A method and system for spectral data completion based on third-dimensional tensor collaborative sparsity and low-rank transformation learning

Abstract

This invention discloses a spectral data completion method and system based on third-dimensional tensor collaborative sparsity and low-rank transform learning. Addressing the complex dynamic changes of the spectral map in the third dimension, this method constructs a tensor completion optimization model including low-rank and sparse branches. It utilizes a learnable low-rank transform matrix to constrain the low-rank property of spatial slices, while simultaneously using a learnable sparse transform matrix to characterize the sparsity of the third-dimensional fibers, forming complementary priors. Through proximal iterative alternation, the low-rank branch, sparse branch, and corresponding transform matrix are collaboratively optimized and weightedly fused, and combined with observation consistency constraints, high-fidelity data reconstruction is achieved. This invention can effectively recover the dynamic details of the spectral map under highly sparse observations, significantly improving completion accuracy and providing reliable data support for spectrum sharing and dynamic access.

Description

Technical Field

[0001] This invention relates to the fields of wireless communication and electromagnetic environment sensing technology, specifically to a spectrum data completion method and system based on third-dimensional tensor cooperative sparsity and low-rank transform learning. Background Technology

[0002] Against the backdrop of accelerated digitalization and intelligentization, the continuous expansion of mobile internet services has driven a significant increase in spectrum occupancy and frequency-based data volume, making the demand for electromagnetic spectrum resources by mobile communication systems increasingly urgent. However, the contradiction between the rigid supply constraints of radio spectrum, a naturally finite and non-renewable strategic resource, and the rapid iteration of communication technologies and the emergence of new application scenarios is becoming increasingly acute. Although research on high-frequency technologies such as visible light communication and terahertz communication continues to advance, their inherent physical characteristics, such as short propagation distance and weak penetration and diffraction capabilities, make it difficult to fully replace mid- and low-frequency bands in terms of wide-area coverage and robust access. Therefore, improving the spectrum utilization efficiency of existing licensed frequency bands has become a key path to alleviate resource scarcity and support sustainable development.

[0003] The current spectrum management model, centered on static fixed licensing, generally suffers from uneven resource allocation, lagging spectrum identification, and low overall utilization efficiency. Constructing high-precision, multi-dimensional dynamic spectrum maps and integrating mechanisms such as opportunistic access and dynamic spectrum sharing can achieve refined perception and efficient reuse of spectrum gaps. Spectrum maps, by integrating multi-dimensional heterogeneous information such as time, frequency, spatial location, and field strength, and deeply coupling with geographic information systems, can more accurately depict the non-uniform distribution and dynamic evolution of electromagnetic signals in three-dimensional geographic space. Given its inherent spatiotemporal coupling and high-order structural characteristics, tensor modeling provides a natural and powerful mathematical framework for the unified representation and processing of such multi-dimensional spectrum data. However, constrained by monitoring site density, sampling duration, sensor deployment costs, and communication bandwidth, the actual collected spectrum observation data often exhibits highly sparse and incomplete characteristics. Tensor completion techniques are urgently needed to robustly reconstruct missing information, thereby ensuring the reliability and accuracy of downstream tasks such as spectrum sharing decisions, interference avoidance, and visualization analysis.

[0004] In recent years, spectrum map reconstruction methods based on low-rank tensor modeling have attracted widespread attention. For example, methods based on the tensor nuclear norm (TNN) and tensor ring low-rank factorization (TRLRF) have improved the reconstruction accuracy of spectrum maps to some extent by characterizing the global correlation of multidimensional spectrum data. However, these methods mainly rely on global low-rank priors, making it difficult to fully characterize local abrupt changes, anomalous frequencies, and non-stationary changes in the third time dimension. Furthermore, their low-rank constraints are usually applied uniformly across all dimensions, lacking the ability to finely model the dynamic evolution characteristics of the time dimension. In scenarios with highly sparse observational data or complex time-dimensional structures, tensor completion methods relying solely on low-rank constraints can easily lead to over-smoothing of time-dimensional details, thus limiting their effectiveness in high-precision spectrum sensing and dynamic spectrum map construction tasks. Summary of the Invention

[0005] The purpose of this invention is to provide a spectrum data completion method and system based on third-dimensional tensor collaborative sparsity and low-rank transform learning, so as to solve the problems of existing tensor completion methods, which rely on the overall low-rank prior and are difficult to characterize local mutations and dynamic non-stationary changes in the third dimension, resulting in excessive smoothing of details.

