Time-vertex joint graph filter bank and design method based on joint graph frequency domain sampling

Abstract

The present invention discloses a time-vertex joint graph filter group based on joint graph frequency domain sampling and a design method. First, sampling is defined based on the joint graph frequency domain; then, the time-vertex joint graph filter group is completely reconstructed based on joint frequency domain critical sampling; finally, joint orthogonal and joint biorthogonal graph filters are designed; the filter group has a symmetrical structure; the redundancy of subband coefficients is reduced, and the data transmission efficiency is improved. The method provided by the present invention designs a time-vertex joint frequency domain spline graph filter group by defining up- and down-sampling operations based on the joint graph frequency domain. The filtering operation and sampling operation of the filter group are both performed on the joint frequency domain, so that the vertex domain and time domain information of the time-varying graph signal can be effectively utilized. The simulation experiment part verifies the reconstruction performance, nonlinear approximation performance and denoising performance of the time-vertex joint frequency domain sampling graph filter group, which is significantly better than the existing frequency domain sampling graph filter group on static graphs.

Description

Technical Field

[0001] The present invention relates to the technical field of graph signal processing, and in particular to a time-vertex joint critical sampling graph filter bank based on joint graph frequency domain sampling and a design method thereof. Background Art

[0002] In recent years, graph signal processing has attracted extensive attention from researchers due to its superiority in analyzing and processing irregular structured data. As core tools in the field of graph signal processing, graph Fourier transform, graph filters and graph filter banks are widely used in various fields. Among them, graph filter banks with multi-resolution analysis capabilities have shown excellent performance in processing large-scale graph signals.

[0003] From the sampling method, most of the existing graph filter banks focus more on vertex domain sampling. Narang and Ortega constructed a two-channel critical sampling graph filter bank for analyzing graph signals based on bipartite graphs. According to the frequency folding phenomenon on the bipartite graph, a set of orthogonal mirror filters are designed to eliminate aliasing, so that the entire system meets the complete reconstruction condition. In addition, Narang and Ortega also proposed a biorthogonal filter with compact support characteristics, and Sakiyama and Watanabe proposed a spectrum filter based on the frequency conversion method and defined by a sine function. Both of the above filters can be applied to a two-channel critical sampling graph filter bank. It is worth noting that for graph signals with non-bipartite graph structures, it is usually necessary to simplify the graph into a bipartite graph. In order to obtain a more detailed spectrum division, Tanaka and Sakiyama designed an M-channel oversampled graph filter bank. In addition, Shumand further proposed an M-channel critical sampling graph filter bank, in which the synthesis part of the filter bank is replaced by an interpolation operation. More importantly, the proposed graph filter bank can be effectively extended to large-scale sparse graph signals. In addition, Ekambaram also proposed a spline filter bank with a novel structure. The integrated part of the filter bank can be regarded as an inverse filtering operation, so it is inevitable to consider the problem of high computational complexity caused by inversion.

[0004] For graph filter banks with vertex domain sampling, the sampling set that leads to complete reconstruction is not unique. The overall performance of the filter bank is heavily dependent on the selected sampling set. To overcome this defect, the concept of frequency domain sampling was introduced into graph filter banks. Sakiyama and Watanabe proposed a two-channel critical frequency domain sampling graph filter bank, which can achieve complete reconstruction of graph signals of any graph structure without simplification, and its complete reconstruction sampling set is unique. In order to obtain a finer spectrum partition, Sakiyama designed an M-channel frequency domain critical sampling graph filter bank, which not only satisfies the complete reconstruction and critical sampling characteristics, but also has a symmetric structure. Subsequently, the concept of frequency domain sampling was also introduced into spline graph filter banks. In order to reduce the computational complexity of the filter bank at the synthesis end, a closed-form expression of inverse filtering was derived. In addition to vertex domain sampling and frequency domain sampling graph filter banks, non-subsampled graph filter banks have attracted the attention of researchers because of their more flexibility in design. Jiang proposed a non-subsampled graph filter bank, which does not contain sampling operations, thereby avoiding the aliasing effect of the signal during transmission. Jiang designed a non-subsampled graph filter bank based on polynomial and lifting structures, which enables the filter bank to obtain better frequency response and complete reconstruction performance.

[0005] However, the above-mentioned graph filter groups are only suitable for analyzing static graph signals, and are not suitable for processing time-varying data that is prevalent in the real world (such as global sea level pressure data, etc.), because these filter groups do not fully utilize the temporal correlation of time-varying data.

[0006] Therefore, it is of practical significance to design a graph filter bank that can fully utilize the spatiotemporal correlation of time-varying data. Yang designed a time-vertex joint non-subsampled graph filter bank, whose analysis filter bank is composed of a series of joint spline filters with rectangular support domains, so as to effectively divide the time-varying graph signal in the frequency domain. However, the support domain type of this filter bank is too single, and it is impossible to perform a more detailed frequency domain division of the time-varying graph signal. Furthermore, Jiang proposed an M-channel time-vertex joint non-subsampled graph filter bank. Its analysis filter bank can be designed as a graph filter with multiple support domain types, such as rectangular, triangular and fan-shaped support domains. At the same time, the computational complexity of the graph filter bank is further reduced by constructing a comprehensive filter bank through a distributed algorithm. However, the non-subsampled graph filter bank inevitably introduces the problem of redundancy, making this type of graph filter bank unsuitable for compressing time-varying data. At the same time, since the sampling operation of the time-vertex joint graph signal is more complicated than that of the static graph signal, it is more difficult to eliminate the aliasing effect caused by the sampling operation in the context of the time-vertex joint sampling graph filter bank.

