributed manner during conjugate gradient iteration. This allows the signal reconstruction process of the filter bank to be implemented in a completely distributed manner. Therefore, during the iteration process, it is not necessary to directly calculate all the data on the graph, but only the data on the current node and the neighboring nodes of the current node. Furthermore, the data on the node is only transmitted between neighboring nodes, which reduces communication costs, is beneficial to the protection of privacy, and is suitable for processing large-scale node graph signals.

[0081] The distributed signal reconstruction system of the M-channel oversampled graph filter bank provided by this invention has independent computing and communication capabilities at each node on the network, and each node participates in the system's signal analysis and processing tasks. This can effectively avoid the problem of the entire network signal being unable to be processed due to the failure of some nodes. In addition, as the number of nodes on the network increases, the number of signals that each node needs to receive, process, and send will not increase accordingly, making it suitable for analyzing and processing graph signals on large-scale networks.

[0082] Other advantages, objectives, and features of the invention will be set forth in part in the description which follows, and in part will be apparent to those skilled in the art from the following examination, or may be learned from practice of the invention. The objectives and other advantages of the invention can be realized and obtained through the following description. Attached Figure Description

[0083] To make the objectives, technical solutions, and beneficial effects of this invention clearer, the following figures are provided for illustration:

[0084] Figure 1 This is a flowchart of a distributed signal reconstruction method for an M-channel oversampled graph filter bank.

[0085] Figure 2 This is a schematic diagram of the structure of the M-channel oversampling filter bank. Detailed Implementation

[0086] The present invention will be further described below with reference to the accompanying drawings and specific embodiments, so that those skilled in the art can better understand and implement the present invention. However, the embodiments described are not intended to limit the present invention.

[0087] Example 1

[0088] like Figure 1 As shown in this embodiment, the distributed signal reconstruction method for an M-channel oversampled graph filter bank avoids the high computational complexity caused by calculating matrix inversion in centralized methods. First, a truncation operator matrix is ​​defined based on the graph's correlation characteristics. This matrix transforms the signal reconstruction problem (global optimization problem) of the M-channel oversampled graph filter bank into a set of local optimization problems. Second, the local optimization problems are solved to obtain local step size and local increment values. These values ​​are then fused, and the fused step size and increment values ​​are used to update the reconstructed graph signal vector, residual vector, and search direction vector. Finally, an iterative process is used to reduce errors and obtain the optimal solution for the reconstructed signal. The specific steps are as follows:

[0089] Step 1, Optimization Problem Reduction: First, define the truncation operator matrix D. i,2rThe truncation operator matrix D i,2r Let D be a diagonal matrix, and let its j-th diagonal element be D. i,2r (j,j) is:

[0090]

[0091] In the formula, C(i,2r) represents the set of node i and all its neighboring nodes within order 2r, and C(i,2r) must satisfy the following condition:

[0092]

[0093] In the formula, This represents the Beurling dimension of the graph. Let represent the Beurling density of the graph, and 2r represent the neighborhood radius. When and When both are very small, |C(i,2r)| is also very small, meaning that the number of all neighboring nodes within 2r order of node i is very small. In this case, the graph... It's a sparse graph, allowing for distributed computations on the graph. Then, the truncation operator matrix D is used. i,2r The signal reconstruction problem (global optimization problem) of the M-channel oversampled map filter bank is transformed into a set of local optimization problems:

[0094]

[0095] in, Represents the square of the 2-norm of a matrix or vector. D represents the reconstructed graph signal vector. i,2r Represents the truncation operator matrix, R b This represents the intermediate matrix of channel M. This represents the subband signal after subband processing, where M represents the number of channels in the filter bank; b represents the channel number of the M channels.

