A method for constructing three-dimensional electromagnetic spectrum map based on tensor completion

Abstract

The present invention relates to the technical field of data visualization, and specifically discloses a method for constructing a three-dimensional electromagnetic spectrum map based on tensor completion, comprising: S1, acquiring a region to be measured in a simulation scene, collecting spectrum data in the region to be measured, and obtaining a missing data set; S2, obtaining a completed tensor based on a tensor singular value decomposition method and the missing data set; S3, constructing a three-dimensional electromagnetic spectrum map based on a Poisson surface reconstruction algorithm and the completed tensor.

Description

Technical Field

[0001] The present invention relates to the technical field of data visualization, and in particular to a method for constructing a three-dimensional electromagnetic spectrum map based on tensor completion. Background Art

[0002] Electromagnetic spectrum maps are of great significance for spectrum sharing and improving spectrum utilization. Especially in the process of building the Internet of Things and 6G communication networks, efficient management of spectrum distribution in areas of interest has become a key issue.

[0003] Traditional two-dimensional spectrum maps can only display spectrum usage in the plane dimension, and cannot fully reflect the dynamic changes and distribution characteristics of the spectrum in three-dimensional space. At the same time, due to the limitations of measurement equipment and interference from environmental factors, spectrum data is often missing or noisy, affecting the integrity and accuracy of spectrum maps. The generation of electromagnetic spectrum maps involves multiple fields, including data collection, data completion, data compression, data visualization and other links.

[0004] The spectrum data sources of electromagnetic spectrum maps are divided into two types: passive sensor collection and active propagation model calculation. The sensor collection scheme relies more on accurate data collection. The spectrum map is constructed by establishing high-density three-dimensional coverage sensors. The quality of the spectrum map depends on the coverage density of the sensors. The advantage of the above method is that the data is real-time and reliable, but it has disadvantages such as high cost and limited range. The collected data is obviously sparse, and the blank area that is not collected is the blind area.

[0005] Therefore, how to use local and scattered electromagnetic spectrum mapping data to construct a global and integrated electromagnetic spectrum map is the main problem that needs to be solved at present. The corresponding generation methods can be divided into two categories: model-based generation methods and data-based generation methods. The electromagnetic spectrum map construction method based on the propagation model needs to take into account the shadow effect caused by building occlusion in the electromagnetic space, the low accuracy of geographic information parameters and the impact of electromagnetic characterization difficulties. The difficulty of achieving accurate propagation prediction can be imagined. The model-based generation method is also called an indirect construction method or a channel model method, which requires more prior environmental information support; the data-based generation method is also called a direct construction method. This type of method mainly relies on the completion, prediction and estimation of measurement data, so it has high requirements on the number of sensors in the surveying area. In actual engineering implementation, the data-based generation method has a more accurate effect than the model-based generation method.

[0006] In summary, in the generation and implementation of three-dimensional electromagnetic spectrum maps, the problem that the industry needs to solve is the problem of completing the blank areas of electromagnetic space sampling in order to achieve the most accurate spectrum map construction possible. Summary of the invention

[0007] In view of the above problems, the purpose of the present invention is to provide a three-dimensional electromagnetic spectrum map construction method based on tensor completion, construct a multi-resolution electromagnetic spectrum map from the perspective of data-driven method, realize the completion of the electromagnetic spectrum map from the perspective of spatial domain, effectively reduce the computational complexity, and effectively ensure the accuracy of the spectrum map.

[0008] The method for constructing a three-dimensional electromagnetic spectrum map based on tensor completion provided by the present invention comprises:

[0009] S1, obtaining a region to be tested in a simulation scene, collecting spectrum data in the region to be tested, and obtaining a missing data set;

[0010] S2, obtaining a completed tensor based on the tensor singular value decomposition method t-SVD and the missing data set;

[0011] S3, constructing a three-dimensional electromagnetic spectrum map based on a Poisson surface reconstruction algorithm and the completed tensor.

[0012] In a possible implementation, the S1 includes:

[0013] S11, obtaining the area to be measured, antenna coordinate position and parameters in the simulation scene; the parameters at least include: the scene to be measured, frequency, antenna directivity, bandwidth, transmission power, antenna height, number of antennas, building height;

[0014] S12, obtaining a grid size for dividing a spatial discrete distribution grid, and determining a resolution of the electromagnetic spectrum map according to the grid size;

[0015] S13, using the drone as a carrier platform for the data receiving device, measuring the electromagnetic spectrum intensity of the grid in the space to obtain spectrum data of the cube;

[0016] S14, based on the area to be measured, antenna coordinate position, parameters and resolution, the spectrum data of the area to be measured in the target time period is summarized to obtain the missing data set.