[0007] In view of the above reasons, it is necessary to conduct further research on the temporal-vertex joint sampling graph filter. Summary of the invention

[0008] In view of this, the purpose of the present invention is to provide a time-vertex joint critical sampling graph filter group based on joint graph frequency domain sampling. This method effectively improves the transmission rate of subband coefficients and reduces data redundancy based on the time-vertex joint graph filter group based on joint graph frequency domain sampling.

[0009] In order to achieve the above object, the present invention provides the following technical solutions:

[0010] The time-vertex joint graph filter bank design method based on joint graph frequency domain sampling provided by the present invention comprises the following steps:

[0011] 1) Obtain a time-varying graph signal X in the time-vertex domain;

[0012] 2) Determine the complete reconstruction conditions for upsampling, downsampling, and time-vertex joint graph filter banks;

[0013] 3) Designing a time-vertex joint graph filter bank using a complete reconstruction condition, wherein the time-vertex joint graph filter bank includes an analysis filter bank and a synthesis filter bank; the analysis filter bank and the synthesis filter bank process the time-varying graph signal X in the following manners:

[0014] 31) Transpose the time-varying graph signal X with the transposed feature matrix of the time-vertex joint graph Calculate the time-varying image signal in the frequency domain of the joint image

[0015] 32) Combine the time-varying image signal in the frequency domain The joint frequency response function with the analysis filter of the kth channel is Calculate and obtain the joint frequency domain signal after the k-th channel analysis and filtering;

[0016] 33) The obtained joint frequency domain signal after analysis filtering is combined with the downsampling matrix Calculate and obtain the signal after the kth channel is jointly downsampled in the frequency domain;

[0017] 34) Combine the obtained signal after the k-th channel joint frequency domain downsampling with the upsampling matrix The signal after joint frequency upsampling of the kth channel is calculated;

[0018] 35) The joint frequency response function of the k-th channel joint frequency up-sampled signal and the k-th channel comprehensive filter is Calculate and obtain the joint frequency domain signal after comprehensive filtering of the kth channel;

[0019] 36) Combine the k-th channel comprehensive filtered joint frequency domain signal with the characteristic matrix of the time-vertex joint graph Calculate and obtain the time-apex domain signal after comprehensive filtering of the kth channel;

[0020] 37) Add the time-vertex domain signals of each channel to obtain the time-varying image signal processed by the filter bank

[0021] Further, the downsampling is performed according to the following steps:

[0022] S11: The original joint frequency domain signal is respectively and The top is evenly divided into two parts;

[0023] S12: Frequency axis The second part of is folded and added to the first part, giving

[0024] S13: Its frequency axis The second part of is folded and added to the first part to get The specific expressions are as follows:

[0025]

[0026]

[0027] Where, t=1,2,…,T / 2; n=1,2,…,N / 2,

[0028] Represents the spectrum value of the nth row and tth column of the spectrum after folding along the time domain; Represents the spectrum value of the nth row and tth column of the spectrum before folding; Represents the spectrum value of the nth row and tth column of the spectrum after folding along the time domain and the image domain;

[0029] a joint frequency domain downsampled image signal representing the time-varying image signal;

[0030] Formula (2) and formula (3) can be expressed in matrix form as follows:

[0031]

[0032] in, and They are subgraphs and The frequency domain downsampling matrix of

[0033] I T / 2 and I N / 2 denote the identity matrices of order T / 2 and N / 2 respectively;

[0034] J T / 2 and J N / 2 denote the inverse identity matrices of order T / 2 and N / 2 respectively;

[0035] According to the properties of Kronecker product and vectorized operation, formula (4) is expressed as:

[0036]

[0037] Among them, X and They are respectively the vectorized original time-varying image signal and the vectorized joint frequency-domain downsampled image signal;

[0038] Then it represents the joint frequency domain downsampling matrix.

[0039] Furthermore, downsampling is a process of mapping the compressed image signal back to the original signal space, which is specifically performed in the following steps:

[0040] make and are the original time-vertex joint graph and the time-vertex joint graph with increased size after upsampling, and their corresponding Laplacian matrices are and

[0041] A set of graph signals on the original time-vertex joint graph is The corresponding joint frequency domain representation is

[0042] Joint frequency domain upsampling of time-varying image signals is through the frequency axis and Repeat the folding of the original frequency domain signal And it is expressed as:

[0043]

[0044]

[0045] Formulas (6) and (7) can be expressed in matrix form as follows:

[0046]

[0047] in, and They are subgraphs and The frequency domain upsampling matrix of

[0048] I T and I N denote the identity matrices of order T and N respectively;

[0049] J T and J N represents the inverse identity matrix of order T and N;

[0050] According to the properties of Kronecker product and vectorized operation, formula (8) is expressed as:

[0051]

[0052] Among them, X and They are respectively the vectorized original time-varying image signal and the vectorized joint frequency domain upsampled image signal;

[0053] Then it represents the joint frequency domain upsampling matrix.