[0096] Step 21, Initialization Operation:

[0097] Step 211: Let the initial sub-band signal vector Let the initial reconstructed graph signal vector It is a zero vector;

[0098] Step 212: Calculate the intermediate matrix {R0,...,R} of the first M / 2 channels. M / 2-1} is: R p =Q L H p ,p=0,...M / 2-1; The intermediate matrix {R} of the M / 2 channels after calculation M / 2 ,...,R M-1} is: Rq = Q H H q , q = M / 2,..., M-1; the coefficient matrix R is calculated as: The coefficient vector is calculated as: In the formula, {Q L , Q H} represents a down-sampling matrix, {H0,...,H M-1} represents an analysis filter matrix, represents a sub-band signal vector after sub-band processing, [·] T represents transposition;

[0099] Step 213, the initial gradient vector is calculated as: The Hessian matrix is calculated as: is calculated as: In the formula, R represents a coefficient matrix of M channels, represents an initial reconstructed graph signal vector of M channels, represents a coefficient vector of M channels;

[0100] Step 214, the initial residual vector is set as: In the formula, represents the initial gradient vector; the initial search direction vector d (0) = r (0) is set; the initial iteration number m = 1 is set; and an iteration termination condition ε is given.

[0101] Step 22, the local step value of the i-th node in the m-th iteration is calculated as: is calculated as:

[0102]

[0103] In the formula, D i,2r represents a truncation operator matrix, r (m-1) represents a residual vector in the m-1-th iteration, d (m-1) represents a search direction vector in the m-1-th iteration, represents a Hessian matrix;

[0104] Step 23, the local step values are fused as:

[0105] Step 231, the maximum value and the minimum value of the local step values of all nodes in the m-th iteration are obtained In the formula, N is the total number of nodes of the graph;

[0106] Step 232, the interval is evenly divided into Ω sub-intervals, and the local step values are divided into Ω groups The part of S located in the l-th interval contains l l = 1, 2, …, Ω.

[0107] Step 233, calculate the average value of the elements in S l denoted as

[0108] Step 234, calculate the weight value ω l of the l-th sub-interval, l = 1, 2, …, Ω, is:

[0109]

[0110] where |S l | represents the number of elements in S l , N is the total number of nodes of the graph, and ζ is a constant.

[0111] Step 235, calculate the fusion step size value under the m-th iteration is:

[0112]

[0113] where ω l represents the weight value of the l-th sub-interval, represents the average value of the elements in S l .

[0114] Step 24: update the reconstructed graph signal vector under the m-th iteration The residual error vector r (m) is:

[0115]

[0116] where represents the reconstructed graph signal vector under the (m-1)-th iteration, r (m-1) represents the residual error vector under the (m-1)-th iteration, represents the fused step size value under the m-th iteration, represents the Hessian matrix, d (m-1) represents the search direction vector under the (m-1)-th iteration.

[0117] Step 25: calculate the local increment value of the i-th node under the m-th iteration is:

[0118]

[0119] where D i,2r represents the truncation operator matrix, r (m-1) and r (m)respectively denote the residual vector under the m-1th and mth iteration;

[0120] Step 26: fusion of local incremental values:

[0121] Step 261, obtain the local incremental values of all nodes under the mth iteration the maximum value and the minimum value in the formula, N is the total number of nodes of the graph;

[0122] Step 262, divide the interval into Ω sub-intervals, and include the part of the local incremental value located in the lth interval in the set S l ', l = 1, 2, …, Ω;

[0123] Step 263, calculate the average value of the elements in the set S l ', denoted as

[0124] Step 264, calculate the weight ω' of the lth sub-interval l , l = 1, 2, …, Ω, as:

[0125]

[0126] In the formula, |S' l | represents the number of elements in the set S' l , N is the total number of nodes of the graph, and ζ' is a constant;

[0127] Step 265, calculate the fusion incremental value under the mth iteration as:

[0128]

[0129] In the formula, ω' l represents the weight of the lth sub-interval, represents the average value of the elements in the set S' l ;

[0130] Step 27: update the search direction vector d (m) under the mth iteration as:

[0131]

[0132] In the formula, r (m) represents the residual vector under the mth iteration; represents the fusion incremental value under the mth iteration, d (m-1) represents the search direction vector under the m-1th iteration;

[0133] Step 28: Determine Is it true or false?

[0134] If the condition is met, the iteration terminates, and the reconstructed graph signal vector from the m-th iteration is reconstructed. Output the final reconstruction result; otherwise, set the iteration count m = m + 1 and return to step 22.