[0017] In a possible implementation, the S2 includes:

[0018] S21, performing spectrum space tensorization on the target space according to a space discretization method to obtain a forward slice, a horizontal slice, and a vertical slice of the spectrum tensor of the target space;

[0019] S22, determining a search area with any cube in the area to be tested as the center, searching for multiple similar cubes similar to the cube in the search area, and summarizing the multiple similar cubes to obtain a matrix and a trajectory tensor;

[0020] S23, performing tensor decomposition on the trajectory tensor according to the t-SVD algorithm to obtain an orthogonal singular value tensor and a singular tuple of the trajectory tensor;

[0021] S24, decomposing the entire spectrum tensor into a low-rank representation according to the orthogonal singular value tensor and the singular tuple to obtain a low-order tensor;

[0022] S25, compressing the front slices of the low-order tensor into a matrix and restoring it to the original space to obtain feature data of the spectrum tensor;

[0023] S26, based on the feature data, using a robust tensor to fill in missing values ​​and outliers to obtain the completed tensor.

[0024] In a possible implementation, the S22 includes:

[0025] When the matrix M i Transformed into trajectory tensor When , the trajectory tensor γ is expressed as follows:

[0026]

[0027] γ∈R L×WH×B

[0028] Where i is a positive integer, i = 1, 2, 3, ..., WH, R L×WH×B is the three-dimensional tensor of the target space, L is the number of similar cubes, W is the length of the target space, H is the height of the target space, and B is the width of the target space;

[0029] The S23 includes: performing Fourier transform on the trajectory tensor γ, and the formula is as follows:

[0030]

[0031] in, Represents the third mode of a three-dimensional tensor The discrete Fourier transform of , the tensor ρ is the Fourier transform result of the trajectory tensor γ, is a 3D tensor;

[0032] The frontal section of the tensor ρ is decomposed according to the t-SVD algorithm, and the formula is as follows:

[0033] [U i , S i , V i ]=SVD(ρ(:,:,i)), i=1,2..B

[0034] Among them, U i represents the left singular vector matrix, S i represents the singular value matrix, Vi represents the right singular vector matrix, B is the width of the target space, (:,:,i) is the tensor ρ, i is a positive integer, and B is the width of the target space;

[0035] Through the inverse Fourier transform, the singular vector matrix U i , singular value matrix S i and the right singular vector matrix V i Converted to the real number domain, the formula is as follows:

[0036]

[0037] where ifft(·, [], 3) represents the inverse Fourier transform of the horizontal slice, u and v are orthogonal singular value tensors, s is a singular tuple, is the left singular vector matrix U i The tensor formed, is the singular value matrix S i The tensor formed, is the right singular vector matrix V i The tensor formed.

[0038] In a possible implementation, the S24 includes:

[0039] Simplify the trajectory tensor γ, the formula is as follows:

[0040] r tubal (γ)<<min(L,WH)

[0041] Simplify the orthogonal singular value tensor u, orthogonal singular value tensor v and singular tuple s, the formula is as follows:

[0042] u′=u(:,1:r tubal , :)

[0043] s′=s(1:r tubal , 1:r tubal , :)

[0044] v′=v(:,1:r tubal , :)

[0045] According to the low-order tensor γ r , the formula is as follows:

[0046] γ r =u′*s′*v′ T ∈R L×WH×B

[0047] Among them, * represents tensor product, γ r is a low-order tensor, parameter r tubal It is used to determine the amount of information in the tensor reconstruction, u′ is the simplification of the singular value tensor u, is the transpose of the orthogonal singular value tensor u, s′ is the simplification of the singular tuple s, L is the number of similar cubes, W is the length of the target space, H is the height of the target space, and B is the width of the target space.

[0048] In a possible implementation, the S25 includes:

[0049] The low-rank tensor γ r The front slice of is compressed into a matrix and restored to the original space, the formula is as follows:

[0050] δ←Rep(γ r )

[0051] Among them, Rep is the reprojection operation, δ is the trajectory tensor characteristic data;

[0052] The S26 includes: based on the feature data, using a robust tensor to fill missing values ​​and outliers, the formula is as follows:

[0053]

[0054] Among them, λ and β are penalty parameters, ‖δ‖ TNN is the nuclear norm of the feature data δ, is the error, ω is the auxiliary variable, u and v are orthogonal singular value tensors, and s is a singular tuple;

[0055] Based on error and auxiliary variable ω to obtain the Lagrangian transformation function The formula is as follows:

[0056]

[0057] Solve and update the error using the soft threshold method And auxiliary variable ω, we get the completed tensor δ0, the formula is as follows:

[0058]

[0059] Among them, γ is the Lagrange multiplier, y1 and y2 are Lagrange dual variables, and δ is the trajectory tensor The characteristic data, F and TNN are positive integers, is the global tensor.

