A method for generating a three-dimensional spectral map

Abstract

The application relates to the technical field of electromagnetic spectrum map, and particularly discloses a three-dimensional spectrum map generation method, which comprises the following steps: dividing a three-dimensional geographic space of a to-be-detected electromagnetic environment into a grid, constructing a three-dimensional spectrum map tensor according to a plurality of spectrum data in the grid; the spectrum data is the average received signal strength of a time slot in the grid; slicing the spectrum map tensor to obtain a tensor slice, and calculating a local factor of the tensor slice to obtain a tensor local factor; applying a smoothing constraint to the tensor local factor to obtain a local smoothing tensor; modeling according to the local smoothing tensor to obtain a completion model; and solving the completion model by using an ADMM algorithm to obtain a completed three-dimensional electromagnetic spectrum map. The three-dimensional spectrum map generation method provided by the application can accurately and efficiently construct a three-dimensional electromagnetic spectrum map by using limited sampling data in a multi-radiation source scene.

Description

Technical Field

[0001] This invention relates to the field of electromagnetic spectrum mapping technology, and specifically to a method for generating three-dimensional spectrum maps. Background Technology

[0002] With the rapid development of wireless communication technology, the number of wireless frequency devices has exploded, making electromagnetic spectrum resources increasingly scarce. In this context, cognitive radio technology has emerged, with its core objective being the efficient management and rational allocation of spectrum resources. In radio systems, spectrum mapping is a crucial task, visualizing the invisible electromagnetic spectrum space and providing an intuitive basis for spectrum resource management and allocation. Through spectrum maps, important information such as the spectrum environment, received signal strength, spectrum busy / idle status, and spectrum access protocols can be clearly understood, thereby enabling functions such as opportunistic spectrum access, spectrum management, and interference coordination, playing a vital role in improving the performance of wireless communication systems.

[0003] In recent years, numerous spectral map reconstruction methods have emerged, mainly categorized into data-driven and model-driven approaches. Data-driven methods typically treat spectral map reconstruction as a matrix completion problem, employing interpolation algorithms. For example, inverse distance weighted interpolation calculates interpolation based on the distance between sampling points and the points to be interpolated, but this method does not consider any physical realities. In urban environments, severe shadow fading significantly impacts the performance of inverse distance weighted interpolation, making it difficult to accurately reflect the true spectral conditions. Other methods include kriging interpolation and optimal data-driven spectral mapping based on distributed kriging. Furthermore, by extending the low-rank properties of matrices to tensors, tensor completion methods combining prediction models have emerged. By imposing different constraints on the spectral map tensor, map reconstruction under various conditions can be achieved. However, most current tensor completion algorithms consider low rank and smoothness globally, neglecting local features and failing to fully utilize known data characteristics. Additionally, most current spectral map generation methods only consider a two-dimensional plane, neglecting the height dimension, making it difficult to achieve a complete representation of the electromagnetic environment of a specific region.

[0004] Model-driven methods primarily rely on radiation source information and electromagnetic propagation models to reconstruct spectrum maps. For example, location estimation-based algorithms estimate the location of major users by analyzing received signal strength values, thereby reconstructing the spectrum map. However, in practical applications, power and path loss models for major users are often difficult to obtain, limiting the applicability of this method. Other indirect model-driven reconstruction methods include schemes that combine minimum absolute shrinkage and the LASSO selection algorithm with path loss models for spectrum map reconstruction. However, these model-driven methods have poor adaptability in multi-radiation source scenarios and cannot accurately complete the spectrum map construction task. Summary of the Invention

[0005] To address the aforementioned problems, the purpose of this invention is to provide a method for generating a three-dimensional spectrum map, which can accurately and efficiently construct a three-dimensional electromagnetic spectrum map in multi-radiation source scenarios using limited sampling data.

[0006] This invention provides a method for generating a three-dimensional spectrum map, comprising:

[0007] The three-dimensional geographic space of the electromagnetic environment under test is divided into grids, and a three-dimensional spectrum map tensor is constructed based on multiple spectrum data in the grids; the spectrum data is the average received signal strength of a time slot in the grid.

[0008] The spectral map tensor is sliced ​​to obtain tensor slices, and the local factors of the tensor slices are calculated to obtain tensor local factors.

[0009] By applying smoothing constraints to the local factors of the tensor, a locally smoothed tensor is obtained.