[0054] Further, the acquisition of the complete reconstruction condition of the time-vertex joint graph filter bank is performed according to the following steps:

[0055] Designing a joint frequency-domain critical sampling time-vertex graph filter bank, the joint frequency-domain critical sampling time-vertex graph filter bank comprising an analysis filter bank, a sampling matrix and a synthesis filter bank;

[0056] The analysis filter bank decomposes the input time-varying image signal into sub-band signals with different frequency domain intervals, the sampling matrix reduces the redundancy of the sub-band coefficients, and the synthesis filter bank reconstructs the processed sub-band signals;

[0057] The kth analysis filter of the analysis filter bank is

[0058] in, is the analysis filter bank;

[0059] is the characteristic matrix of the Laplacian matrix of the original time-vertex joint graph;

[0060] is the joint filter kernel of the kth analysis filter;

[0061] The kth analysis filter of the synthesis filter bank is

[0062] in, is a comprehensive filter bank;

[0063] is the characteristic matrix of the Laplacian matrix of the original time-vertex joint graph;

[0064] is the joint filter kernel of the kth integrated filter;

[0065] The input-output relationship of the time-vertex joint critical sampling graph filter bank is determined as:

[0066]

[0067] Among them, X and are respectively the vectorized input time-varying graph signal and the output time-varying graph signal;

[0068] and Respectively represent the frequency domain downsampling and upsampling matrices of the kth channel;

[0069] because is an orthogonal matrix corresponding to the original time-vertex joint graph, and the transmission matrix of the framework is simplified to:

[0070]

[0071] in, represents the transmission matrix;

[0072] Will and Substituting into formula (11) we can get:

[0073]

[0074] When the transmission matrix satisfies When , the time-vertex joint critical sampling graph filter bank satisfies the complete reconstruction condition; using the properties of the Kronecker product, similar terms are combined into:

[0075]

[0076]

[0077]

[0078]

[0079] At this time, the overall transmission matrix is ​​decomposed into the above four parts, namely When all frequency responses of the joint graph filter satisfy:

[0080]

[0081]

[0082]

[0083]

[0084] Available

[0085] The transfer function of the time-vertex joint critical sampling graph filter bank satisfies That is, the graph filter bank satisfies the complete reconstruction condition, where c represents a constant.

[0086] Further, the joint orthogonal graph filter bank is designed according to the following steps:

[0087] Given a prototype filter The remaining filters are derived from this filter, namely:

[0088]

[0089] Substituting this filter design into the graph filter bank yields the following orthogonal condition:

[0090]

[0091] in, represents the T-i+1th eigenvalue in the time domain, Represents the N-j+1th eigenvalue of the vertex domain.

[0092] Further, the biorthogonal image filter bank is designed according to the following steps:

[0093] For the biorthogonal image filter bank, the high-frequency joint analysis filter and the joint synthesis filter bank are derived from the low-frequency joint filter bank, that is:

[0094]

[0095] The graph filter bank satisfies the following complete reconstruction conditions, namely:

[0096]

[0097] in, Represents the N-j+1th eigenvalue of the vertex domain.

[0098] The time-vertex joint critical sampling graph filter group based on joint graph frequency domain sampling provided by the present invention comprises an input unit, an analysis filter group, a synthesis filter group and an output unit;

[0099] The input unit is used to obtain a time-varying graph signal X in the time-vertex domain;

[0100] The analysis filter bank is used to divide the input time-varying image signal into different joint frequency domain regions to obtain subband coefficients of each channel;

[0101] The integrated filter bank is used to reconstruct the processed sub-band signals;

[0102] The output unit is used to output the time-varying image signal processed by the filter bank.

[0103] Further, the analysis filter bank includes an analysis filter characteristic matrix module, a joint frequency response function processing module, and a downsampling module;

[0104] The feature matrix module is used to transform the feature matrix of the time-vertex joint graph After being processed by this module, the time-varying image signal can be transformed into a joint frequency domain image signal;

[0105] The analysis filter joint frequency response function processing module is used to process the joint frequency response function of the analysis filter of the kth channel Perform spectrum division processing on the time-varying image signal;

[0106] The downsampling module is used to downsample the matrix of the joint image frequency domain The signal processed by the analysis filter combined with the frequency response function processing module is compressed.

[0107] Further, the comprehensive filter group includes an upsampling module, a comprehensive filter joint frequency response function processing module, and a comprehensive filter characteristic matrix module;

[0108] The upsampling module is used to upsample the matrix in the frequency domain of the joint image Map the processed subband coefficients back to the original image signal space;

[0109] The integrated filter joint frequency response function processing module is used to process the joint frequency response function of the integrated filter of the kth channel. Process the image signal after being processed by the upsampling module;

[0110] The comprehensive filter feature matrix module is used to pass the feature matrix of the time-vertex joint graph The processed joint frequency domain time-varying image signal is converted into a time-varying image signal in the time-vertex domain.