[0060] In a possible implementation, S3 includes:

[0061] S31, performing data preprocessing on the completed tensor to obtain point cloud data; the data preprocessing at least includes: removing outliers, filtering and smoothing;

[0062] S32, constructing an indicator function according to the point cloud data;

[0063] S33, constructing a Poisson equation according to the divergence of the gradients of the screening function and the indicator function;

[0064] S34, solving the Poisson equation using a finite difference method or a fast Fourier transform method of the Poisson equation;

[0065] S35, extracting isosurfaces according to the Poisson equation, and constructing a three-dimensional electromagnetic spectrum map according to the Marching Cubes algorithm and the isosurface values.

[0066] In a possible implementation, the S32 includes:

[0067] Determine each point P according to the following formula i Gaussian distribution Φ i (x):

[0068]

[0069] Superimpose all points P i The contribution of is used to obtain the global indicator function Φ(x), the formula is as follows:

[0070]

[0071] Among them, w i For point P i The relevant weight, σ is the standard deviation of the Gaussian kernel, i is a positive integer, and x represents a point in three-dimensional space.

[0072] In a possible implementation, the S33 includes:

[0073] The Poisson equation is constructed based on the divergence of the gradients of the screening function and the indicator function;

[0074] Among them, the screening function The following formula:

[0075]

[0076] The source term ρ(x) is given by the following formula:

[0077]

[0078] The gradient calculation formula of the scalar function Φ(x) in three-dimensional space is as follows:

[0079]

[0080] The Poisson equation Δu(x) is as follows:

[0081] Δu(x)=ρ(x)

[0082] Among them, Δ is the Laplace operator, ∈ is the screening parameter, is the gradient operator, for The Euclidean norm of , that is, the length (magnitude) of the gradient vector, is the partial derivative of the function Φ(x) with respect to the coordinate x, is the partial derivative of the function Φ(x) with respect to the coordinate y, is the partial derivative of the function Φ(x) with respect to the coordinate z.

[0083] In a possible implementation, the S34 includes:

[0084] Convert the source term from the real domain to the frequency domain, and calculate the frequency domain result of the Poisson equation in the frequency domain based on the source term;

[0085] The frequency domain result is inversely transformed back to the real domain to obtain the solution of Poisson's equation.

[0086] The method for constructing a three-dimensional electromagnetic spectrum map based on tensor completion provided by the present invention avoids invalid surveying and mapping and the inability of parallel coordinates to fully display the changing laws of the electromagnetic situation, reduces the complexity of engineering implementation, selects passive sensor collection as the data source, and selects a data-based generation method as the generation method of the electromagnetic spectrum map. It can significantly improve the processing efficiency and analysis accuracy of electromagnetic spectrum data, provide a comprehensive, intuitive, and real-time decision support platform for spectrum managers, help the refined management and optimal configuration of spectrum resources, and promote the efficient operation and sustainable development of wireless communication networks. BRIEF DESCRIPTION OF THE DRAWINGS

[0087] Figure 1 A schematic diagram of a process for constructing a three-dimensional electromagnetic spectrum map provided by an embodiment of the present invention;

[0088] Figure 2 A schematic diagram of electromagnetic spectrum map data collection and spatial segmentation provided by an embodiment of the present invention;

[0089] Figure 3 The radiation source position and simulation scenario provided by the embodiment of the present invention;

[0090] Figure 4 A three-dimensional spectrum map after tensor completion provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0091] The following detailed description of the embodiments of the present invention is further described in detail in conjunction with the accompanying drawings and examples. The detailed description of the following embodiments and the accompanying drawings are used to exemplarily illustrate the principles of the present invention, but cannot be used to limit the scope of the present invention, that is, the present invention is not limited to the preferred embodiments described, and the scope of the present invention is defined by the claims.