[0010] Modeling is performed based on the local smoothing tensor, and the low-rank constraint of the tensor truncation nuclear norm is combined to obtain the complete model;

[0011] The ADMM algorithm is used to solve the completed model to obtain the completed three-dimensional electromagnetic spectrum map.

[0012] In one possible implementation, constructing a three-dimensional spectral map tensor based on multiple spectral data in the grid includes:

[0013] The spectrum map tensor is determined using the following formula. :

[0014] ;

[0015] in, For the true value of the spectrum map tensor, For tensor elements, For known grid indexes, The x-coordinate of the grid, The ordinate of the grid, The vertical coordinates of the grid, In order to be in Average received signal strength of the grid at the location, This is the set of all grids and the average received signal strength.

[0016] In one possible implementation, calculating the local factors of the tensor slice to obtain the tensor local factors includes:

[0017] The local factors of tensor slices are calculated using the following formula:

[0018] ;

[0019] ;

[0020] ;

[0021] ;

[0022] ;

[0023] in, For the first Tensor slices, For orthogonal tensors, For diagonal tensors, For singular value decomposition operations, For the first The pre-local factor of a tensor slice For the first Post-local factors of tensor slices For orthogonal tensors Extract the first R columns, To truncate the first R rows and first R columns of the diagonal tensor For orthogonal tensors Extract the first R columns, For the pre-local factor of the tensor, For the post-local factor of the tensor, Represents the inverse process of the Fourier transform. Indicates the tensor slice number, For the estimated tensor rank, This indicates the conjugate transpose.

[0024] In one possible implementation, applying smoothing constraints to the local factors of the tensor to obtain a locally smooth tensor includes:

[0025] The smoothing constraint for the local factor is determined using the following formula:

[0026] ;

[0027] ;

[0028] ;

[0029] ;

[0030] in, For the pre-local factor of the tensor, For the post-local factor of the tensor, Indicates the tensor slice number, Represents the three-modal total variational operator, For first-order matrix difference operators, For the spectrum map tensor.

[0031] In one possible implementation, the difference between consecutive rows of a matrix is ​​determined according to the following formula. and :

[0032] ;

[0033] ;

[0034] in, For first-order matrix difference operators, The first local factor of the tensor A slice, The th local factor of the tensor A slice, Represents the first of the matrix OK, This indicates the number of the tensor slice.

[0035] In one possible implementation, the modeling based on the local smooth tensor, combined with the low-rank constraint of the tensor truncation nuclear norm, to obtain the completed model includes:

[0036] The completed model is obtained using the following formula:

[0037] ;

[0038] in, , For tensor local factors, For the observations of the spectrum map tensor, Indicates the tensor slice number, For first-order matrix difference operators, The first local factor of the tensor A slice, The th local factor of the tensor A slice, and For weighting coefficients, representing the mode of a tensor To represent the auxiliary variables of the tensor expansion with magnitude -n, For the cutoff coefficient, For Frobenius norm, For operations to retrieve values ​​from known grid indices, For the true value of the spectrum map tensor, For known grid indices.

[0039] In one possible implementation, the step of using the ADMM algorithm to solve the completed model to obtain the completed three-dimensional electromagnetic spectrum map includes:

[0040] Update according to the following formula :

[0041] ;

[0042] ;

[0043] Update according to the following formula :

[0044] ;

[0045] ;

[0046] Update the auxiliary variable according to the following formula. :

[0047] ;

[0048] Update the auxiliary variable according to the following formula. :

[0049] ;

[0050] in, Indicates the tensor slice number. , Indicates the number of loops, The transpose of the first-order matrix difference operator Indicates transpose, and These are additional slack variables used for algorithm convergence. , , , The coefficient variables introduced for the ADMM algorithm constraint coefficients, and They are used to replace and , Indicates according to tensor module The reverse process of unfolding involves refolding the matrix back into tensor form. For Frobenius norm, .

[0051] In one possible implementation, the step of using the ADMM algorithm to solve the completed model to obtain the completed three-dimensional electromagnetic spectrum map includes:

[0052] Update the auxiliary variable according to the following formula. :

[0053] ;

[0054] Update the auxiliary variable according to the following formula. :

[0055] ;

[0056] Update the auxiliary variable according to the following formula. :

[0057] ;

[0058] Update the auxiliary variable according to the following formula. :

[0059] ;

[0060] Update the 3D electromagnetic spectrum map according to the following formula. :

[0061] ;

[0062] ;

[0063] Update according to the following formula :

[0064] ;

[0065] in, Indicates conjugate transpose, , , For constraint term coefficients, Auxiliary variables representing tensor expansion of magnitude -n For the cutoff coefficient, For the multipliers of the constraints, For the set of all grids and average received signal strength, For the wide-area singular value thresholding operator related to minimizing the truncated nuclear norm.