[0135] In the above iteration process, the method provided in this embodiment only receives data from neighboring nodes within 2r order and performs local calculations. The calculation results are then sent to neighboring nodes within 2r order. Therefore, the calculation is performed only on the current node and its neighboring nodes within 2r order, without requiring a fusion center to collect data from the entire graph and then process it centrally. The computational complexity is only O(|C(i,2r)|). 2 ). Due to the diagram It is a sparse graph. For a small radius 2r, |C(i,2r)| is a very small value and does not change with the increase of the total number of nodes N in the graph.

[0136] Example 2

[0137] like Figure 2 As shown, Figure 2 This is a schematic diagram of the structure of the M-channel oversampling filter bank. The filter bank is represented as {H0,…,H...} M-1 The sampling diagonal matrix is ​​represented as {J0, J1}, and the synthesis filter bank is represented as {G0, ..., G}. M-1 When a graph signal vector x is input, the output is the reconstructed graph signal vector. It can be represented as:

[0138]

[0139]

[0140] In the formula, T represents the system's transmission matrix, I represents the identity matrix, and M represents the number of channels in the filter bank.

[0141] For a bipartite graph The graph nodes are divided into two disjoint sets L = {i1, i2, ..., i...} |L|} and H={i1,i2…,i |H|}, and satisfy |L|+|H|=N, where N is the total number of nodes in the graph and E represents the set of edges in the graph.

[0142] Define the downsampling matrix {Q} L Q H}for:

[0143]

[0144]

[0145] The up-sampling matrix is the transpose of the down-sampling matrix, denoted as The down-sampling followed by up-sampling operation is denoted as:

[0146]

[0147] The sub-band signal vector {z0,...,z M-1} is:

[0148] z p = Q L H p x, p = 0,..., M / 2 - 1

[0149] z q = Q H H q x, q = M / 2,..., M - 1

[0150] The perfect reconstruction condition is:

[0151]

[0152] The analysis filter bank {H0,...,H M-1} employs a concatenation of spline filter banks, where the spline filter bank is denoted as:

[0153]

[0154] where L is a graph normalized Laplacian matrix and K is the order of the filter bank. The sub-band signal vector after sub-band processing is denoted as where the sub-band processing can be signal denoising, compression, etc.

[0155] Embodiment 3

[0156] Given the sub-band signal vector The signal reconstruction can be implemented using a centralized optimization method, which equates the output reconstructed graph signal to the solution of the following least squares problem:

[0157]

[0158] where represents the square of the two-norm of a matrix or vector, and the global optimal solution of the optimization problem is:

[0159]

[0160] where

[0161]

[0162]

[0163] R p =Q L H p p = 0, ..., M / 2-1

[0164] R q =Q H H q ,q=M / 2,…,M-1

[0165] Simultaneously, to satisfy the complete reconstruction condition, the synthesis filter bank matrix {G0,...,G...} is used. M-1} Requires the following to be met:

[0166]

[0167] Among them, G b Let b represent the b-th synthesized filter matrix.

[0168] As can be seen from the global optimal solution formula of the centralized method, the solution to the signal reconstruction problem of the M-channel oversampled image filter bank requires inverting matrix R, with a computational complexity of O(N). 3 When the total number of nodes N in the graph is very large, the computational complexity is very high. Furthermore, when the condition number of a large-dimensional matrix R is large, the subband signal... Even subtle changes can cause a solution Significant changes result in poor robustness.

[0169] The performance of this embodiment will be illustrated below through specific simulation examples.

[0170] This embodiment uses a smoothed graph signal on a random sensor network as the original input signal, and applies the Centralized Processing Method (CPM) and the Distributed Conjugate Gradient (DCG) method to reconstruct and denoise the input signal, respectively. The centralized method accurately calculates the matrix inverse.

[0171] Simulation Example 1: Signal Reconstruction

[0172] In this example, the original input signal is input into the analysis filter bank, and then processed by downsampling to obtain the subband signal. The subband signal is then upsampled and input into the synthesis filter bank to achieve signal reconstruction. The reconstruction performance is measured by the mean square error (MSE) and the signal-to-noise ratio (SNR), expressed as follows:

[0173]

[0174] where x represents an input signal, represents a reconstructed signal, and N represents the total number of nodes in the graph.