[0092] In the description of the present invention, it should be noted that, unless otherwise specified, “plurality” means two or more than two; the terms “first”, “second”, etc. are only used for descriptive purposes and cannot be understood as indicating or implying relative importance; for ordinary technicians in this field, the specific meanings of the above terms in the present invention can be understood according to the specific circumstances.

[0093] The purpose of the present invention is to solve the defects of low precision and deviation of restored data from real data in the current electromagnetic spectrum map construction method. Specifically, the following steps are included: first, a spatial grid granularity division method is designed for the selected scene, and the collected data is pre-processed and stored in the database; then, the spectrum tensorization method is used to complete the missing data based on the low-rank characteristics of the spectrum tensor and the tensor singular value decomposition method is used to restore the integrity of the spectrum data; finally, in order to obtain detailed and smooth electromagnetic spectrum data, a three-dimensional electromagnetic spectrum map is constructed based on the completed tensor data through the Poisson surface reconstruction algorithm.

[0094] Combination Figure 1-Figure 4 The present invention provides a method for constructing a three-dimensional electromagnetic spectrum map based on tensor completion, comprising:

[0095] S1, obtaining the area to be tested in the simulation scene, collecting spectrum data in the area to be tested, and obtaining the missing data set;

[0096] In a possible implementation, S1 includes:

[0097] S11, obtaining the area to be measured, antenna coordinate position and parameters in the simulation scene; the parameters at least include: the scene to be measured, frequency, antenna directivity, bandwidth, transmission power, antenna height, number of antennas, building height;

[0098] In a possible implementation, the length W of the scene to be measured is 1000 m, the width B is 1000 m, and the environmental parameters and transmitter parameters in the scene to be measured are shown in Table 1. The antenna position coordinates are (100,824,100), (150,500,15), (260,160,35), (650,755,65), (750,260,15).

[0099] Table 1

[0100] parameter Numeric Test scene (W×B) 1000m×1000m Resolution 1m, 10m, 50m frequency 2000MHz Antenna directivity Omnidirectional bandwidth 200KHz Transmission power 30mW Noise spectral density -174dBm / Hz Antenna height 1.5m-75m Number of antennas 5 Building height (H) 5m-100m

[0101] S12, obtaining a grid size for dividing a spatial discrete distribution grid, and determining a resolution of the electromagnetic spectrum map according to the grid size;

[0102] In one possible implementation, the electromagnetic spectrum map is resolved with 10 m as the side length of the cube grid.

[0103] S13, using the drone as a carrier platform for the data receiving device, measuring the electromagnetic spectrum intensity of the grid in the space to obtain the spectrum data of the cube;

[0104] In a possible implementation, if the electromagnetic spectrum intensity is not collected, it is considered that the spectrum intensity of the cube grid is unknown.

[0105] S14, based on the area to be measured, antenna coordinate position, parameters and resolution, the spectrum data of the area to be measured in the target time period is summarized to obtain the missing data set.

[0106] After the above steps, the electromagnetic spectrum information obtained in space is incomplete. In order to obtain complete spectrum space information, it is necessary to complete the missing electromagnetic spectrum information.

[0107] S2, based on the tensor singular value decomposition method (t-SVD) and the missing data set, the completed tensor is obtained;

[0108] In a possible implementation, S2 includes:

[0109] S21, performing spectrum space tensorization on the target space according to a space discretization method to obtain a forward slice, a horizontal slice, and a vertical slice of the spectrum tensor of the target space;

[0110] In one possible implementation, according to the spatial discretization method, we can obtain a A three-dimensional tensor of , where W, H, and B represent the length, height, and width of the target space respectively. The missing data set is collected as a tensor The input data, and Represent the frontal, transverse and vertical slices of the spectrum tensor respectively.

[0111] S22, determining a search area with any cube in the area to be tested as the center, searching for multiple similar cubes similar to the cube in the search area, and summing up the multiple similar cubes to obtain a matrix and a trajectory tensor;

[0112] In one example, for a point (cube) x in the area to be measured i ∈R B×1 (i=1,2,3,…,WH), take it as the center, select a search area of ​​size w×w, and search for the i Similar points (cubes), get L similar points, and further get the matrix M i ∈R L×B .

[0113] In a possible implementation, S22 includes:

[0114] When the matrix M i Transformed into trajectory tensor When , the trajectory tensor γ is expressed as follows:

[0115]

[0116] γ∈R L×WH×B

[0117] Where i is a positive integer, i = 1, 2, 3, ..., WH, R L×WH×B is the three-dimensional tensor of the target space, L is the number of similar cubes, W is the length of the target space, H is the height of the target space, and B is the width of the target space;

[0118] S23, performing tensor decomposition on the trajectory tensor according to the t-SVD algorithm to obtain an orthogonal singular value tensor and a singular tuple of the trajectory tensor;

[0119] Compared with CP decomposition and Tucker decomposition, the t-SVD-based method can effectively improve the computational efficiency by solving the singular values ​​in the Fourier domain to obtain the best fit of the tensor.