[0066] In one possible implementation, the step of using the ADMM algorithm to solve the completed model to obtain the completed three-dimensional electromagnetic spectrum map includes:

[0067] Update the multipliers of the constraints according to the following formula. :

[0068] ;

[0069] ;

[0070] ;

[0071] ;

[0072] ;

[0073] in, , , , , For the multipliers of the constraints, Indicates the tensor slice number, Indicates the number of loops, , For tensor local factors;

[0074] Update the constraint term coefficients according to the following formula. :

[0075] ;

[0076] ;

[0077] ;

[0078] in, The increment factor of the coefficient in each iteration. , , These are the coefficients of the constraint terms.

[0079] In one possible implementation, the step of using the ADMM algorithm to solve the completed model to obtain the completed three-dimensional electromagnetic spectrum map includes:

[0080] Determine whether the convergence condition is met or the preset number of iterations has been reached;

[0081] When the convergence condition is met or the preset number of iterations is reached, the completion model outputs a completed three-dimensional electromagnetic spectrum map.

[0082] If the convergence condition is not met and the preset number of iterations has not been reached, continue iterating and updating the parameters.

[0083] The three-dimensional spectrum map generation method provided by this invention solves the problem that existing spectrum map reconstruction technologies only focus on global low-rank properties and ignore the role of local smoothness in improving map reconstruction capabilities. It can accurately and efficiently construct three-dimensional electromagnetic spectrum maps in multi-radiation source scenarios using limited sampling data, thereby meeting the growing needs for spectrum monitoring and management and improving the performance and reliability of cognitive radio systems in complex electromagnetic environments. Attached Figure Description

[0084] Figure 1 A flowchart illustrating the three-dimensional spectrum map generation method provided in an embodiment of the present invention;

[0085] Figure 2 The spectrum map tensor model provided in the embodiments of the present invention. Detailed Implementation

[0086] The embodiments of the present invention will be further described in detail below with reference to the accompanying drawings and examples. The following detailed description of the embodiments and the accompanying drawings are used to illustrate the principles of the present invention by way of example, but should not be used to limit the scope of the present invention. That is, the present invention is not limited to the described preferred embodiments, and the scope of the present invention is defined by the claims.

[0087] In the description of this invention, it should be noted that, unless otherwise stated, "a plurality of" means two or more; the terms "first," "second," etc., are used for descriptive purposes only and should not be construed as indicating or implying relative importance; those skilled in the art can understand the specific meaning of the above terms in this invention as appropriate.

[0088] To facilitate understanding of this invention, the proprietary terms involved will first be explained.

[0089] The Alternating Multiplier Method (ADMM) is a classic algorithm for solving constrained minimization problems. Consider the following constrained minimization problem:

[0090] ;

[0091] in, and All are convex functions. The ADMM algorithm considers an augmented Lagrangian function, as follows:

[0092] ;

[0093] Where y is the Lagrangian multiplier vector. The ADMM iterative mechanism is:

[0094] ;

[0095] in, It is a positive penalty parameter.

[0096] Tensor completion algorithms, as an extension of matrices to higher orders, mainly include methods based on tensor decomposition and low-rank tensor completion based on rank minimization without decomposition structure.

[0097] The tensor decomposition-based method can be transformed into the following optimization problem:

[0098] ;

[0099] in, It is the observation tensor. It is to recover the tensor. The index set representing the observation entries. Represents the rank of the tensor. Given a low-rank tensor The rank limit. The differences between different tensor decomposition methods mainly lie in the definition of rank, using the rank obtained by different decomposition methods to replace the rank of tensor completion, such as CP-based decomposition, Tucker-based decomposition, tensor chain-based decomposition, tensor ring-based decomposition, etc.