[0111] The beneficial effects of the present invention are:

[0112] The present invention provides a design method for a time-vertex joint critical sampling graph filter group based on joint graph frequency domain sampling. By defining up and down sampling operations based on the joint graph frequency domain, a time-vertex joint frequency domain spline graph filter group is constructed. The filtering operation and sampling operation of the filter group are both performed on the joint frequency domain, so that the vertex domain and time domain information of the time-varying graph signal can be effectively utilized. The simulation experiment part verifies the reconstruction performance, nonlinear approximation performance and denoising performance of the time-vertex joint frequency domain sampling graph filter group. The experimental results show that when processing time-varying graph signals, the performance of the filter group is significantly better than the existing frequency domain sampling graph filter group on static graphs.

[0113] The method provided by the present invention first defines sampling based on the joint graph frequency domain; secondly, a complete reconstruction of the time-vertex joint graph filter group based on joint frequency domain critical sampling is performed; then, graph filters are constructed based on joint orthogonality and joint biorthogonality respectively; therefore, the filter group has a symmetrical structure; the time-vertex joint graph filter group based on joint graph frequency domain sampling designed by this method reduces the redundancy of subband coefficients and improves data transmission efficiency.

[0114] The method provided by the present invention effectively improves the transmission rate of subband coefficients of a time-vertex joint graph filter bank based on joint graph frequency domain sampling and reduces data redundancy.

[0115] Other advantages, objectives and features of the present invention will be described in the following description to some extent, and to some extent, will be obvious to those skilled in the art based on the following examination and study, or can be taught from the practice of the present invention. The objectives and other advantages of the present invention can be realized and obtained through the following description. BRIEF DESCRIPTION OF THE DRAWINGS

[0116] In order to make the purpose, technical solution and beneficial effects of the present invention clearer, the present invention provides the following drawings for illustration:

[0117] Figure 1 Schematic diagram of a four-channel joint frequency-domain critical sampling filter bank in an embodiment.

[0118] Figure 2 Flowchart of the method for designing time-vertex joint critical sampling graph filter banks based on joint graph frequency domain sampling.

[0119] Figure 3 : Nonlinear approximation performance of different filter banks under synthetic data sets in the embodiments.

[0120] Figure 4 : Nonlinear approximation performance of different filter banks in the embodiment under the sea surface temperature dataset. DETAILED DESCRIPTION

[0121] The present invention is further described below in conjunction with the accompanying drawings and specific embodiments so that those skilled in the art can better understand the present invention and implement it, but the embodiments are not intended to limit the present invention.

[0122] Example 1

[0123] like Figure 1 and Figure 2 As shown, the time-vertex joint graph filter bank design method based on joint graph frequency domain sampling provided in this embodiment includes the following steps:

[0124] 1) Obtain a time-varying graph signal X in the time-vertex domain;

[0125] 2) determine the complete reconstruction conditions for upsampling and downsampling as well as time-vertex joint critical sampling graph filter banks;

[0126] 3) Designing a time-vertex joint graph filter bank using the complete reconstruction condition, wherein the time-vertex joint graph filter bank includes a plurality of analysis filters and a plurality of synthesis filters; the analysis filters and the synthesis filters process the time-varying graph signal X in the following manners respectively:

[0127] 31) Transpose the time-varying graph signal X with the transposed feature matrix of the time-vertex joint graph Calculate the time-varying image signal in the frequency domain of the joint image

[0128] 32) Combine the time-varying image signal in the frequency domain The joint frequency response function with the analysis filter of the kth channel is Calculate and obtain the joint frequency domain signal after the k-th channel analysis and filtering;

[0129] 33) The obtained joint frequency domain signal after analysis filtering is combined with the downsampling matrix Calculate and obtain the signal after the kth channel is jointly downsampled in the frequency domain;

[0130] 34) Combine the obtained signal after the k-th channel joint frequency domain downsampling with the upsampling matrix The signal after joint frequency upsampling of the kth channel is calculated;

[0131] 35) The joint frequency response function of the k-th channel joint frequency up-sampled signal and the k-th channel comprehensive filter is Calculate and obtain the joint frequency domain signal after comprehensive filtering of the kth channel;

[0132] 36) Combine the k-th channel comprehensive filtered joint frequency domain signal with the characteristic matrix of the time-vertex joint graph Calculate and obtain the time-apex domain signal after comprehensive filtering of the kth channel;

[0133] 37) Add the time-vertex domain signals of each channel to obtain the time-varying image signal processed by the filter bank

[0134] In this embodiment, the time-vertex joint graph filter group uses the complete reconstruction condition to design a joint orthogonal graph filter group and a joint biorthogonal graph filter. The time-vertex joint graph filter group includes a joint orthogonal graph filter group and a joint biorthogonal graph filter. The design is specifically performed according to the following steps:

[0135] S1) determining a sampling operation process based on downsampling and upsampling definitions in the frequency domain of the joint graph;

[0136] The downsampling operation compresses the filtered sub-band signal, thereby improving data transmission efficiency.

