Graph data compression method based on linear canonical transformation of two-dimensional graph

Abstract

The invention discloses a graph data compression method based on linear canonical transformation of a two-dimensional graph, and belongs to the technical field of digital data processing. According to the method, the underlying structure features of the to-be-compressed data are utilized, the to-be-compressed data are modeled into the two-dimensional diagram data on the Cartesian product diagram, the two-dimensional diagram data are distributed to two different dimensions, representation of the to-be-compressed data is achieved, and local structure information of the to-be-compressed data is fully utilized. According to the method, the linear regular transformation of the two-dimensional diagram is defined through the definition of the Fourier transformation of the diagram and the matrix decomposition property of the linear regular transformation, the flexibility of the linear regular parameters of the two-dimensional diagram is utilized, so that the transformation captures the local features and detail information of the data of the two-dimensional diagram, and the compression and processing are performed in the linear regular domain of the two-dimensional diagram; and effective compression of two-dimensional diagram data under a low compression ratio is realized. According to the method, parameter optimization and selection of linear regular transformation of the two-dimensional diagram are carried out through the normalized root mean square error and the peak signal-to-noise ratio, so that different application scenes are matched, and the compression performance of the two-dimensional diagram is further improved.

Description

Technical Field

[0001] The present invention relates to a graph data compression method, in particular to a graph data compression method based on linear canonical transformation of a two-dimensional graph, and belongs to the technical field of digital data processing. Background Art

[0002] With the rapid development of information technology, a large amount of various signal data has emerged, including data from sensors, social networks, and mobile devices. These data are usually high-dimensional, discrete, and irregular. Traditional signal processing methods have certain limitations in processing these complex data because they often fail to fully consider the topological structure characteristics between data. However, the emergence of graph signal processing technology fills this gap. Graph signal processing can analyze and process the structure, pattern, and relationship in complex data, so as to better understand the data and extract useful information. Graph signal processing has a wide range of applications in computer vision, image processing, and social network analysis. The development of these technologies has not only promoted scientific and technological innovation, but also promoted social progress, bringing more development opportunities and well-being to mankind.

[0003] Similar to classical signal processing, a major research direction in graph signal processing is to design transformation methods to exploit the structure of signals on the graph. Among them, the graph Fourier transform (GFT) is a hot topic. From the existing research results, the graph Fourier transform is mainly defined in two ways: based on the adjacency matrix and based on the graph Laplace matrix. However, similar to the classical Fourier transform, the graph Fourier transform has limitations in certain applications or mathematical problems. In order to overcome the shortcomings of the graph Fourier transform, researchers have proposed some new transformation methods, such as spectral graph wavelet transform, windowed graph Fourier transform, joint (time and graph) Fourier transform, graph fractional Fourier transform, windowed graph fractional Fourier transform and graph linear canonical transform (GLCT). These transformations play a vital role in the processing and analysis of graph data and have been applied to social networks, sensor networks, and extended to tasks in machine learning and deep neural networks. Through graph data processing, scientists can reveal high-order structures in the network and extract effective information from the data, including shape features. Compared with traditional data methods, these new methods have obvious advantages in extracting and inferring high-order information.

[0004] As the scale and complexity of data continue to increase, traditional one-dimensional graph transformation methods are unable to cope with multidimensional graph data. Therefore, it is of great significance to study multidimensional transformation. Multidimensional transformation can not only process complex high-dimensional graph data, but also better capture the multidimensional features in graph data, thereby improving the accuracy and efficiency of data analysis and processing. In particular, the introduction of multidimensional graph Fourier transform solves the problem that graph Fourier transform cannot correctly define the spectrum of multidimensional graph signals on the Cartesian product of graphs. Multidimensional graph Fourier transform can rearrange the one-dimensional spectrum obtained by graph Fourier transform into the multidimensional frequency domain, thereby providing the multidimensional spectrum of the signal. The methods and algorithms for studying multidimensional graph transformation have broad prospects in practical applications. For example, in the field of computer vision, multidimensional graph transformation can be used to improve the compression and processing efficiency of image and video data; in social network analysis, it can be used to reveal the multidimensional features and patterns in social relationships; in sensor networks, it can optimize the transmission and processing flow of sensor data. With the continuous development and application of these technologies, it will bring new impetus to the development of information science and technology and provide strong support for innovation in various fields.