[0006] To achieve the above objectives, the technical solution provided by this invention is: a spectral data completion method based on third-dimensional tensor cooperative sparsity and low-rank transform learning, comprising the following steps: S1: Obtain the 3D spectral map tensor and its observation index set for incomplete observations, and determine the size of the tensor to be completed; wherein, the first and second dimensions of the 3D spectral map tensor correspond to spatial location, and the third dimension corresponds to time series; construct a tensor completion optimization model based on third-dimensional collaborative sparsity and low-rank constraints, the tensor completion optimization model including low-rank branch and sparse branch, by introducing the kernel norm and... The norm regularization term constrains the low-rank property of the space slice and the sparsity of the third-dimensional fiber. Combined with the observation consistency condition, an objective function to be optimized is established. S2: Initialize and fill the data at unobserved locations in the 3D spectrum map tensor to obtain the initialized tensor to be completed; initialize the low-rank branch variables, sparse branch variables, and the low-rank transformation matrix and sparse transformation matrix of the third dimension; S3: Starting with the initial values ​​of each variable in step S2, iteratively optimize each variable using a proximal iteration method with alternating updates of multiple variables according to the objective function established in step S1. In each iteration, update the low-rank branch variable, the sparse branch variable and their corresponding transformation matrix respectively; perform weighted fusion on the updated low-rank branch and sparse branch, update the tensor to be completed, and correct the tensor to be completed according to the observation consistency constraint. Use the corrected tensor to be completed as the input for the next iteration optimization, and calculate the peak signal-to-noise ratio to evaluate the completion effect. S4: Repeat the iterative optimization until the convergence threshold or the maximum number of iterations is reached, and output the final completed spectrum map tensor.

[0007] To optimize the above technical solution, the specific measures also include: In step S1, the objective function of the tensor completion optimization model jointly optimizes the low-rank branch, the sparse branch, and their corresponding transformation matrices, introducing a proximal penalty term to constrain the update magnitude of each variable in adjacent iterations. The expression is:

[0008] in, For block diagonal operators; and These represent the reconstruction weights of the low-rank branch variable and the sparse branch variable, respectively. Indicates low-rank branch variables; Represents sparse branch variables; The tensor to be completed represents incomplete observations; Represents a three-dimensional spectrum map tensor; Indicates the observation index set A projection operator that takes a value at one position and sets it to zero at the other positions; symbol Represents the third modulo product; For nuclear norm; for Norm; Denotes the Frobenius norm; These are the sparse regularization coefficients; , They represent , The List; Represents a low-rank transformation matrix; Represents a sparse transformation matrix; This is the penalty coefficient; The inertial smoothing coefficient; Indicates the first The low-rank branch variables after the round of updates; Indicates the first The sparse branch variables after each round of updates; Indicates the first The low-rank transformation matrix after the round of updates; Indicates the first The sparse transformation matrix after the round of updates; Indicates the first The tensor to be completed after the round of updates.

[0009] In step S2, the initial filling of data at unobserved locations in the three-dimensional spectrum map tensor is specifically performed as follows: the boundary of the three-dimensional spectrum map tensor is extended along the spatial dimension, spatial interpolation is performed on the unobserved locations in the three-dimensional spectrum map tensor on each third-dimensional slice, the interpolation results are weighted and fused with the observed values, and the observed values ​​are forcibly backfilled at the observed locations in the three-dimensional spectrum map tensor to obtain the initial tensor to be filled.