[0137] Order diagram and Figure are the original time-vertex joint graph before downsampling and the time-vertex joint graph with reduced size after downsampling, and their corresponding Laplacian matrices are and

[0138] Pair and Figure The Laplace matrix of is decomposed into:

[0139] and

[0140] in, and denote the joint eigenvector matrix and the joint eigenvalue matrix respectively,

[0141] and Indicates that the dimension of the feature matrix is ​​NT×NT;

[0142] remember For the picture A set of time-varying graph signals on , whose joint graph Fourier transform is:

[0143]

[0144] in, represents the frequency value of the time-varying signal X at the joint frequency,

[0145] Represents the feature matrix in the time domain; represents the transposed feature matrix on the graph domain; Indicates that the dimension of the feature matrix is ​​N×T;

[0146] Joint frequency domain downsampling of time-varying image signals It is obtained through the following three steps:

[0147] First, the original joint frequency domain signal is respectively and The top is evenly divided into two parts;

[0148] Then, the frequency axis The second part of is folded and added to the first part, giving

[0149] Finally, for For example, the frequency axis The second part of is folded and added to the first part to get It is expressed as follows:

[0150]

[0151]

[0152] Where, t=1,2,…,T / 2; n=1,2,…,N / 2,

[0153] Represents the spectrum value of the nth row and tth column of the spectrum after folding along the time domain; Represents the spectrum value of the nth row and tth column of the spectrum before folding; Represents the spectrum value of the nth row and tth column of the spectrum after folding along the time domain and the image domain;

[0154] Formula (2) and formula (3) can be expressed in matrix form as follows:

[0155]

[0156] in, and They are subgraphs and The frequency domain downsampling matrix,

[0157] And I T / 2 and I N / 2 denote the identity matrices of order T / 2 and N / 2 respectively,

[0158] J T / 2 and J N / 2 denote the inverse identity matrices of order T / 2 and N / 2 respectively.

[0159] According to the properties of Kronecker product and vectorized operation, formula (4) can be expressed as:

[0160]

[0161] Among them, x and are the vectorized original time-varying image signal and the vectorized joint frequency domain downsampled image signal, respectively.

[0162] and Then it represents the joint frequency domain downsampling matrix.

[0163] The upsampling operation maps the compressed image signal back to the original signal space.

[0164] make and are the original time-vertex joint graph and the time-vertex joint graph with increased size after upsampling, and their corresponding Laplacian matrices are and

[0165] A set of graph signals on the original time-vertex joint graph is The corresponding joint frequency domain representation is

[0166] Joint frequency domain upsampling of time-varying image signals is through the frequency axis and Repeat the folding of the original frequency domain signal And it is expressed as:

[0167]

[0168]

[0169] Formulas (6) and (7) can be expressed in matrix form as follows:

[0170]

[0171] in, and They are subgraphs and The frequency domain upsampling matrix,

[0172] And I T and I N denote the identity matrices of order T and N respectively,

[0173] J T and J N represents the inverse identity matrix of order T and N.

[0174] According to the properties of Kronecker product and vectorized operation, formula (8) can be expressed as:

[0175]

[0176] Among them, X and They are respectively the vectorized original time-varying image signal and the vectorized joint frequency domain upsampled image signal;

[0177] Then it represents the joint frequency domain upsampling matrix.

[0178] S2) determining a complete reconstruction condition for a time-vertex joint critical sampling graph filter bank;

[0179] like Figure 1 As shown, Figure 1 Schematic diagram of a four-channel joint frequency-domain critical sampling graph filter bank, which includes an analysis filter bank, a sampling matrix and a synthesis filter bank.

[0180] The analysis filter in this embodiment is a four-channel

[0181] Comprehensive filter bank; the analysis filter in this embodiment is a four-channel

[0182] The analysis filter group decomposes the input time-varying image signal into sub-band signals with different frequency domain intervals, the sampling matrix reduces the redundancy of the sub-band coefficients, and the synthesis filter group reconstructs the processed sub-band signals.

[0183] The kth analysis filter of the analysis filter bank is

[0184] in, is the characteristic matrix of the Laplacian matrix of the original time-vertex joint graph,

[0185] is the joint filter kernel of the kth analysis filter;

[0186] The kth analysis filter of the synthesis filter bank is

[0187] in, is the characteristic matrix of the Laplacian matrix of the original time-vertex joint graph,

[0188] is the joint filter kernel of the kth integrated filter;

[0189] The input-output relationship of the four-channel joint frequency-domain critical sampling graph filter bank is:

[0190]

[0191] Among them, X and are the vectorized input time-varying graph signal and output time-varying graph signal, respectively.