[0100] In practice, tensor rank bounds may be unavailable in some applications. When observations are few, choosing a high-rank bound can lead to overfitting. To avoid this, another approach is to borrow the idea of ​​minimizing the nuclear norm from matrix completion models and directly minimize the tensor's rank. For the low-rank tensor completion problem, the general model for directly minimizing the tensor rank can be expressed as:

[0101] ;

[0102] Depending on the definition of tensor rank, there are various optimization models for the tensor completion problem. However, due to... Since it is a non-convex function, the optimization problem becomes an NP-hard problem. Different methods may employ tensor rank such as CP rank, Tucker rank, TT rank, TR rank, etc.

[0103] Suppose we have a third-order tensor Therefore, the tensor t-svd is decomposed as follows:

[0104] ;

[0105] in, and They are all orthogonal tensors. It is a diagonal tensor.

[0106] Figure 1A flowchart illustrating the three-dimensional spectrum map generation method provided in the embodiments of the present invention is shown below. Figure 1 As shown, the present invention provides a method for generating a three-dimensional spectrum map, comprising:

[0107] Step S1: Divide the three-dimensional geographic space of the electromagnetic environment to be tested into a grid, and construct a three-dimensional spectrum map tensor based on multiple spectrum data in the grid.

[0108] The spectral data represents the average received signal strength in one time slot within the grid; the three-dimensional geographic space is divided into a set of... A cube mesh, in which The spectral map tensor represents the x, y, and z axes in space, respectively. .

[0109] In one possible implementation, the spectral map tensor is determined according to the following formula. :

[0110] ;

[0111] in, For the true value of the spectrum map tensor, For tensor elements, For known grid indexes, The x-coordinate of the grid, The ordinate of the grid, The vertical coordinates of the grid, In order to be in Average received signal strength of the grid at the location, This is the set of all grids and the average received signal strength.

[0112] Step S2: Slice the spectrum map tensor to obtain tensor slices, and calculate the local factors of the tensor slices to obtain tensor local factors.

[0113] In one possible implementation, the local factors of the tensor slice are calculated, and the tensor local factors include:

[0114] The local factors of tensor slices are calculated using the following formula:

[0115] ;

[0116] ;

[0117] ;

[0118] ;

[0119] ;

[0120] in, For the first Tensor slices, For orthogonal tensors, For diagonal tensors, For singular value decomposition operations, For the first The pre-local factor of a tensor slice For the first Post-local factors of tensor slices For orthogonal tensors Extract the first R columns, To truncate the first R rows and first R columns of the diagonal tensor For orthogonal tensors Extract the first R columns, For the pre-local factor of the tensor, For the post-local factor of the tensor, Represents the inverse process of the Fourier transform. Indicates the tensor slice number, For the estimated tensor rank, This indicates the conjugate transpose.

[0121] Step S3: Apply smoothing constraints to the local factors of the tensor to obtain the locally smoothed tensor;

[0122] In one possible implementation, in order to impose smoothing constraints on the local factors of the tensor, Expanding along each of the three dimensions yields the matrix. and .

[0123] Apply the difference operator to the expanded matrix :

[0124] ;

[0125] Therefore, it can be done and The difference between consecutive rows of the expanded matrix is ​​obtained.

[0126] The difference between consecutive rows of a matrix is ​​determined using the following formula. and :

[0127] ;

[0128] ;

[0129] in, For first-order matrix difference operators, The first local factor of the tensor A slice, The th local factor of the tensor A slice, Represents the first of the matrix OK, This indicates the number of the tensor slice.

[0130] The smoothing constraint for the local factor is determined using the following formula:

[0131] ;

[0132] ;

[0133] ;

[0134] ;

[0135] in, For the pre-local factor of the tensor, For the post-local factor of the tensor, Indicates the tensor slice number, Represents the three-modal total variational operator, For first-order matrix difference operators, For the spectrum map tensor.

[0136] Step S4: Model the model based on the local smoothing tensor, and combine it with the low-rank constraint of the tensor truncation nuclear norm to obtain the complete model;

[0137] In one possible implementation, the general spectrum map generation problem can be modeled as a multidimensional tensor completion problem:

[0138] ;

[0139] in This represents the tensor truncation nuclear norm, which is approximately equal to the tensor rank.

[0140] The problem of generating spectral maps based on observations can be represented by the tensor local factor smoothness constraint:

[0141] ;

[0142] Equivalent to:

[0143] .

[0144] Since the truncated nuclear norm of any given tensor is equal to the weighted sum of the truncated nuclear norms on the tensor expansion matrix, therefore It can be written in the following form:

[0145] ;

[0146] in, express The expansion matrix modulo n.