[0120] S23 includes: performing Fourier transform on the trajectory tensor γ, the formula is as follows:

[0121]

[0122] in, Represents the third mode of a three-dimensional tensor The discrete Fourier transform of , the tensor ρ is the Fourier transform result of the trajectory tensor γ, is a 3D tensor;

[0123] According to the t-SVD algorithm, the front section of the tensor ρ is decomposed as follows:

[0124] [U i ,S i ,V i ]=SVD(ρ(:,:,i)),i=1,2..B

[0125] Among them, U i represents the left singular vector matrix, S i represents the singular value matrix, V i represents the right singular vector matrix, B is the width of the target space, (:,:,i) is the tensor ρ, i is a positive integer, and B is the width of the target space;

[0126] Through the inverse Fourier transform, the singular vector matrix U i , singular value matrix S i and the right singular vector matrix V i Converted to the real number domain, the formula is as follows:

[0127]

[0128] where ifft(·, [], 3) represents the inverse Fourier transform of the horizontal slice, u and v are orthogonal singular value tensors, s is a singular tuple, is the left singular vector matrix U i The tensor formed, is the singular value matrix S i The tensor formed, is the right singular vector matrix V i The tensor formed.

[0129] S24, decomposing the entire spectrum tensor into a low-rank representation according to the orthogonal singular value tensor and the singular tuple to obtain a low-rank tensor;

[0130] In a possible implementation, S24 includes:

[0131] In the process of tensor decomposition, the entire spectrum tensor is decomposed instead of separately. In order to more effectively perform low-rank representation, the trajectory tensor γ is simplified as follows:

[0132] r tubal (γ)<<min(L,WH)

[0133] Further utilizing the low-rank property of the tensor, we simplify the orthogonal singular value tensor u, orthogonal singular value tensor v, and singular tuple s as follows:

[0134] u′=u(:,1:r tubal , :)

[0135] s′=s(1:r tubal , 1:r tubal , :)

[0136] v′=v(:,1:r tubal , :)

[0137] According to the low-order tensor γ r , the formula is as follows:

[0138]

[0139] Among them, * represents tensor product, γ r is a low-order tensor, parameter r tubal It is used to determine the amount of information in the tensor reconstruction, u′ is the simplification of the singular value tensor u, is the transpose of the orthogonal singular value tensor u, s′ is the simplification of the singular tuple s, L is the number of similar cubes, W is the length of the target space, H is the height of the target space, and B is the width of the target space.

[0140] S25, compressing the front slices of the low-order tensor into a matrix and restoring it to the original space to obtain the characteristic data of the spectrum tensor;

[0141] In a possible implementation, S25 includes:

[0142] The low-rank tensor γ r The front slice of is compressed into a matrix and restored to the original space, the formula is as follows:

[0143] δ←Rep(γ r )

[0144] Among them, Rep is the reprojection operation, δ is the trajectory tensor characteristic data;

[0145] S26, based on the feature data, using the robust tensor to fill the missing values ​​and outliers to obtain the completed tensor.

[0146] In a possible implementation, S26 includes: based on the feature data, using a robust tensor to fill in missing values ​​and outliers, the formula is as follows:

[0147]

[0148] Among them, λ and β are penalty parameters, ‖δ‖ TNN is the nuclear norm of the feature data δ, is the error, ω is the auxiliary variable, u and v are orthogonal singular value tensors, and s is a singular tuple;

[0149] The specific solution process is as follows:

[0150] 1. The Lagrange transform of this process can be expressed as:

[0151]

[0152] Among them, γ is the Lagrange multiplier, y1 and y2 represent the Lagrange dual variables respectively.

[0153] 2. Further, we get The Lagrangian function of is solved using the soft threshold method:

[0154]

[0155] get Solution:

[0156]

[0157] 3. Get the Lagrangian function of ω:

[0158]

[0159] Setting the partial derivative of ω to 0 gives the solution for ω:

[0160]

[0161] 4. Update and ω, set the partial derivative of δ to 0, and further obtain the completed tensor δ0:

[0162]

[0163] Among them, γ is the Lagrange multiplier, y1 and y2 are Lagrange dual variables, and δ is the trajectory tensor The characteristic data, F and TNN are positive integers, is the global tensor.