[0192] and Respectively represent the frequency domain downsampling and upsampling matrices of the kth channel;

[0193] because is an orthogonal matrix corresponding to the original time-vertex joint graph. At this time, the transmission matrix of the framework can be simplified to:

[0194]

[0195] in, represents the transfer matrix of the graph filter bank;

[0196] Will and Substituting into formula (11) we can get:

[0197]

[0198] When the transmission matrix satisfies When , the graph filter bank satisfies the complete reconstruction condition.

[0199] Using the properties of the Kronecker product, we can combine similar terms to obtain:

[0200]

[0201]

[0202]

[0203]

[0204] At this time, the overall transmission matrix is ​​decomposed into the above four parts, namely

[0205] When all frequency responses of the joint graph filter satisfy:

[0206]

[0207]

[0208]

[0209]

[0210] Available

[0211] Therefore, the transfer function of the graph filter bank satisfies That is, the graph filter bank satisfies the complete reconstruction condition.

[0212] S3) Design of Joint Graph Filter Bank:

[0213] (Design of Joint Orthogonal Graph Filter Banks) Given a prototype filter The remaining filters can be derived from this filter, namely:

[0214]

[0215] Substituting this filter design into the graph filter bank yields the following orthogonal condition:

[0216]

[0217] (Design of Joint Biorthogonal Image Filter Bank) For the biorthogonal image filter bank, the high-frequency joint analysis filter and the joint synthesis filter bank are derived from the low-frequency joint filter bank, that is:

[0218]

[0219] This makes the graph filter bank satisfy the following complete reconstruction condition, namely:

[0220]

[0221] Under the perfect reconstruction condition of the time-vertex joint critical sampling graph filter bank, the frequency response of the joint orthogonal filter is guaranteed to be constant.

[0222] Example 2

[0223] like Figure 2 As shown, the framework of the time-vertex joint graph filter bank provided in this embodiment is a time-vertex joint critical sampling graph filter bank based on joint graph frequency domain sampling, including

[0224] Input unit, analysis filter bank, synthesis filter bank and output unit;

[0225] The input unit is used to obtain a time-varying graph signal X in the time-vertex domain;

[0226] The analysis filter bank is used to divide the input time-varying image signal into different joint frequency domain regions, so as to facilitate subsequent different operations on the sub-band coefficients of each channel;

[0227] The integrated filter bank is used to reconstruct the processed sub-band signal. If the sub-band coefficients are not processed, the input signal can be reconstructed at the output end;

[0228] The output unit is used to output the time-varying image signal processed by the filter bank.

[0229] The analysis filter bank includes an analysis filter characteristic matrix module, a joint frequency response function processing module, and a downsampling module;

[0230] The feature matrix module is used to transform the feature matrix of the time-vertex joint graph After being processed by this module, the time-varying image signal can be transformed into a joint frequency domain image signal;

[0231] The analysis filter joint frequency response function processing module is used to process the joint frequency response function of the analysis filter of the kth channel Perform spectrum division processing on the time-varying image signal;

[0232] The downsampling module is used to downsample the matrix of the joint image frequency domain The signal processed by the analysis filter combined with the frequency response function processing module is compressed.

[0233] The comprehensive filter group includes an up-sampling module, a comprehensive filter joint frequency response function processing module, and a comprehensive filter characteristic matrix module;

[0234] The upsampling module is used to upsample the matrix in the frequency domain of the joint image Map the processed subband coefficients back to the original image signal space;

[0235] The integrated filter joint frequency response function processing module is used to process the joint frequency response function of the integrated filter of the kth channel. Process the image signal after being processed by the upsampling module;

[0236] The comprehensive filter feature matrix module is used to pass the feature matrix of the time-vertex joint graph The processed joint frequency domain time-varying image signal is converted into a time-varying image signal in the time-vertex domain.

[0237] like Figure 2 As shown, the first dotted line box represents the analysis filter bank of the graph filter bank, and the second dotted line box represents the synthesis filter bank of the graph filter bank.

[0238] in, Represents the transposed feature matrix of the time-vertex joint graph. The time-varying graph signal can be transformed into a joint frequency domain graph signal after being processed by this module.

[0239] is the joint frequency response function of the analysis filter of the kth channel. This module realizes the spectrum division of the time-varying image signal;

[0240] The downsampling matrix of the joint image frequency domain is used to compress the subband coefficients.

[0241] It is the upsampling matrix of the joint image frequency domain. This module maps the processed subband coefficients back to the original image signal space; is the joint frequency response function of the integrated filter of the kth channel;

[0242] Represents the feature matrix of the time-vertex joint graph. This module converts the processed joint frequency domain time-varying graph signal into a time-vertex domain time-varying graph signal.

[0243] Example 3

[0244] This embodiment uses the designed time-vertex joint critical sampling graph filter bank based on joint graph frequency domain sampling to perform simulation experiments. The specific process is as follows:

[0245] In the nonlinear approximation experiment, some subband coefficients with high absolute values ​​are retained, while the remaining subband coefficients are set to zero.

[0246] This experiment compares the time-vertex joint frequency domain spline graph filter with the existing frequency domain sampling graph filter group on static graphs. When processing time-varying graph signals, this type of static graph filter group needs to process the graph signal at each moment independently and then take the average.