[0005] Graph linear canonical transforms provide flexibility in processing data and can improve performance. However, graph linear canonical transforms have the same disadvantage as graph Fourier transforms in displaying the spectrum of multidimensional graph signals, that is, they ignore the product structure in the graph. Cartesian product graphs are tools for representing multi-domain data, while multidimensional graph data is a special type of graph data defined on Cartesian product graphs. Therefore, in order to solve the compression problem of multidimensional graph data and enable graph linear canonical transforms to pay attention to the product structure in the graph, a new graph data processing algorithm is urgently needed. Summary of the invention

[0006] In order to solve the problem of low compression accuracy of graph data under compression ratio, the purpose of the present invention is to provide a graph data compression method based on two-dimensional graph linear regularization transformation, by modeling the data to be compressed as two-dimensional graph data on a Cartesian product graph, allocating the data to two different dimensions, and defining the two-dimensional graph linear regularization transformation based on matrix decomposition of the linear regularization transformation, capturing the local features and detail information of the two-dimensional graph signal, compressing and processing in the two-dimensional graph linear regularization domain, and realizing effective compression of two-dimensional graph data under low compression ratio.

[0007] In order to achieve the above object, the present invention adopts the following technical solutions:

[0008] The present invention discloses a graph data compression method based on two-dimensional graph linear canonical transformation, comprising the following steps:

[0009] Step 1: For the data to be compressed, model it as a two-dimensional graph data on the Cartesian product graph. This two-dimensional graph data is different from the actual two-dimensional data. It is a two-dimensional graph signal distributed on the nodes of the Cartesian product graph. Map the data to be compressed to the Cartesian product graph of two factor graphs , modeled as a two-dimensional graph data on a Cartesian product graph

[0010] Step 1.1: Analyze the structure of the data to be compressed and construct two factor graphs to represent the information of the data to be compressed and

[0011] The specific implementation method of step 1.1 is:

[0012] For an undirected graph in represents a vertex set, is the number of vertices, Denotes the edge set, W denotes the weighted adjacency matrix, i, j∈{0, 1, …, N-1}. Defined in an undirected graph The signal on

[0013]

[0014] in is a set of real numbers. The graph signal is represented by a real-valued vector as Where T represents transpose. The Laplacian matrix of Where d represents the degree matrix of the graph and W represents the weighted adjacency matrix. The eigendecomposition of the Laplacian matrix is Where H represents the Hermitian transpose, U = [u0, u1, ..., u N-1 ] represents the eigenvector matrix, Represents the eigenvalue matrix, and the related eigenvalues ​​are represented and arranged as follows:

[0015] For two undirected graphs and make and represents the corresponding Laplace matrix, then the Cartesian product graph of these two undirected graphs is defined as Undirected Graph and are called the two factor graphs of the Cartesian product graph. The vertex set of the Cartesian product graph is in and N1 and N2 are factor graphs and The number of vertices of when Sometimes when Sometimes Where i1, j1∈{0, 1, ..., N1-1}, i2, j2∈{0, 1, ..., N2-1}; the weight is Where δ represents the Delta function; the Laplace operator is defined as In the formula and denote the identity matrices of length N1 and N2 respectively.

[0016] For the data to be compressed, two factor graphs are constructed to represent the information of the data to be compressed by analyzing the data structure. and

[0017] Step 1.2: Map the data to be compressed into the Cartesian product graph of two factor graphs , modeled as a two-dimensional graph data on a Cartesian product graph in Representing Factor Graphs The vertex of Representing Factor Graphs vertices, n1∈{0,…,N1-1}, n2∈{0,…,N2-1}.

[0018] Step 2: Based on the matrix decomposition properties of graph Fourier transform and linear canonical transformation, define the linear canonical transformation of the two-dimensional graph. The two-dimensional graph data obtained in step 1 is obtained Calculate the linear canonical transformation of its two-dimensional graph and obtain the linear canonical transformation coefficient of the two-dimensional graph

[0019] The specific implementation method of step 2 is:

[0020] For undirected graphs The eigendecomposition of its Laplacian matrix is Where U=[u0,u1,…,u N-1 ] is the eigenvector matrix, is an eigenvalue matrix, and the associated eigenvalues ​​satisfy The Fourier transform of a graph is defined as

[0021]

[0022] where u k represents the kth eigenvector in U, and * represents conjugation. The vector form of the Fourier transform of the graph is

[0023]

[0024] The inverse Fourier transform of a graph is defined as

[0025]

[0026] The vector form of the inverse Fourier transform of a graph is

[0027]

[0028] For the signal Its linear canonical transformation is defined as:

[0029]

[0030] in represents the linear regularization parameter matrix, where ab-bc = 1. The linear regularization transformation is implemented by the following matrix decomposition:

[0031]

[0032] in represents the chirp modulation parameters, represents the scale parameter, represents the fractional order parameter.