[0164] After the above steps, the complete electromagnetic space spectrum data tensor is obtained, that is, the completed tensor δ0.

[0165] S3, constructs a three-dimensional electromagnetic spectrum map based on the Poisson surface reconstruction algorithm and the completed tensor.

[0166] By reconstructing the three-dimensional electromagnetic spectrum map through the Poisson surface reconstruction algorithm, detailed and smooth electromagnetic spectrum data can be obtained.

[0167] In a possible implementation, S3 includes:

[0168] S31, performing data preprocessing on the completed tensor to obtain point cloud data; the data preprocessing at least includes: removing outliers, filtering and smoothing;

[0169] S32, constructing an indicator function according to the point cloud data;

[0170] The indicator function is positive inside the point cloud, negative outside, and close to zero on the surface of the point cloud. This step is usually achieved by placing a weighted Gaussian kernel around each point. The indicator function is constructed and defined in three-dimensional space to distinguish between the inside and outside of the point cloud. Inside the point cloud, the indicator function Φ(x)>0, and outside the point cloud, the indicator function Φ(x)<0, and the value near the surface of the point cloud is close to zero.

[0171] In a possible implementation, S32 includes:

[0172] Determine each point P according to the following formula i Gaussian distribution Φi (x):

[0173]

[0174] Superimpose all points P i The contribution of is used to obtain the global indicator function Φ(x), the formula is as follows:

[0175]

[0176] Among them, w i For point P i The relevant weight, σ is the standard deviation of the Gaussian kernel, i is a positive integer, and x is a point in three-dimensional space.

[0177] S33, constructing a Poisson equation according to the divergence of the gradients of the screening function and the indicator function;

[0178] In a possible implementation, S33 includes: constructing a Poisson equation according to the divergence of the gradients of the screening function and the indicator function;

[0179] The source term ρ(x) of the Poisson equation is related to the indicator function Φ(x). In order to ensure that the reconstructed surface is close to the point cloud, the source term is constructed by calculating the divergence of the gradient of Φ(x). In practical engineering applications, it is difficult to use Φ(x) directly. The concept of the Poisson surface equation is introduced, in which a screening function is introduced. To suppress the influence of distant points.

[0180] Among them, the screening function The following formula:

[0181]

[0182] The source term ρ(x) is given by the following formula:

[0183]

[0184] The gradient calculation formula of the scalar function Φ(x) in three-dimensional space is as follows:

[0185]

[0186] The Poisson equation Δu(x) is as follows:

[0187] Δu(x)=ρ(x)

[0188] Among them, Δ is the Laplace operator, ∈ is the screening parameter, is the gradient operator, for The Euclidean norm of , that is, the length (magnitude) of the gradient vector, is the partial derivative of the function Φ(x) with respect to the coordinate x, is the partial derivative of the function Φ(x) with respect to the coordinate y, is the partial derivative of the function Φ(x) with respect to the coordinate z.

[0189] S34, solving the Poisson equation using a finite difference method or a fast Fourier transform method of the Poisson equation;

[0190] Finite difference methods or fast Fourier transforms (FFTs) of Poisson's equation are used to solve a field function that minimizes its Laplace energy when its gradient field is most similar to the gradient field of the indicator function. The solution process produces a voxel grid, and the value at the center of each voxel represents an estimate of the field function.

[0191] In a possible implementation, FFT is used to solve u(x), and S34 includes:

[0192] The source term ρ(x) is converted from the real domain to the frequency domain, and since the Poisson equation in the frequency domain becomes a simple multiplication, the frequency domain result of the Poisson equation u(x) in the frequency domain is calculated based on the source term;

[0193] The frequency domain results are inversely transformed back to the real domain to obtain the solution to Poisson's equation.

[0194] S35, extracting isosurfaces according to the Poisson equation, and constructing a three-dimensional electromagnetic spectrum map through the Marching Cubes algorithm and the isosurface values.

[0195] In a possible implementation, the surface reconstruction is completed by the Marching Cubes algorithm, and the isosurface u(x)=0 is extracted from u(x). The process includes:

[0196] 1. Divide the space into multiple small volume units, such as a cube grid.

[0197] 2. For each volume cell, determine the way the isosurface passes through the cell based on the values ​​of u(x) at the eight vertices of the cell, and generate a set of triangles based on predefined rules to approximate the surface portion passing through the cell.