[0247] The nonlinear approximation effects of different filter banks on synthetic datasets and sea level pressure datasets are shown in Figure 2. Figure 3 and Figure 4 As shown in the figure, "GraphSS_ideal", "GraphSS_orth" and "GraphSS_bior" respectively indicate that the filters in the two-channel frequency domain sampling graph filter bank (GraphSS) are ideal filters, orthogonal mirror filters and biorthogonal filters.

[0248] “SGFBSS_ideal”, “SGFBSS_B10” and “SGFBSS_B20” respectively indicate that the filters in the frequency domain spline filter bank (SGFBSS) are ideal filters and Butterworth filters of order 10 and 20.

[0249] from Figure 3 and Figure 4It is not difficult to see that when retaining the same proportion of sub-band coefficients, the nonlinear approximation effect of the time-vertex joint frequency domain spline graph filter group is much better than the existing frequency domain sampling graph filter group on static graphs. Its advantage is more obvious on smooth data sets (such as sea level pressure data sets). This is because the time-vertex joint frequency domain spline graph filter group not only captures the evolution characteristics of the time-varying graph signal in the vertex domain, but also makes full use of the correlation information in the time domain.

[0250] The above-described embodiments are only preferred embodiments for fully illustrating the present invention, and the protection scope of the present invention is not limited thereto. Equivalent substitutions or changes made by those skilled in the art based on the present invention are within the protection scope of the present invention. The protection scope of the present invention shall be subject to the claims.

Claims

1. A time-vertex joint graph filter bank design method based on joint graph frequency domain sampling, characterized by: The following steps are involved: 1) Obtain a time-varying graph signal X in the time-vertex domain; 2) Determine the complete reconstruction conditions for upsampling, downsampling, and time-vertex joint graph filter banks; 3) Designing a time-vertex joint graph filter bank using a complete reconstruction condition, wherein the time-vertex joint graph filter bank includes an analysis filter bank and a synthesis filter bank; the analysis filter bank and the synthesis filter bank process the time-varying graph signal X in the following manners: 31) Transpose the time-varying graph signal X with the transposed feature matrix of the time-vertex joint graph Calculate the time-varying image signal in the frequency domain of the joint image 32) Combine the time-varying image signal in the frequency domain The joint frequency response function with the analysis filter of the kth channel is Calculate and obtain the joint frequency domain signal after the k-th channel analysis and filtering; 33) The obtained joint frequency domain signal after analysis filtering is combined with the downsampling matrix Calculate and obtain the signal after the kth channel is jointly downsampled in the frequency domain; 34) Combine the obtained signal after the k-th channel joint frequency domain downsampling with the upsampling matrix The signal after joint frequency upsampling of the kth channel is calculated; 35) The joint frequency response function of the k-th channel joint frequency up-sampled signal and the k-th channel comprehensive filter is Calculate and obtain the joint frequency domain signal after comprehensive filtering of the kth channel; 36) Combine the k-th channel comprehensive filtered joint frequency domain signal with the characteristic matrix of the time-vertex joint graph Calculate and obtain the time-apex domain signal after comprehensive filtering of the kth channel; 37) Add the time-vertex domain signals of each channel to obtain the time-varying image signal processed by the filter bank 2. The method for designing a time-vertex joint graph filter bank based on joint graph frequency domain sampling according to claim 1, characterized in that: The downsampling is performed according to the following steps: S11: The original joint frequency domain signal is respectively and The top is evenly divided into two parts; S12: Frequency axis The second part of is folded and added to the first part, giving S13: Its frequency axis The second part of is folded and added to the first part to get The specific expressions are as follows: Where, t=1,2,…,T / 2; n=1,2,…,N / 2, Represents the spectrum value of the nth row and tth column of the spectrum after folding along the time domain; Represents the spectrum value of the nth row and tth column of the spectrum before folding; Represents the spectrum value of the nth row and tth column of the spectrum after folding along the time domain and the image domain; a joint frequency domain downsampled image signal representing the time-varying image signal; Formula (2) and formula (3) can be expressed in matrix form as follows: in, and They are subgraphs and The frequency domain downsampling matrix of I T / 2 and I N / 2 denote the identity matrices of order T / 2 and N / 2 respectively; J T / 2 and J N / 2 denote the inverse identity matrices of order T / 2 and N / 2 respectively; According to the properties of Kronecker product and vectorized operation, formula (4) is expressed as: Among them, x and They are respectively the vectorized original time-varying image signal and the vectorized joint frequency-domain downsampled image signal; Then it represents the joint frequency domain downsampling matrix.

3. The method for designing a time-vertex joint graph filter bank based on joint graph frequency domain sampling according to claim 2, characterized in that: Downsampling is the process of mapping the compressed image signal back to the original signal space, which is specifically performed in the following steps: make and are the original time-vertex joint graph and the time-vertex joint graph with increased size after upsampling, and their corresponding Laplacian matrices are and A set of graph signals on the original time-vertex joint graph is The corresponding joint frequency domain representation is Joint frequency domain upsampling of time-varying image signals is through the frequency axis and Repeat the folding of the original frequency domain signal And it is expressed as: Formulas (6) and (7) can be expressed in matrix form as follows: in, and They are subgraphs and The frequency domain upsampling matrix of I T and I N denote the identity matrices of order T and N respectively; J T and J N represents the inverse identity matrix of order T and N; According to the properties of Kronecker product and vectorized operation, formula (8) is expressed as: Among them, x and They are respectively the vectorized original time-varying image signal and the vectorized joint frequency domain upsampled image signal; Then it represents the joint frequency domain upsampling matrix.