[0033] Due to the flexibility of linear canonical transformation parameters, linear canonical transformation has shown superior performance in the field of traditional signal processing. In graph signal processing, by introducing graph fractional Fourier transform, graph scaling transform and graph chirp modulation, the graph linear canonical transformation has stronger data representation capabilities. The graph Fourier transform operator is decomposed into where Q = [q0, q1, ..., q N-1 ]express The eigenvector matrix and R = diag([r0, r1, ..., r N-1 ])express The eigenvalue matrix of the graph is defined as:

[0034]

[0035] in The scaled Laplace matrix is ​​S = YXY H , and Y H =PΞP H , where P = [p0, p1, ..., p N-1 ] indicates Y H The eigenvector matrix and Ξ=diag([ι0,ι1,…,ι N-1 ]) indicates Y H The eigenvalue matrix of the graph scaling transformation is defined as:

[0036] ST σf=PQ H f (9)

[0037] in Define a signal in the spectral domain as:

[0038]

[0039] 1 of them N Represents an N-dimensional column vector whose elements are all 1. Performing an inverse Fourier transform on the signal yields The chirp modulation of the image signal is defined as:

[0040] CM ξ f=Υf (11)

[0041] in Represents the diagonal matrix of s(n).

[0042] Introducing graph fractional order parameters Graph scale parameters Figure chirp modulation parameters The graph linear canonical transformation is defined as:

[0043]

[0044] The vector form of the linear canonical transformation of the graph is:

[0045]

[0046] The corresponding eigenvalue is Λ M = diga([λ0,λ1,…,λ N-1 ]), The vector form of the inverse linear canonical transformation of a graph is defined as:

[0047]

[0048] The flexibility of graph linear regularization transformation parameters is utilized to capture subtle features of the data and achieve better data representation of subtle features.

[0049] For the graph and The Laplace matrix is Then the graph linear canonical Laplace operator is defined as

[0050]

[0051] And γ i , P i , Q i It is given by the following formula (16):

[0052]

[0053] in P=[p0,p1,…,p N-1 ], Ξ=diag([ι0, ι1,…, ι N-1 ]), Λ M =diag([λ0,λ1,…,λ N-1 ]). Cartesian product graph The Laplace operator A characteristic value of The corresponding eigenvector is and And meet:

[0054]

[0055] Where k1=0,…,N1-1, k2=0,…,N2-1.

[0056] For a Cartesian product graph A two-dimensional graph signal on , whose two-dimensional graph linear canonical transformation is defined as:

[0057]

[0058] in The inverse linear canonical transform of a two-dimensional graph is defined as:

[0059]

[0060] Where n1=0,…,N1-1, n2=0,…,N2-1.

[0061] For the Cartesian product graph in step 1 And the two-dimensional graph data on it Compute the linear canonical transformation of its two-dimensional graph:

[0062]

[0063] Get the linear regular transformation coefficients of the two-dimensional graph

[0064] Step 3: Linear regular transformation coefficients of the two-dimensional graph obtained in step 2 Expand it into a column vector and sort the elements in the column vector from large to small. Set the compression factor to 0<μ<1. For the sorted column vector, keep the part with coefficients less than the compression factor μ and set the remaining coefficients to zero to obtain the linear canonical domain compression coefficient of the two-dimensional graph. Adjust the linear canonical domain compression coefficient of the two-dimensional graph to the same as The same shape, denoted as And perform a two-dimensional linear canonical inverse transformation on it to obtain the compressed graph data That is, compression of two-dimensional graph data at a low compression ratio is achieved based on the linear canonical transformation of the two-dimensional graph.