[0010] In step S3, the low-rank branch variable and the sparse branch variable are updated, specifically as follows: The expression for updating the low-rank branch variable is:

[0011] The expression for updating the sparse branch variable is:

[0012] in, Indicates the first The low-rank transformation matrix slice variables of the wheel; Indicates the first Slice variables of the sparse transformation matrix of the wheel; For the proximal operator corresponding to the nuclear norm regularization term; Low-rank transformation matrix The Column vector; Low-rank transformation matrix The Column vector; for The proximal operator corresponding to the regularization term; These represent the current tensor to be completed along the first rank, with the remaining low-rank components or sparse components fixed. The third-dimensional residual matrix of the low-rank transformation direction and the sparse transformation direction; Indicates the first The low-rank transformation matrix slice variables of the wheel; Indicates the first Slice variables of the sparse transformation matrix of the wheel; These are the sparse regularization coefficients; The vector reconstruction operator is used to rearrange data in vector form into matrix form according to a preset dimension; it is a vectorization operator. The inverse operation.

[0013] In step S3, the transformation matrices corresponding to the low-rank branch variables and sparse branch variables are updated, specifically as follows: Based on the current low-rank branch variables and sparse branch variables, the columns of the matrix are optimized by minimizing the reconstruction error; after each update, the resulting matrix column vectors are scaled to eliminate the scale uncertainty between the transformation matrix and the representation coefficients.

[0014] The expression for updating the low-rank transformation matrix is:

[0015] The expression for updating the sparse transformation matrix is:

[0016] in, Indicates the column normalization operator; For the first Low-rank transformation matrix of the round The Column vector; For the first Round sparse transformation matrix The Column vector.

[0017] In step S3, the updated low-rank branch and sparse branch are merged using a weighted summation method, and an inertial smoothing coefficient is introduced to enhance iteration stability, specifically as follows:

[0018] in, Indicates the first The low-rank transformation matrix obtained during the iteration; Indicates the first The sparse transformation matrix is ​​obtained through iterative updates.

[0019] In step S3, the observation consistency constraint is implemented in each iteration by forcibly replacing the value of the tensor to be completed at the observation location with the original observation value. Specifically:

[0020] in, Represents the set of unobserved locations; Represents the set of observation indices; Indicates the first The final tensor to be completed in the round is... and Together constitute; Indicates the tensor to be completed In the set of unobserved locations The element value on; Represents a three-dimensional spectrum map tensor In the observation index set The observed value at that location.

[0021] The corrected tensor to be completed is used as the input for the next iteration of optimization, and the peak signal-to-noise ratio is calculated to evaluate the completion effect; wherein, a preheating process is performed before the formal iteration; the preheating process includes a coefficient preheating stage and a dictionary preheating stage.

[0022] As another important technical solution, this invention also provides a spectrum data completion system based on third-dimensional tensor cooperative sparsity and low-rank transform learning, comprising: The model building module is used to obtain the 3D spectral map tensor and its observation index set of incomplete observations, and determine the size of the tensor to be completed. The first and second dimensions of the 3D spectral map tensor correspond to spatial location, and the third dimension corresponds to time series. A tensor completion optimization model based on third-dimensional cooperative sparsity and low-rank constraints is constructed. The tensor completion optimization model includes low-rank branches and sparse branches. By introducing regularization terms to constrain the low-rank property of spatial slices and the sparsity of third-dimensional fibers, combined with the observation consistency condition, the objective function to be optimized is established. The initialization module is used to initialize and fill the data at unobserved locations in the 3D spectrum map tensor to obtain the initialized tensor to be completed; it initializes the low-rank branch variables, sparse branch variables, and the low-rank transformation matrix and sparse transformation matrix of the third dimension. The iterative optimization module starts with the initial values ​​of each variable in the initialization module and iteratively optimizes each variable using a proximal iterative approach with alternating updates of multiple variables based on the objective function established in the model building module. In each iteration, it updates the low-rank branch variables, sparse branch variables, and their corresponding transformation matrices. The updated low-rank and sparse branches are then weighted and fused to update the tensor to be completed. The tensor to be completed is then corrected according to the observation consistency constraint. The corrected tensor to be completed is used as the input for the next iterative optimization, and the peak signal-to-noise ratio is calculated to evaluate the completion effect. The output module is used to iterate and optimize until the convergence threshold or the maximum number of iterations is reached, and outputs the final completed spectrum map tensor.