[0147] The problem to be solved can be expressed as the following formula, and the complete model can be obtained based on the following formula:

[0148] ;

[0149] in, , For tensor local factors, For the observations of the spectrum map tensor, Indicates the tensor slice number, For first-order matrix difference operators, The first local factor of the tensor A slice, The th local factor of the tensor A slice, and For weighting coefficients, representing the mode of a tensor To represent the auxiliary variables of the tensor expansion with magnitude -n, For the cutoff coefficient, For Frobenius norm, For operations to retrieve values ​​from known grid indices, For the true value of the spectrum map tensor, For known grid indices.

[0150] Step S5: Use the ADMM algorithm to solve the complete model and obtain the complete three-dimensional electromagnetic spectrum map.

[0151] In one possible implementation, an auxiliary variable is introduced. And order The tensor completion problem can then be written as:

[0152] ;

[0153] The augmented Lagrangian function for the above problem can be written as:

[0154] ;

[0155] in, The multiplier is a constraint multiplier.

[0156] The solution of the unknown variables is decomposed into several sub-problems based on the obtained augmented Lagrangian function.

[0157] Update according to the following formula :

[0158] ;

[0159] ;

[0160] Update according to the following formula :

[0161] ;

[0162] ;

[0163] Update the auxiliary variable according to the following formula. :

[0164] ;

[0165] Update the auxiliary variable according to the following formula. :

[0166] ;

[0167] in, Indicates the tensor slice number. , Indicates the number of loops, The transpose of the first-order matrix difference operator Indicates transpose, and These are additional slack variables used for algorithm convergence. , , , The coefficient variables introduced for the ADMM algorithm constraint coefficients, and They are used to replace and , Indicates according to tensor module The reverse process of unfolding involves refolding the matrix back into tensor form. For Frobenius norm, .

[0168] In one possible implementation, the ADMM algorithm is used to solve the completed model, resulting in a completed three-dimensional electromagnetic spectrum map, including:

[0169] Update the auxiliary variable according to the following formula. :

[0170] ;

[0171] Update the auxiliary variable according to the following formula. :

[0172] ;

[0173] Update the auxiliary variable according to the following formula. :

[0174] ;

[0175] Update the auxiliary variable according to the following formula. :

[0176] ;

[0177] Update the 3D electromagnetic spectrum map according to the following formula. :

[0178] ;

[0179] ;

[0180] Update according to the following formula :

[0181] ;

[0182] in, Indicates conjugate transpose, , , For constraint term coefficients, Auxiliary variables representing tensor expansion of magnitude -n For the cutoff coefficient, For the multipliers of the constraints, For the set of all grids and average received signal strength, For the wide-area singular value thresholding operator related to minimizing the truncated nuclear norm.

[0183] In one possible implementation, the ADMM algorithm is used to solve the completed model, resulting in a completed three-dimensional electromagnetic spectrum map, including:

[0184] Update the multipliers of the constraints according to the following formula. :

[0185] ;

[0186] ;

[0187] ;

[0188] ;

[0189] ;

[0190] in, , , , , For the multipliers of the constraints, Indicates the tensor slice number, Indicates the number of loops, , For tensor local factors;

[0191] Update the constraint term coefficients according to the following formula. :

[0192] ;

[0193] ;

[0194] ;

[0195] in, The increment factor of the coefficient in each iteration. , , These are the coefficients of the constraint terms. .

[0196] In one possible implementation, the ADMM algorithm is used to solve the completed model, resulting in a completed three-dimensional electromagnetic spectrum map, including:

[0197] Determine whether the convergence condition is met or the preset number of iterations has been reached;

[0198] When the convergence condition is met or the preset number of iterations is reached, the completed model outputs a completed 3D electromagnetic spectrum map.

[0199] If the convergence condition is not met and the preset number of iterations has not been reached, continue iterating and updating the parameters.

[0200] Compared with existing spectrum map reconstruction technologies, the three-dimensional spectrum map generation method provided by this invention has the following significant advantages:

[0201] (1) Multidimensional data feature fusion improves reconstruction accuracy. By innovatively combining global low-rank and local smoothness constraints, multidimensional feature fusion is achieved in the tensor completion framework. This preserves the overall spatial correlation of the spectral data while enhancing the refined expression of electromagnetic features in local areas, effectively solving the problem of detail loss caused by traditional methods that only focus on global characteristics, and significantly improving the accuracy and physical realism of map reconstruction.