[0065] Step 3.1: Linear regular transformation coefficients of the two-dimensional graph obtained in step 2 Expand it into a column vector and sort the elements in the column vector from large to small. Set the compression factor to 0<μ<1. For the sorted column vector, keep the part where the linear regular transformation coefficient μ of the two-dimensional graph is less than the compression factor, that is, take the first N cp elements, then μ=N cp / N × , where N × Represents the total number of elements in the column vector, and sets the remaining coefficients to zero to obtain the linear canonical field compression coefficients of the two-dimensional graph.

[0066] Step 3.2: Adjust the linear regular domain compression coefficient of the two-dimensional graph obtained in step 3.1 to The same shape, denoted as And perform a two-dimensional linear canonical inverse transform on it, and reconstruct the coefficients retained by compression, that is:

[0067]

[0068] Get the compressed graph data That is, compression of two-dimensional graph data at a low compression ratio is achieved based on the linear canonical transformation of the two-dimensional graph.

[0069] Step 4: Through the two evaluation indicators of normalized root mean square error NRMS and peak signal-to-noise ratio PSNR, the parameters of the linear canonical transformation of the two-dimensional image are optimized and selected to further improve the compression effect of the two-dimensional image data.

[0070] Step 4.1: The normalized root mean square error is defined as follows:

[0071]

[0072] in, Represents the mean of the original signal. By subtracting the mean from the denominator, the effect of translation is eliminated, allowing for a more accurate assessment of compression quality. The smaller the normalized root mean square error value, the better the compression performance. Adjust the parameters of the linear canonical transformation of the two-dimensional graph so that the parameters with the smallest normalized root mean square error are the optimal parameters.

[0073] Step 4.2: The peak signal-to-noise ratio is defined as follows:

[0074]

[0075] A higher peak signal-to-noise ratio value indicates that the compressed signal retains more original signal information, thus reflecting a higher fidelity. The parameters of the linear canonical transformation of the two-dimensional graph are adjusted so that the parameters with the largest peak signal-to-noise ratio are the optimal parameters, thus achieving compression of the two-dimensional graph data at a low compression ratio.

[0076] Beneficial effects:

[0077] 1. The present invention discloses a graph data compression method based on linear canonical transformation of a two-dimensional graph. The data to be compressed is modeled as two-dimensional graph data on a Cartesian product graph by utilizing the underlying structural features of the data to be compressed, and is distributed into two different dimensions, thereby achieving effective representation of the data to be compressed and making full use of its local structural information.

[0078] 2. The present invention discloses a graph data compression method based on two-dimensional graph linear regularization transformation. The two-dimensional graph linear regularization transformation is defined through the definition of graph Fourier transform and the matrix decomposition property of linear regularization transformation. Due to the flexibility of the two-dimensional graph linear regularization parameters, the transformation captures the local features and detail information of the two-dimensional graph data, and compresses and processes it within the two-dimensional graph linear regularization domain, thereby achieving effective compression of the two-dimensional graph data at a low compression ratio.

[0079] 3. The present invention discloses a graph data compression method based on linear canonical transformation of two-dimensional graphs, which optimizes and selects parameters of linear canonical transformation of two-dimensional graphs by normalizing root mean square error and peak signal-to-noise ratio to match different application scenarios and further improve the compression performance of two-dimensional graphs. BRIEF DESCRIPTION OF THE DRAWINGS

[0080] Figure 1 This is a flow chart of a graph data compression method based on two-dimensional graph linear canonical transformation of the present invention.

[0081] Figure 2 The kNN graph of 250 meteorological observation stations used in the embodiment of the present invention;

[0082] Figure 3 It is a schematic diagram of the entire compression process in the embodiment used in the present invention;

[0083] Figure 3 (a) represents two-dimensional graph data

[0084] Figure 3 (b) indicates The spectrum after the linear canonical transformation of the two-dimensional graph, where the parameters are σ=0.5, ξ=0, α=1, μ=0.05;

[0085] Figure 3 (c) shows the compressed two-dimensional image data under the parameters σ = 0.5, ξ = 0, α = 1, μ = 0.05

[0086] Figure 4 The normalized root mean square error and peak signal-to-noise ratio results of the present invention under different two-dimensional graph linear regularization parameters;

[0087] Figure 4 (a) represents the normalized root mean square error under different linear regularization parameters of two-dimensional graphs;

[0088] Figure 4 (b) Peak signal-to-noise ratio under different linear regularization parameters of two-dimensional graphs. DETAILED DESCRIPTION

[0089] The present invention will be further described below in conjunction with the accompanying drawings and embodiments.