[0023] The present invention also proposes an electronic device, comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements a spectral data completion method based on third-dimensional tensor cooperative sparsity and low-rank transform learning as described above.

[0024] The present invention also proposes a computer-readable storage medium storing a computer program that enables a computer to execute a spectrum data completion method based on third-dimensional tensor cooperative sparsity and low-rank transform learning as described above.

[0025] Compared with the prior art, the beneficial effects of the present invention are: This invention characterizes the global low-rank structure of spatial slices through a learnable low-rank transformation matrix, while capturing local mutations, anomalous frequencies, and non-stationary changes on the third-dimensional fiber through a learnable sparse transformation matrix. This collaborative mechanism enables the completion results to ensure the stability of the overall structure and restore dynamic details with high fidelity under highly sparse observations.

[0026] This invention employs a data-driven transform learning strategy. The low-rank and sparse transform matrices are adaptively updated during optimization based on data distribution characteristics, thereby enabling more accurate matching of the diverse signal distribution features under different electromagnetic environments. Compared to fixed transform methods, this invention's adaptive learning mechanism significantly improves the model's generalization ability and reconstruction accuracy in complex scenarios.

[0027] This invention effectively suppresses scale drift and numerical oscillation problems in the transformation matrix update process by introducing a proximal penalty term into the objective function to constrain the update magnitude of variables in adjacent iterations and introducing an inertial smoothing coefficient for weighted fusion. At the same time, through a two-stage strategy of coefficient preheating and dictionary preheating, stable initial values ​​are provided for the variables, further improving the convergence reliability and numerical stability of the algorithm. Attached Figure Description

[0028] Figure 1 : Schematic diagram of the overall process of the method of the present invention.

[0029] Figure 2 : A schematic diagram of the peak signal-to-noise ratio comparison curves under different sampling rates in this embodiment of the invention.

[0030] Figure 3 : A schematic diagram of the completed spectrum map slice in an embodiment of the present invention. Detailed Implementation

[0031] The present invention will be further described in detail below through specific embodiments, but it should not be construed as limiting the scope of the subject matter of the present invention to the following embodiments. All technologies implemented based on the above content of the present invention fall within the scope of the present invention.

[0032] like Figure 1 As shown, this invention provides a spectral data completion method based on third-dimensional tensor cooperative sparsity and low-rank transform learning, comprising the following steps: S1: Obtain the 3D spectral map tensor and its observation index set for incomplete observations, and determine the size of the tensor to be completed; wherein, the first and second dimensions of the 3D spectral map tensor correspond to spatial location, and the third dimension corresponds to time series; construct a tensor completion optimization model based on third-dimensional collaborative sparsity and low-rank constraints, the tensor completion optimization model including low-rank branch and sparse branch, by introducing the kernel norm and... The norm regularization term constrains the low-rank property of the space slice and the sparsity of the third-dimensional fiber. Combined with the observation consistency condition, an objective function to be optimized is established. To establish observation consistency conditions, the observed data and index set are used to constrain the recovery results to not shift at the observed positions, thus ensuring that the tensor to be completed is consistent with the original sampling information:

[0033] in, Indicates the observation index set The projection operator, which sets the value at one location and zero at the others, is used to ensure that the tensor to be completed is consistent with the observed value at the observed location.