[0202] (2) Enhanced 3D electromagnetic environment modeling capabilities. The height dimension is incorporated into the spectrum map construction system, and a 3D tensor model is established to fully represent the spatial electromagnetic field distribution. Compared with traditional 2D planar mapping, it can more realistically reflect the propagation and attenuation characteristics of electromagnetic waves in urban three-dimensional space, providing accurate 3D visualization support for spectrum resource management in densely populated high-rise building areas.

[0203] (3) High robustness under finite sampling. The developed local feature enhancement mechanism can extract deep spatial correlations from sparse sampled data. Simulation results show that, under the same sampling rate, the reconstruction error of this scheme is significantly reduced compared to the traditional Kriging interpolation method.

[0204] The above description is merely a specific embodiment of the present invention, but the scope of protection of the present invention is not limited thereto. Any variations or substitutions that can be easily conceived by those skilled in the art within the technical scope disclosed in the present invention should be included within the scope of protection of the present invention. Therefore, the scope of protection of the present invention should be determined by the scope of the claims.

Claims

1. A method for generating a three-dimensional spectrum map, characterized in that, include: The three-dimensional geographic space of the electromagnetic environment to be measured is divided into grids, and a three-dimensional spectrum map tensor is constructed based on multiple spectrum data in the grids. The spectrum data is the average received signal strength of one time slot in the grid; The spectral map tensor is sliced ​​to obtain tensor slices, and the local factors of the tensor slices are calculated to obtain tensor local factors. By applying smoothing constraints to the local factors of the tensor, a locally smoothed tensor is obtained. Modeling is performed based on the local smoothing tensor, and the low-rank constraint of the tensor truncation kernel norm is combined to obtain the complete model; The ADMM algorithm is used to solve the completed model to obtain the completed three-dimensional electromagnetic spectrum map; The calculation of the local factors of the tensor slice, to obtain the tensor local factors, includes: The local factor of a tensor slice is calculated using the following formula: in, Let U and V be the i-th tensor slice, U and V be orthogonal tensors, S be the diagonal tensor, and SVD() be the singular value decomposition operation. For the i-th tensor slice, the pre-local factor, Let U(:,1:R) be the local factor of the i-th tensor slice, U(:,1:R) be the first R columns of the orthogonal tensor U, S(1:R,1:R) be the first R rows and first R columns of the diagonal tensor, and (V(:,1:R)) be the first R columns of the orthogonal tensor V. For the pre-local factor of the tensor, is the post-local factor of the tensor, ifft() denotes the inverse process of the Fourier transform, i represents the tensor slice number, R is the estimated tensor rank, and c represents the conjugate transpose.

2. The method for generating a three-dimensional spectrum map according to claim 1, characterized in that, The construction of a three-dimensional spectrum map tensor based on multiple spectrum data in the grid includes: The spectrum map tensor is determined using the following formula. in, For the true value of the spectrum map tensor, d x',y',z' Let Ω be a tensor element, Ω be a known grid index, x' be the x-coordinate of the grid, y' be the y-coordinate of the grid, and z' be the z-coordinate of the grid. The average received signal strength of the grid at position (x',y',z') This is the set of average received signal strengths across all grids.

3. The method for generating a three-dimensional spectrum map according to claim 1, characterized in that, Applying smoothing constraints to the local factors of the tensor to obtain a locally smoothed tensor includes: The smoothing constraint for the local factor is determined using the following formula: in, For the pre-local factor of the tensor, For the post-local factors of the tensor, || || 3MTV Represents the three-modal total variational operator, L i For first-order matrix difference operators, For the spectrum map tensor, α i For weighting coefficients, A i (i) is the matrix formed by expanding the tensor along the i-th dimension from its pre-local factors, B i (i) is the matrix formed by expanding the post-local factors of the tensor along the i-th dimension.

4. The method for generating a three-dimensional spectrum map according to claim 3, characterized in that, The difference L between consecutive rows of a matrix is ​​determined using the following formula. i A i (i) and L i B i (i): Among them, L i For the first-order matrix difference operator, A i (i) represents the i-th slice of the pre-local factor of the tensor, A i (i) is the matrix formed by expanding the tensor along the i-th dimension from its pre-local factors, B i (i) is the matrix formed by expanding the tensor along the i-th dimension using the post-local factors, and (n) i ,:) represents the nth digit of the matrix. i OK.