[0090] Embodiment 1:

[0091] In real life, there is a lot of data that needs to be compressed to reduce storage space, improve transmission efficiency, and reduce processing costs. For example, pixel data in digital images, measurement values ​​in sensor networks, relationship data in social networks, or temperature data in meteorological observations.

[0092] like Figure 1 As shown, the present embodiment discloses a graph data compression method based on linear canonical transformation of a two-dimensional graph, using the public temperature data of the National Center for Environmental Information (NCEI) of the National Oceanic and Atmospheric Administration (NOAA) of the United States, randomly selecting 250 meteorological observation stations, and their corresponding temperature records for 2023. The present embodiment discloses a graph data compression method based on linear canonical transformation of a two-dimensional graph, and the specific implementation steps are as follows:

[0093] Step 1: For the temperature data of 250 meteorological observation stations in 2023, model them as two-dimensional graph data on the Cartesian product graph, which is a two-dimensional graph signal distributed on the nodes of the Cartesian product graph.

[0094] Step 1.1: The data to be compressed is 365 days of temperature records from 250 meteorological observation stations. Its underlying structure can be divided into two dimensions: one dimension is the number of days in a year as the time axis, and the other dimension is the location of different temperature measurement stations.

[0095] First, a natural path graph consisting of 365-day temperature records of the temperature observation station is constructed, denoted as in, represents the vertex set corresponding to each day, ε1={(v i , v i+1 )|0≤i<364} represents the edge set connecting consecutive days. The adjacency matrix W1 is defined as follows:

[0096]

[0097] This path graph structure captures the sequential nature of daily temperature observations, with each node connected to its predecessor and successor nodes.

[0098] Then, the k-nearest neighbor (kNN) method is used to construct a graph model for the 250 meteorological observation stations, namely the kNN graph, denoted as For each observation station, the spherical distance is calculated using their latitude and longitude coordinates. Specifically, let (θ i ,φ i ) and (θ j ,φ j ) represent two observation stations v i and v j The coordinates (in radians) of , where i, j∈{0, 1, ..., 249}. The spherical distance is defined as

[0099] d ij =arccos(cos(θ i )cos(θ j )cos(φ i -φ j )+sin(θ i )sin(θ j )).

[0100] Based on these spherical distances, the neighbors of each observation station can be identified. kNN graph It is constructed by setting each node as an observation station, denoted as The edge set ε2 consists of the edges connecting each vertex to its k nearest neighbors. i (0≤i≤249), connecting it with the k closest vertices based on spherical distance. The weight matrix W2 is defined by applying the spherical distance to the Gaussian kernel function. For two nodes i and j connected by an edge, the weight is defined as

[0101]

[0102] Among them, ~ means that two nodes are connected by an edge, ε 2 is a preset parameter. It should be noted that when calculating the spherical distance, in order to avoid significant precision errors when using the Gaussian kernel function, the radius of the earth R = 6,357 kilometers is omitted. Let k = 5, and the kNN graph constructed like Figure 2 shown.

[0103] Step 1.2: Path diagram based on temperature records And kNN diagram of meteorological observation stations Constructing a Cartesian product graph The product plot captures the temporal and spatial relationships inherent in the data. The 365-day temperature records of the 250 weather stations are then represented as a two-dimensional plot on a Cartesian product plot, where each element Indicates observation station In the The temperature of the day, where n1∈{0, 1, ..., 364}, n2∈{0, 1, ..., 249}, such as Figure 3 (a) This representation method enables a comprehensive analysis of temperature changes in time and space.

[0104] Step 2: For the Cartesian product graph in step 1 And the two-dimensional graph data on it Compute the linear canonical transformation of its two-dimensional graph:

[0105]

[0106] Get the linear regular transformation coefficients of the two-dimensional graph Figure 3 (b) indicates The spectrum after linear canonical transformation of the two-dimensional graph, where the parameters are σ=0.5, ξ=0, α=1, μ=0.05.