[0198] 3. Connect the triangles generated by all units to form the final watertight surface model by removing unnecessary triangles, optimizing the topology and adding subdivision operations.

[0199] The method for constructing a three-dimensional electromagnetic spectrum map based on tensor completion provided by the present invention avoids invalid surveying and mapping and the inability of parallel coordinates to fully display the changing laws of the electromagnetic situation, reduces the complexity of engineering implementation, selects passive sensor collection as the data source, and selects a data-based generation method as the generation method of the electromagnetic spectrum map. It can significantly improve the processing efficiency and analysis accuracy of electromagnetic spectrum data, provide a comprehensive, intuitive, and real-time decision support platform for spectrum managers, help the refined management and optimal configuration of spectrum resources, and promote the efficient operation and sustainable development of wireless communication networks.

[0200] The above is only a specific embodiment of the present invention, but the protection scope of the present invention is not limited thereto. Any person skilled in the art can easily think of changes or substitutions within the technical scope disclosed by the present invention, which should be included in the protection scope of the present invention. Therefore, the protection scope of the present invention should be based on the protection scope of the claims.

Claims

1. A method for constructing a three-dimensional electromagnetic spectrum map based on tensor completion, characterized in that: include: S1, obtaining a region to be tested in a simulation scene, collecting spectrum data in the region to be tested, and obtaining a missing data set; S2, obtaining a completed tensor based on the tensor singular value decomposition method t-SVD and the missing data set; S3, constructing a three-dimensional electromagnetic spectrum map based on a Poisson surface reconstruction algorithm and the completed tensor; The S2 includes: S21, performing spectrum space tensorization on the target space according to a space discretization method to obtain a forward slice, a horizontal slice, and a vertical slice of the spectrum tensor of the target space; S22, determining a search area with any cube in the area to be tested as the center, searching for multiple similar cubes similar to the cube in the search area, and summarizing the multiple similar cubes to obtain a matrix and a trajectory tensor; S23, performing tensor decomposition on the trajectory tensor according to the t-SVD algorithm to obtain an orthogonal singular value tensor and a singular tuple of the trajectory tensor; S24, decomposing the entire spectrum tensor into a low-rank representation according to the orthogonal singular value tensor and the singular tuple to obtain a low-order tensor; S25, compressing the front slices of the low-order tensor into a matrix and restoring it to the original space to obtain feature data of the spectrum tensor; S26, based on the feature data, using a robust tensor to fill in missing values ​​and outliers to obtain the filled tensor; The S25 includes: The low-rank tensor γ r The front slice of is compressed into a matrix and restored to the original space, the formula is as follows: δ←Rep(Υ r ) Among them, Rep is the reprojection operation, δ is the trajectory tensor characteristic data; The S26 includes: based on the feature data, using a robust tensor to fill missing values ​​and outliers, the formula is as follows: Among them, λ and β are penalty parameters, ‖δ‖ TNN is the nuclear norm of the feature data δ, is the error, ω is the auxiliary variable, u and v are orthogonal singular value tensors, and s is a singular tuple; Based on error and auxiliary variable ω to obtain the Lagrangian transformation function The formula is as follows: Solve and update the error using the soft threshold method And auxiliary variable ω, we get the completed tensor δ0, the formula is as follows: Among them, γ is the Lagrange multiplier, y1 and y2 are Lagrange dual variables, and δ is the trajectory tensor The characteristic data, F and TNN are positive integers, is the global tensor.

2. The method for constructing a three-dimensional electromagnetic spectrum map according to claim 1, characterized in that: The S1 includes: S11, obtaining the area to be measured, antenna coordinate position and parameters in the simulation scene; the parameters at least include: the scene to be measured, frequency, antenna directivity, bandwidth, transmission power, antenna height, number of antennas, building height; S12, obtaining a grid size for dividing a spatial discrete distribution grid, and determining a resolution of the electromagnetic spectrum map according to the grid size; S13, using the drone as a carrier platform for the data receiving device, measuring the electromagnetic spectrum intensity of the grid in the space to obtain spectrum data of the cube; S14, based on the area to be measured, antenna coordinate position, parameters and resolution, the spectrum data of the area to be measured in the target time period is summarized to obtain the missing data set.