[0034] To fully utilize the third-dimensional structural information, low-rank branches and sparse branches are established simultaneously only in the third module direction:

[0035]

[0036] Among them, low-rank branch variables sparse branch variables and the third-dimensional low-rank transformation matrix With sparse transformation matrix ;symbol This represents the third-order product, which involves expanding the three-dimensional tensor into a matrix form along the third dimension, performing matrix multiplication with the third-dimensional transformation matrix, and then reconstructing it back into a three-dimensional tensor. In the sense of the third-dimensional transformation, it is expressed and reconstructed by a set of low-rank and sparse representations of spatial slices.

[0037] In the third model, we coordinate constraints on low rank and sparsity to construct the objective function of the tensor completion optimization model. Under the condition of observation consistency, we jointly optimize the low-rank branch, the sparse branch, and their corresponding transformation matrices:

[0038] Introducing a proximal penalty term to constrain the update magnitude of each variable in adjacent iterations, the expression is:

[0039] in, For block diagonal operators; and These represent the reconstruction weights of the low-rank branch variable and the sparse branch variable, respectively. The tensor to be completed represents incomplete observations; Represents a three-dimensional spectrum map tensor; For nuclear norm; for Norms, to facilitate sparsity constraints; This represents the Frobenius norm, used to measure the difference between a variable and the result of its previous iteration; These are the sparse regularization coefficients; , They represent , The Column normalization is used to eliminate scale uncertainty and improve numerical stability. This is the penalty coefficient; The inertial smoothing coefficient; Indicates the first The low-rank branch variables after the round of updates; Indicates the first The sparse branch variables after each round of updates; Indicates the first The low-rank transformation matrix after the round of updates; Indicates the first The sparse transformation matrix after the round of updates; Indicates the first The tensor to be completed after the round of updates.

[0040] S2: Initialize and fill the data at unobserved locations in the 3D spectrum map tensor to obtain the initialized tensor to be completed; initialize the low-rank branch variables, sparse branch variables, and the low-rank transformation matrix and sparse transformation matrix of the third dimension; Boundary expansion processing is performed on the 3D spectral map tensor along the spatial dimensions to mitigate the instability of edge regions under random missing conditions. Subsequently, spatial interpolation is performed on unobserved locations on each third-dimensional slice to obtain initial interpolation results. The interpolation results are then compared with the observation consistency results using fixed weighting coefficients. and Weighted fusion is performed, and observations are forcibly backfilled at the observation locations.

[0041] After obtaining the initial tensor to be completed Then, it is expanded along the third dimension to initialize the representation tensors of the low-rank and sparse branches. and During the initialization phase, the low-rank representation tensor and the sparse representation tensor are directly set to zero tensors, allowing the model to enter the subsequent proximal alternation iteration process without prior bias, thereby avoiding the interference of unstable initial values ​​on the convergence process under random sampling conditions.

[0042] Based on the observation completeness of each fiber in the third dimension, the third-dimensional index is statistically analyzed and sorted. The third-dimensional fibers with the most observations are selected as the candidate set, and a low-rank transformation matrix is ​​constructed accordingly. With sparse transformation matrix The initial column vectors are then normalized to satisfy the unit norm constraint.

[0043] S3: Starting with the initial values ​​of each variable in step S2, iteratively optimize each variable using a proximal iteration method with alternating updates of multiple variables according to the objective function established in step S1. In each iteration, update the low-rank branch variable, the sparse branch variable and their corresponding transformation matrix respectively; perform weighted fusion on the updated low-rank branch and sparse branch, update the tensor to be completed, and correct the tensor to be completed according to the observation consistency constraint. Use the corrected tensor to be completed as the input for the next iteration optimization, and calculate the peak signal-to-noise ratio to evaluate the completion effect. A warm-up process is performed before the formal iteration; the warm-up process includes a coefficient warm-up stage and a dictionary warm-up stage; the coefficient warm-up stage is as follows: Fixed Tensor to be Completed and in fixed , Under the condition of low-rank branch variables With sparse branch variables Perform proximal updates to obtain a stable initial representation in the third-dimensional transform domain. The dictionary warm-up phase is as follows: [Follow-up steps are needed for a complete translation.] During the round, tensors to be completed are allowed. Participate in the fusion update, while updating the third-dimensional transformation matrix at a lower frequency or under stable conditions. , And continuously impose observation consistency constraints.