[0107] Step 3: Linear regular transformation coefficients of the two-dimensional graph obtained in step 2 Expand it into a column vector and sort the elements in the column vector from large to small. Set the compression factor to 0<μ<1. For the sorted column vector, keep the part with coefficients less than the compression factor and set the remaining coefficients to zero to obtain the linear canonical domain compression coefficient of the two-dimensional graph. Adjust the linear canonical domain compression coefficient of the two-dimensional graph to the same as The same shape, denoted as And perform a two-dimensional linear canonical inverse transformation on it to obtain the compressed graph data

[0108] Step 3.1: Linear regular transformation coefficients of the two-dimensional graph obtained in step 2 Expand it into a column vector and sort the elements in the column vector from large to small. Set the compression factor to 0<μ<1. For the sorted column vector, keep the part of the two-dimensional graph whose linear regular transformation coefficient is less than the compression factor, that is, take the first N cp elements, then μ=N cp / N × , where N × Represents the total number of elements in the column vector, and sets the remaining coefficients to zero to obtain the linear canonical field compression coefficients of the two-dimensional graph.

[0109] Step 3.2: Adjust the linear regular domain compression coefficient of the two-dimensional graph obtained in step 3.1 to The same shape, denoted as And perform a two-dimensional linear canonical inverse transform on it, and reconstruct the coefficients retained by compression, that is:

[0110]

[0111] Get the compressed graph data Based on the two-dimensional graph linear regularization parameters σ=0.5,ξ=0,α=1 and the compression factor μ=0.05, the compressed two-dimensional graph data is obtained. like Figure 3 (c) as shown.

[0112] Step 4: Optimize and select the parameters of the linear canonical transformation of the two-dimensional graph through the two evaluation indicators of normalized root mean square error NRMS and peak signal-to-noise ratio PSNR to further improve the compression performance.

[0113] Step 4.1: The normalized root mean square error is defined as follows:

[0114]

[0115] in, Represents the mean of the original signal. By subtracting the mean from the denominator, the effect of translation is eliminated, allowing for a more accurate assessment of compression quality. The smaller the normalized root mean square error value, the better the compression performance. Adjust the parameters of the linear canonical transformation of the two-dimensional graph so that the parameters with the smallest normalized root mean square error are the optimal parameters.

[0116] Step 4.2: The peak signal-to-noise ratio is defined as follows:

[0117]

[0118] A higher peak signal-to-noise ratio value indicates that the compressed signal retains more original signal information, thus reflecting a higher fidelity. The parameters of the linear canonical transformation of the two-dimensional graph are adjusted so that the parameters with the largest peak signal-to-noise ratio are the optimal parameters.

[0119] When the linear regularization parameters of the two-dimensional graph are set to σ = 0.5, ξ = 0, and α = 1, the normalized root mean square error and peak signal-to-noise ratio performance are examined when the compression factors are 0.01, 0.05, 0.1, 0.15, 0.2, 0.25, 0.3, and 0.35, respectively. The results are shown in the following table.

[0120]

[0121] Set ξ = 0 and the compression factor to μ = 0.05, choose different σ and α, Figure 4(a) shows the normalized root mean square error under different parameters. It can be seen that when α = 0.9 and σ = 1.4, the normalized root mean square error is the smallest, that is, the compression performance is the best; Figure 4 (b) shows the peak signal-to-noise ratio under different parameters. It can be seen that when α = 0.9 and σ = 2, the normalized root mean square error is the smallest, that is, the compression performance is the best. The above results show that due to the adjustability of the linear regularization transformation parameters of the two-dimensional image, more detailed information can be captured during the compression process, thus improving the compression performance of the two-dimensional image.

[0122] The specific description above further illustrates the purpose, technical solutions and beneficial effects of the invention in detail. It should be understood that the above is only a specific embodiment of the present invention and is not intended to limit the scope of protection of the present invention. Any modifications, equivalent substitutions, improvements, etc. made within the spirit and principles of the present invention should be included in the scope of protection of the present invention.