[0175] Table 1 shows the signal reconstruction performance obtained by different methods, where the input signal is a random sensor network graph signal with N = 4096 nodes. As can be seen from Table 1, the distributed conjugate gradient algorithm of the present method achieves almost the same mean square error and signal-to-noise ratio as the centralized algorithm within a small error range, which shows that both the centralized method and the distributed conjugate gradient algorithm can achieve complete reconstruction of the M-channel oversampling graph filter bank.

[0176] Table 1 Signal reconstruction performance obtained by different methods

[0177] Mean square error Signal to noise ratio / dB CPM 5.04e-30 293.16 The invention 6.19e-30 292.34

[0178] In addition, in the present example, the running time of the centralized algorithm and the distributed conjugate gradient algorithm for signal reconstruction as the number of nodes of the random sensor network graph increases is also compared. The running time of the above two algorithms is given in Table 2, where “--” represents that the running time exceeds 100000 seconds or the method is not applicable. As can be seen from Table 2, the centralized algorithm cannot run to obtain results when the number of nodes N ≥ 160000, but the distributed conjugate gradient algorithm of the present method can still calculate the results, which shows that the distributed conjugate gradient algorithm of the present method has low computational complexity and better robustness, and is suitable for analyzing and processing large-scale node graph signals on the network.

[0179] Table 2 Running time (seconds) of centralized and distributed reconstruction algorithms

[0180] N 10000 20000 40000 80000 160000 320000 CPM 0.057 0.148 0.345 1.077 -- -- The invention 29.91 87.48 76.71 906.50 5608.45 21472.18

[0181] Simulation Example 2: Signal denoising

[0182] The graph signal denoising on the network is one of the applications of the distributed signal reconstruction system of the M-channel oversampled graph filter bank. In this example, the graph signal denoising process implemented on the system is simulated. The input signal is a noisy signal superimposed on the original input signal with a uniformly distributed random noise. In this example, the input signal is decomposed into M sub-band signals by analyzing the filter bank and the down-sampling operation, then the sub-band signals are processed by the hard threshold value, the processed sub-band signals are up-sampled, and finally the signal reconstruction is completed by the synthesis filter bank to realize the denoising of the graph signal. The denoising performance is measured by the input signal noise ratio (ISNR) and the output signal noise ratio (OSNR), and the expressions are as follows:

[0183]

[0184] In the formula, x0 is the noise-free test signal, x is the noisy signal, is the reconstructed signal.

[0185] Table 3 shows the denoising performance comparison of the centralized algorithm and the distributed conjugate gradient method under different noise standards η, wherein the input signal is a random sensor network graph signal with a node number N = 4096, and the input signal noise ratio and the output signal noise ratio are the average values of multiple experiments. As can be seen from Table 3, both methods can well denoise the noisy signal, and the distributed conjugate gradient algorithm of the present method is only inferior to the centralized method in denoising performance when the noise standard η = 1 / 32, and is superior to the centralized method in denoising performance under other noise standards.

[0186] Table 3 shows the denoising performance of different methods under different noise standards, and the index is the signal-to-noise ratio (dB).

[0187]

[0188]

[0189] Theoretical analysis and simulation example results show that, compared with existing methods, the present method can achieve almost the same reconstruction performance as the centralized method, has low computational complexity, better robustness, and is suitable for analyzing and processing large-scale node graph signals on the network; at the same time, the system has better denoising performance.

[0190] The above-described embodiments are only preferred embodiments of the present application, and the protection scope of the present application is not limited thereto. Equivalent substitutions or transformations made by those skilled in the art on the basis of the present application are within the protection scope of the present application. The protection scope of the present application is subject to the claims.