3. The method for constructing a three-dimensional electromagnetic spectrum map according to claim 1, characterized in that: The S22 includes: When the matrix M i Transformed into trajectory tensor When , the trajectory tensor γ is expressed as follows: γ∈R L×WH×B Where i is a positive integer, i = 1, 2, 3, ..., WH, R L×WH×B is the three-dimensional tensor of the target space, L is the number of similar cubes, W is the length of the target space, H is the height of the target space, and B is the width of the target space; The S23 includes: performing Fourier transform on the trajectory tensor γ, and the formula is as follows: in, Represents the third mode of a three-dimensional tensor The discrete Fourier transform of the trajectory tensor Υ is obtained by using the tensor ρ. is a 3D tensor; The frontal section of the tensor ρ is decomposed according to the t-SVD algorithm, and the formula is as follows: [U i ,S i ,V i ]=SVD(ρ(:,:,i)),i=1,2..B Among them, U i represents the left singular vector matrix, S i represents the singular value matrix, V i represents the right singular vector matrix, B is the width of the target space, (:,:,i) is the tensor ρ, i is a positive integer, and B is the width of the target space; Through the inverse Fourier transform, the singular vector matrix U i , singular value matrix S i and the right singular vector matrix V i Converted to the real number domain, the formula is as follows: where ifft(·,[],3) represents the inverse Fourier transform of the horizontal slice, u and v are orthogonal singular value tensors, s is a singular tuple, is the left singular vector matrix U i The tensor formed, is the singular value matrix S i The tensor formed, Right singular vector matrix V i The tensor formed.

4. The method for constructing a three-dimensional electromagnetic spectrum map according to claim 1, characterized in that: The S24 includes: Simplify the trajectory tensor Υ, the formula is as follows: r tubal (c)<<min(L,WH) Simplify the orthogonal singular value tensor u, orthogonal singular value tensor v and singular tuple s, the formula is as follows: u′=u(:,1:r tubal ,:) s′=s(1:r tubal ,1:r tubal ,:) v′=v(:,1:r tubal ,:) According to the low-order tensor γ r , the formula is as follows: Among them, * represents tensor product, γ r is a low-order tensor, parameter r tubal It is used to determine the amount of information in the tensor reconstruction, u′ is the simplification of the singular value tensor u, is the transpose of the orthogonal singular value tensor u, s′ is the simplification of the singular tuple s, L is the number of similar cubes, W is the length of the target space, H is the height of the target space, and B is the width of the target space.

5. The method for constructing a three-dimensional electromagnetic spectrum map according to claim 1, characterized in that: The S3 includes: S31, performing data preprocessing on the completed tensor to obtain point cloud data; the data preprocessing at least includes: removing outliers, filtering and smoothing; S32, constructing an indicator function according to the point cloud data; S33, constructing a Poisson equation according to the divergence of the gradients of the screening function and the indicator function; S34, solving the Poisson equation using a finite difference method or a fast Fourier transform method of the Poisson equation; S35, extracting isosurfaces according to the Poisson equation, and constructing a three-dimensional electromagnetic spectrum map according to the Marching Cubes algorithm and the isosurface values.

6. The method for constructing a three-dimensional electromagnetic spectrum map according to claim 5, characterized in that: The S32 includes: Determine each point P according to the following formula i Gaussian distribution Φ i (x): Superimpose all points P i The contribution of is used to obtain the global indicator function Φ(x), the formula is as follows: Among them, w i For point P i The relevant weight, σ is the standard deviation of the Gaussian kernel, i is a positive integer, and x represents a point in three-dimensional space.

7. The method for constructing a three-dimensional electromagnetic spectrum map according to claim 5, characterized in that: The S33 includes: The Poisson equation is constructed based on the divergence of the gradients of the screening function and the indicator function; Among them, the screening function The following formula: The source term ρ(x) is given by the following formula: The gradient calculation formula of the scalar function Φ(x) in three-dimensional space is as follows: The Poisson equation Δu(x) is as follows: Δu(x)=ρ(x) Among them, Δ is the Laplace operator, ∈ is the screening parameter, is the gradient operator, for The Euclidean norm of , that is, the length (magnitude) of the gradient vector, is the partial derivative of the function Φ(x) with respect to the coordinate x, is the partial derivative of the function Φ(x) with respect to the coordinate y, is the partial derivative of the function φ(x) with respect to the coordinate z.

8. The method for constructing a three-dimensional electromagnetic spectrum map according to claim 7, characterized in that: The S34 includes: Convert the source term from the real domain to the frequency domain, and calculate the frequency domain result of the Poisson equation in the frequency domain based on the source term; The frequency domain result is inversely transformed back to the real domain to obtain the solution of Poisson's equation.

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