Claims

1. A graph data compression method based on two-dimensional graph linear canonical transformation, characterized by: The following steps are included: Step 1: For the data to be compressed, model it as two-dimensional graph data on a Cartesian product graph, where the two-dimensional graph data is a two-dimensional graph signal distributed on the nodes of the Cartesian product graph; Map the data to be compressed into the Cartesian product of two factor graphs , modeled as a two-dimensional graph data on a Cartesian product graph Step 2: Based on the matrix decomposition properties of graph Fourier transform and linear canonical transformation, define the linear canonical transformation of two-dimensional graph; According to step 1, we get the two-dimensional graph data Calculate the linear canonical transformation of its two-dimensional graph and obtain the linear canonical transformation coefficient of the two-dimensional graph Step 3: Linear regular transformation coefficients of the two-dimensional graph obtained in step 2 Expand it into a column vector, and sort the elements in the column vector from large to small; set the compression factor to 0<μ<1, for the sorted column vector, retain the part with coefficients less than the compression factor μ, and set the remaining coefficients to zero, to obtain the linear canonical domain compression coefficient of the two-dimensional graph; Adjust the linear regular domain compression factor of the two-dimensional graph to The same shape, denoted as And perform a two-dimensional linear canonical inverse transformation on it to obtain the compressed graph data That is, compression of two-dimensional graph data at a low compression ratio is achieved based on the linear canonical transformation of the two-dimensional graph.

2. The method for compressing graph data based on linear canonical transformation of two-dimensional graphs according to claim 1, characterized in that: Step 1 is implemented as follows: Step 1.1: Analyze the structure of the data to be compressed and construct two factor graphs to represent the information of the data to be compressed and The specific implementation method of step 1.1 is: For an undirected graph in represents a vertex set, is the number of vertices, represents the edge set, W represents the weighted adjacency matrix, i, j∈{0, 1, ..., N-1}; defined in an undirected graph The signal on in is a real number set; the graph signal is represented by a real-valued vector Where T represents transpose; the undirected graph The Laplacian matrix of is defined as Where D represents the degree matrix of the graph, W represents the weighted adjacency matrix; the eigendecomposition of the Laplacian matrix is where H represents the Hermitian transpose, U∈[u0,u1,…,u N-1 ] represents the eigenvector matrix, Represents the eigenvalue matrix, and the related eigenvalues ​​are represented and arranged as follows: For two undirected graphs and make and represents the corresponding Laplace matrix, then the Cartesian product graph of these two undirected graphs is defined as Undirected Graph and are called the two factor graphs of the Cartesian product graph; the vertex set of the Cartesian product graph is in And N1 and N2 are factor graphs and The number of vertices of when Sometimes when Sometimes Where i1, j1∈{0, 1, ..., N1-1}, i2, j2∈{0, 1, ..., N2-1}; The weight is Where δ represents the Delta function; the Laplace operator is defined as In the formula and denote the identity matrices of length N1 and N2 respectively; For the data to be compressed, two factor graphs are constructed to represent the information of the data to be compressed by analyzing the data structure. and Step 1.2: Map the data to be compressed into the Cartesian product graph of two factor graphs , modeled as a two-dimensional graph data on a Cartesian product graph in Representing Factor Graphs The vertex of Representing Factor Graphs vertices, n1∈{0,…,N1-1}, n2∈{0,…,N2-1}.