Claims

1. A distributed signal reconstruction method for M-channel oversampled graph filter banks, characterized in that: The method comprises the following steps: Step 1, first define a truncation operator matrix according to the relevant characteristics of the graph, and convert the signal reconstruction problem of the M-channel oversampled graph filter bank into a series of local optimization problems through the truncation operator matrix; Step 2, solve the local optimization problem to obtain a local step value and a local increment value, fuse the local step value and the local increment value, and update the reconstructed graph signal vector, the residual vector and the search direction vector by using the fused step value and the increment value; Step 3, through an iterative process until a preset error threshold is met, and an optimal solution of the reconstructed signal is obtained at this time; The local optimization problem in step 1 is performed according to the following steps: First define the truncation operator matrix D i,2r whose jth diagonal element D i,2r (j, j) is: In the formula, C(i, 2r) represents a set of nodes i and all neighbor nodes of node i within 2r orders; and C(i, 2r) needs to satisfy the following conditions: wherein denotes the Beurling dimension of the graph, denotes the Beurling density of the graph, 2r denotes the neighborhood radius; with the truncation operator matrix D i,2r The signal reconstruction problem for an M-channel oversampled filter bank is converted into a set of local optimization problems: wherein represents the square of the two-norm of a matrix or vector, denotes the reconstructed picture signal vector, D i,2r denotes a truncation operator matrix, R b denotes an intermediate matrix of M channels, denotes a subband signal vector after subband processing, M denotes the number of channels of the filter bank; The solving of the local optimization problem in step 2 is performed according to the following steps: Step 21: initialization operation: Step 22: Calculate the local step size value of the ith node in the mth iteration is: where D i,2r denotes the truncated operator matrix, r (m-1) denotes the residual vector at the m-1th iteration, d (m-1) denotes the search direction vector at the m-1th iteration, denotes the Hessian matrix; Step 23: fuse the local step value: Step 24: Update the reconstructed image signal vector at the mth iteration Residual vector r (m) is: wherein denotes the reconstructed image signal vector at the m-1th iteration, r (m-1) denotes the residual vector at the m-1th iteration, denotes the fused step size at the mth iteration, denotes the Hessian matrix, d (m-1) denotes the search direction vector at the m-1th iteration; Step 25: Calculate the local increment value of the ith node in the mth iteration is: where D i,2r denotes the truncation operator matrix, r (m-1) and r (m) denote the residual vectors at the (m-1)th and mth iterations, respectively. Step 26: fuse the local increment value: Step 27: Update the search direction vector d at the mth iteration (m) is: wherein r (m) represents the residual vector at the mth iteration, represents the fused delta value at the mth iteration, d (m-1) represents the search direction vector at the m-1th iteration; Step 28: judge ||r (m) || ∞ <ε is established: if the condition is established, terminate the iteration, output the reconstructed image signal vector under the mth iteration as the final reconstruction result; otherwise, let the iteration number m = m + 1, and return to step 22.

2. The distributed signal reconstruction method of an M-channel oversampled graph filter bank as recited in claim 1, wherein: The initialization operation in step 21 is performed according to the following steps: Step 211, let the initial subband signal vector b = 0, 1,... M - 1; let the initial reconstructed image signal vector be a zero vector; Step 212, compute the intermediate matrix {R0,...,R M / 2-1} of the first M / 2 channels as: p L p M / 2 M-1 q H q Compute the coefficient vector as: L H M-1 T [·]denotes the transpose.​​​​​​​​​​​​ Step 213, calculating an initial gradient vector is calculated as: calculating the Hessian matrix is calculated as: where R denotes a coefficient matrix of M channels, denotes an initial reconstructed image signal vector of M channels, denotes a coefficient vector of M channels; Step 214, let the initial residual vector be: wherein, denotes the initial gradient vector; let the initial search direction vector d (0) = r (0) ; Let the initial iteration number m = 1; Give an iteration termination condition ε.

3. The distributed signal reconstruction method of an M-channel oversampled graph filter bank as recited in claim 1, wherein: The fusion of the local step value in step 23 is performed according to the following steps: Step 231, obtaining the local step value of all nodes in the mth iteration the maximum value in the set i = 1, 2,... N and the minimum value i = 1, 2,... N, wherein N is the total number of nodes of the graph; Step 232, dividing the interval into Ω sub-intervals, and dividing the local step value into Ω parts, where the part located in the l-th interval is included in the set S l , l = 1, 2, …, Ω; Step 233, compute the set S l the average of the inner elements, denoted as Step 234, calculating the weight ω of the lth sub-interval l l = 1, 2,..., Ω is: where |S l | denotes the number of elements in the set S l N is the total number of nodes in the graph, and ζ is a constant. Step 235, calculate the fusion step size value in the mth iteration is: where ω l denotes the weight of the lth sub-interval, denotes the average value of the elements of the set S l .