5. The method for generating a three-dimensional spectrum map according to claim 1, characterized in that, The modeling based on the local smooth tensor, combined with the low-rank constraint of the tensor truncation nuclear norm, yields the completed model, including: The completed model is obtained using the following formula: in, For the pre-local factor of the tensor, For the post-local factor of the tensor, Here, represents the observations of the spectral map tensor, i represents the tensor slice number, and L represents the tensor slice number. i For the first-order matrix difference operator, A i (i) is the matrix formed by expanding the tensor along the i-th dimension from its pre-local factors, B i (i) is the matrix formed by expanding the tensor along the i-th dimension using the post-local factors, α i and β p Here, p represents the weighting coefficients, and p represents the mode of the tensor. Let r be the matrix formed by expanding the tensor along the i-th dimension, r be the truncation coefficient, F be the Frobenius norm, and P be the matrix formed by expanding the tensor along the i-th dimension. Ω For operations to retrieve values ​​from known grid indices, Ω represents the truth value of the spectrum map tensor, and Ω represents the known grid index.

6. The method for generating a three-dimensional spectrum map according to claim 1, characterized in that, The process of solving the completed model using the ADMM algorithm to obtain the completed three-dimensional electromagnetic spectrum map includes: Update according to the following formula Update according to the following formula Update the auxiliary variable P according to the following formula. i : Update the auxiliary variable Q according to the following formula. i : Where i represents the tensor slice number, i = 1, 2, 3, k represents the loop count, and L... i T T denotes the transpose of the first-order matrix difference operator, and I denotes the transpose. i (i) k and J i (i) k Additional slack variables are used for algorithm convergence and M. i (i) k N i (i) k W k Z i k The coefficient variable introduced for the ADMM algorithm, μ k For constraint term coefficients, and They were used to replace A respectively i (i) and B i (i) fold i () denotes the inverse process of expanding the tensor modulus i, which refolds the matrix back into tensor form, where F is the Frobenius norm, and prox l ,(v,β)=max(0,v-β)+min(0,v+β), The auxiliary variables formed by expanding the pre-local factors of the tensor along the i-th dimension. The auxiliary variable is formed by expanding the former local factors of the tensor along the i-th dimension.

7. The method for generating a three-dimensional spectrum map according to claim 6, characterized in that, The process of solving the completed model using the ADMM algorithm to obtain the completed three-dimensional electromagnetic spectrum map includes: Update the tensor's prelocal factor according to the following formula. Update the postlocal factor of the tensor according to the following formula. Update the auxiliary variable according to the following formula. Update the auxiliary variable according to the following formula. Update the 3D electromagnetic spectrum map according to the following formula. Update according to the following formula Where c represents the conjugate transpose, β, γ, and σ are the coefficients of the constraint terms. Let r represent the matrix formed by expanding the tensor along the i-th dimension, where r is the truncation coefficient. For the multipliers of the constraints, D is the set of average received signal strengths across all grids. r For wide-area singular value thresholding operators related to minimizing the truncated nuclear norm, For the multipliers of the constraints, For the wide-area singular value thresholding operator related to minimizing the truncated nuclear norm, β p / ρ represents the truncation parameters.

8. The method for generating a three-dimensional spectrum map according to claim 7, characterized in that, The process of solving the completed model using the ADMM algorithm to obtain the completed three-dimensional electromagnetic spectrum map includes: Update the multipliers of the constraints according to the following formula. in, W, Z, Here, is the multiplier for the constraint condition, i represents the tensor slice number, and k represents the loop count. For the pre-local factor of the tensor, The post-local factor of the tensor; Update the constraint term coefficients {μ,β,σ} according to the following formula: μ=min(μ max (day) β=min(β max (hb) σ=min(σ max (it) Where η is the increment factor of the coefficient in each iteration, and β, σ, and μ are the constraint coefficients.

9. The method for generating a three-dimensional spectrum map according to claim 1, characterized in that, The process of solving the completed model using the ADMM algorithm to obtain the completed three-dimensional electromagnetic spectrum map includes: Determine whether the convergence condition is met or the preset number of iterations has been reached; When the convergence condition is met or the preset number of iterations is reached, the completion model outputs a completed three-dimensional electromagnetic spectrum map. If the convergence condition is not met and the preset number of iterations has not been reached, continue iterating and updating the parameters.

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