3. The method for compressing graph data based on linear canonical transformation of two-dimensional graphs as claimed in claim 2, characterized in that: Step 2 is implemented as follows: For undirected graphs The eigendecomposition of its Laplacian matrix is Where U=[u0,u1,…,u N-1 ] is the eigenvector matrix, is an eigenvalue matrix, and the associated eigenvalues ​​satisfy The Fourier transform of a graph is defined as where u k Indicates the first eigenvectors, * indicates conjugation; the vector form of the Fourier transform of the graph is The inverse Fourier transform of a graph is defined as The vector form of the inverse Fourier transform of a graph is For the signal Its linear canonical transformation is defined as: in represents the linear regularization parameter matrix, where ad-bc = 1; The linear canonical transformation is implemented by the following matrix decomposition: in represents the chirp modulation parameters, represents the scale parameter, represents the fractional order parameter; In graph signal processing, by introducing graph fractional Fourier transform, graph scaling transform and graph chirp modulation, the graph linear canonical transform is defined, which has stronger data representation ability; the graph Fourier transform operator is decomposed into where Q = [q0, q1, ..., q N-1 ]express The eigenvector matrix and R = diag([r0, r1, ..., r N-1 ])express The eigenvalue matrix of the graph; the fractional Fourier transform is defined as: in The scaled Laplace matrix is ​​S = YXY H , and Y H =PΞP H , where P = [p0, p1, ..., p N-1 ] indicates Y H The eigenvector matrix and Ξ=diag([ι0,ι1,…,ι N-1 ]) indicates Y H The eigenvalue matrix of the graph scaling transformation is defined as: ST σ f=PQ H f, (9) in Define a signal in the spectral domain as: 1 of them N Represents an N-dimensional column vector whose elements are all 1; the inverse Fourier transform of the signal is obtained The chirp modulation of the image signal is defined as: CM ξ f=γf, (11) in express The diagonal matrix of ; Introducing graph fractional order parameters Graph scale parameters Figure chirp modulation parameters The graph linear canonical transformation is defined as: The vector form of the linear canonical transformation of the graph is: The corresponding eigenvalue is Λ M =diag([λ0,λ1,…,λ N-1 ]), The vector form of the inverse linear canonical transformation of a graph is defined as: The flexibility of graph linear regularization transformation parameters is used to capture subtle features of data and achieve better data representation of subtle features; For the graph and The Laplace matrix is Then the graph linear canonical Laplace operator is defined as and It is given by the following formula (16): in P=[p0,p1,…,p N-1 ], Ξ=diag([ι0, ι1,…, ι N-1 ]), Λ M =diag([λ0,λ1,…,λ N-1 ]); Cartesian product graph The Laplace operator A characteristic value of The corresponding eigenvector is and And meet: in For a Cartesian product graph A two-dimensional graph signal on , whose two-dimensional graph linear canonical transformation is defined as: in The inverse linear canonical transform of a two-dimensional graph is defined as: Where n1=0, ..., N1-1, n2=0, ..., N2-1; For the Cartesian product graph in step 1 And the two-dimensional graph data on it Compute the linear canonical transformation of its two-dimensional graph: Get the linear regular transformation coefficients of the two-dimensional graph 4. The method for compressing graph data based on linear canonical transformation of two-dimensional graphs as claimed in claim 3, characterized in that: Step 3 is implemented as follows: Step 3.1: Linear regular transformation coefficients of the two-dimensional graph obtained in step 2 Expand it into a column vector, and sort the elements in the column vector from large to small; set the compression factor to 0<μ<1, and for the sorted column vector, retain the part of the two-dimensional graph linear regular transformation coefficient μ that is less than the compression factor, that is, take the first N cp elements, then μ=N cp / N × , where N × Represents the total number of elements in the column vector, and sets the remaining coefficients to zero to obtain the linear canonical field compression coefficient of the two-dimensional graph; Step 3.2: Adjust the linear regular domain compression coefficient of the two-dimensional graph obtained in step 3.1 to The same shape, denoted as And perform a two-dimensional linear canonical inverse transform on it, and reconstruct the coefficients retained by compression, that is: Get the compressed graph data That is, compression of two-dimensional graph data at a low compression ratio is achieved based on the linear canonical transformation of the two-dimensional graph.

5. A graph data compression method based on two-dimensional graph linear canonical transformation as claimed in claim 1, 2, 3 or 4, characterized in that: The method also includes step 4: optimizing and selecting parameters of the linear canonical transformation of the two-dimensional image through two evaluation indicators, namely, the normalized root mean square error NRMS and the peak signal-to-noise ratio PSNR, so as to further improve the compression effect of the two-dimensional image data.

6. The method for compressing graph data based on linear canonical transformation of two-dimensional graphs as claimed in claim 5, characterized in that: Step 4 is implemented as follows: Step 4.1: The normalized root mean square error is defined as follows: in, Represents the mean of the original signal; by subtracting the mean from the denominator, the effect of translation is eliminated, and the compression quality is evaluated more accurately; the smaller the normalized root mean square error value, the better the compression performance; adjust the parameters of the linear regularization transformation of the two-dimensional graph, and the parameters with the smallest normalized root mean square error are the optimal parameters; Step 4.2: The peak signal-to-noise ratio is defined as follows: A high peak signal-to-noise ratio value indicates that the compressed signal retains more original signal information, reflecting a higher fidelity; the parameters of the linear canonical transformation of the two-dimensional graph are adjusted so that the parameters with the largest peak signal-to-noise ratio are the optimal parameters, thereby achieving compression of the two-dimensional graph data at a low compression ratio.

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