4. The distributed signal reconstruction method of an M-channel oversampled graph filter bank as recited in claim 1, wherein: The fusion of the local increment value in step 26 is performed according to the following steps: Step 261, obtaining the local incremental value of all nodes in the mth iteration the maximum value in the set i = 1, 2,... N and the minimum value i = 1, 2,... N, wherein N is the total number of nodes of the graph; Step 262, dividing the interval into Ω sub-intervals, and dividing the local increment value located in the l-th interval into the set S l , l = 1, 2, …, Ω; Step 263, compute the set S l the average of the inner elements, denoted as Step 264, calculating the weight ω of the lth sub-interval l , l = 1, 2,..., Ω is: where |S l denotes the number of elements in the set S l N is the total number of nodes in the graph, and ζ is a constant. Step 265, calculate the fusion increment value under the mth iteration is: where ω l denotes the weight of the lth sub-interval, denotes the average value of the elements in the set S l .

5. A distributed signal reconstruction system of M-channel oversampled graph filter banks, comprising a memory module, a processor module, a communication module, a time synchronization module and a data verification module, characterized in that, The memory module stores data on the network, the communication module completes the transmission of data between nodes on the network, the time synchronization module ensures that the time of data transmission between each node on the network is synchronized, the data verification module ensures that the network data acquisition, transmission and storage are correct values, and the processor module implements the method according to any one of claims 1 to 4 when executing a processing program.

6. The distributed signal reconstruction system of an M-channel oversampled graph filter bank as defined in claim 5, wherein: The processor module comprises an analysis filter bank, a sampling operation and a synthesis filter bank; the sampling operation comprises up-sampling and down-sampling operations; The graph signal is input into the analysis filter bank, and after being processed by the down-sampling operation, a sub-band signal is obtained, the sub-band signal is input into the synthesis filter bank after being processed by the up-sampling operation, and a reconstructed signal is obtained after processing; the graph signal and the reconstructed signal satisfy the following formula: In the formula, x and Let {G0,...,G} represent the input signal vector and the reconstructed signal vector, respectively, and let T represent the system's transfer matrix. M-1 } represents the synthesis filter matrix, {H0,...,H M-1 } represents the analysis filter matrix, {J0,J1} represents the sampling diagonal matrix, I represents the identity matrix, p represents the channel number of the first M / 2 channels, q represents the channel number of the last M / 2 channels, and M represents the number of channels in the filter bank.

7. The distributed signal reconstruction system of an M-channel oversampled graph filter bank as defined in claim 6, wherein: The sampling operations include up-sampling and down-sampling operations and define a down-sampling matrix {Q L ,Q H} and an up-sampling matrix The down-sampling matrix {Q L ,Q H} is defined according to the following equation: wherein the bipartite graph is split into two disjoint sets L = {i1, i2, …, i |L|} and H = {i1, i2, …, i |H|}, satisfying |L| + |H| = N, N being the total number of nodes of the graph, E denoting the edge set of the graph; the up-sampling matrix is the transpose of the down-sampling matrix {Q L , Q H}; the sampling operation of the system is performed according to a down-sampling followed by an up-sampling operation, in particular as follows: .

8. The distributed signal reconstruction system of an M-channel oversampled graph filter bank as defined in claim 6, wherein: The vector {z0,...,z M-1} of the subband signals is: z p = Q L H p x, p = 0,..., M / 2 - 1 z q = Q H H q x, q = M / 2,..., M-1 where x represents an input signal vector, {Q L H} represents a down-sampling matrix, {H0,...,H M-1} represents an analysis filter bank matrix, {z0,...,z M-1} represents an M-channel sub-band signal vector; or​ The complete reconstruction condition of the system is: where {G0,...,G M-1} denotes a synthesis filter matrix, {H0,...,H M-1} denotes an analysis filter matrix, {Q L , Q H} denotes a down-sampling matrix, denotes an up-sampling matrix, I denotes an identity matrix, p denotes a channel number of the first M / 2 channels, q denotes a channel number of the last M / 2 channels, and M denotes a channel number of a filter bank.

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