4. The method for designing a time-vertex joint graph filter bank based on joint graph frequency domain sampling according to claim 3, characterized in that: The acquisition of the complete reconstruction condition of the time-vertex joint graph filter bank is performed according to the following steps: Designing a joint frequency-domain critical sampling time-vertex graph filter bank, the joint frequency-domain critical sampling time-vertex graph filter bank comprising an analysis filter bank, a sampling matrix and a synthesis filter bank; The analysis filter bank decomposes the input time-varying image signal into sub-band signals with different frequency domain intervals, the sampling matrix reduces the redundancy of the sub-band coefficients, and the synthesis filter bank reconstructs the processed sub-band signals; The kth analysis filter of the analysis filter bank is in, is the analysis filter bank; is the characteristic matrix of the Laplacian matrix of the original time-vertex joint graph; is the joint filter kernel of the kth analysis filter; The kth analysis filter of the synthesis filter bank is in, is a comprehensive filter bank; is the characteristic matrix of the Laplacian matrix of the original time-vertex joint graph; is the joint filter kernel of the kth integrated filter; The input-output relationship of the time-vertex joint critical sampling graph filter bank is determined as: Among them, x and are respectively the vectorized input time-varying graph signal and the output time-varying graph signal; and Respectively represent the frequency domain downsampling and upsampling matrices of the kth channel; because is an orthogonal matrix corresponding to the original time-vertex joint graph, and the transmission matrix of the framework is simplified to: in, represents the transmission matrix; Will and Substituting into formula (11) we can get: When the transmission matrix satisfies When , the time-vertex joint critical sampling graph filter bank satisfies the complete reconstruction condition; using the properties of the Kronecker product, similar terms are combined into: At this time, the overall transmission matrix is ​​decomposed into the above four parts, namely When all frequency responses of the joint graph filter satisfy: Available The transfer function of the time-vertex joint critical sampling graph filter bank satisfies That is, the graph filter bank satisfies the complete reconstruction condition, where c represents a constant.

5. The method for designing a time-vertex joint graph filter bank based on joint graph frequency domain sampling according to claim 4, characterized in that: The joint orthogonal graph filter bank is designed according to the following steps: Given a prototype filter The remaining filters are derived from this filter, namely: Substituting this filter design into the graph filter bank yields the following orthogonal condition: in, represents the T-i+1th eigenvalue in the time domain, Represents the N-j+1th eigenvalue of the vertex domain.

6. The method for designing a time-vertex joint graph filter bank based on joint graph frequency domain sampling according to claim 5, characterized in that: The biorthogonal graph filter bank is designed according to the following steps: For the biorthogonal image filter bank, the high-frequency joint analysis filter and the joint synthesis filter bank are derived from the low-frequency joint filter bank, that is: The graph filter bank satisfies the following complete reconstruction conditions, namely: in, Represents the N-j+1th eigenvalue of the vertex domain.

7. A time-vertex joint critical sampling graph filter bank based on joint graph frequency domain sampling, characterized in that: It includes an input unit, an analysis filter bank, a synthesis filter bank and an output unit; The input unit is used to obtain a time-varying graph signal X in the time-vertex domain; The analysis filter bank is used to divide the input time-varying image signal into different joint frequency domain regions to obtain subband coefficients of each channel; The integrated filter bank is used to reconstruct the processed sub-band signals; The output unit is used to output the time-varying image signal processed by the filter bank. The analysis filter bank includes an analysis filter characteristic matrix module, a joint frequency response function processing module, and a downsampling module; The feature matrix module is used to transform the feature matrix of the time-vertex joint graph After being processed by this module, the time-varying image signal can be transformed into a joint frequency domain image signal; The analysis filter joint frequency response function processing module is used to process the joint frequency response function of the analysis filter of the kth channel Perform spectrum division processing on the time-varying image signal; The downsampling module is used to downsample the matrix of the joint image frequency domain After compressing the signal processed by the analysis filter combined with the frequency response function processing module; The comprehensive filter group includes an up-sampling module, a comprehensive filter joint frequency response function processing module, and a comprehensive filter characteristic matrix module; The upsampling module is used to upsample the matrix in the frequency domain of the joint image Map the processed subband coefficients back to the original image signal space; The integrated filter joint frequency response function processing module is used to process the joint frequency response function of the integrated filter of the kth channel. Process the image signal after being processed by the upsampling module; The comprehensive filter feature matrix module is used to pass the feature matrix of the time-vertex joint graph The processed joint frequency domain time-varying image signal is converted into a time-varying image signal in the time-vertex domain.

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