[0044] After the preheating process is complete, proceed to the main iteration step:

[0045] in, This indicates that the current tensor to be completed will be... The matrix obtained by expanding along the third dimension; They represent from , Remove the first The matrix obtained after column division These represent removing the first... The rest of the line , Represents the set of coefficient vectors. These represent the current tensor to be completed along the first rank, with the remaining low-rank components or sparse components fixed. The third-dimensional residual matrix of the low-rank transformation direction and the sparse transformation direction.

[0046] Based on the Residual matrix of the wheel By constructing input terms for proximal updates through projection and rearrangement, the low-rank branch variables are updated:

[0047] in, Indicates the first The low-rank transformation matrix slice variables of the wheel; For the proximal operator corresponding to the nuclear norm regularization term, This represents the singular value soft thresholding parameter. It reduces the effective rank of the matrix by performing soft thresholding on the singular values ​​of the input matrix one by one, suppressing small singular value components while keeping the singular vector unchanged. Low-rank transformation matrix The Column vectors. The SVT operation along the third dimension... Singular value decomposition is performed on each of the two-dimensional slices, and soft thresholding is applied to their singular values.

[0048] Update sparse branch variables:

[0049] in, Indicates the first Slice variables of the sparse transformation matrix of the wheel; for The proximal operator corresponding to the regularization term performs element-wise soft thresholding on the amplitude of each element of the input variable, continuously suppressing small amplitude components and retaining significant components, promoting the sparsity of the representation result, and guiding the sparse branch to highlight local mutations, abnormal frequency points or non-stationary change features in the third-dimensional structure. Low-rank transformation matrix The Column vector.

[0050] Update the low-rank transformation matrix and normalize its column vectors to ensure scaling consistency and numerical stability:

[0051] in, This represents the column normalization operator, which normalizes a vector to a unit vector according to the L2 norm. The low-rank representation indicates the tensor's rank in the third dimension. A two-dimensional slice The low-rank coefficient vector obtained by vectorization.

[0052] Update the sparse transformation matrix and normalize its column vectors:

[0053] in, The sparsity representation of the tensor in the third dimension is the first... A two-dimensional slice The sparse coefficient vector obtained after vectorization. , The column normalization process can eliminate scale uncertainty and suppress numerical oscillations.

[0054] The updated low-rank and sparse branches are merged, and a weighted summation method is used, with an inertial smoothing coefficient introduced to enhance iteration stability:

[0055] in, , Control the contribution of low-rank branches and sparse branches to the update. The inertial smoothing coefficient. The corrected tensor to be completed obtained from the fusion update. Simultaneously considering the prior knowledge of the third-dimensional structure of both low-rank and sparse branches, and through... Inertial smoothing is introduced to enhance iterative stability.

[0056] The observation consistency constraint is implemented in each iteration of optimization by forcibly replacing the value of the recovered tensor at the observation location with the original observation value, specifically as follows:

[0057] in, Represents the set of unobserved locations; Represents the set of observation indices; Indicates the first The final tensor to be completed in the round is... and Together constitute; Indicates the tensor to be completed In the set of unobserved locations The element value on; Represents a three-dimensional spectrum map tensor In the observation index set The observed value at that location.

[0058] Based on the convergence threshold With maximum number of iterations Stop the process. Replace the observed values ​​directly at the observed locations, and use the fusion results to complete the data at the unobserved locations. Calculate the peak signal-to-noise ratio to evaluate the completion effect.

[0059] S4: Repeat the iterative optimization until the convergence threshold or the maximum number of iterations is reached, and output the final completed spectrum map tensor.

[0060] Example 1 This example uses simulation data. Eight dynamic radiation sources at 2.45 GHz were set up within a 1.25 km × 1.25 km × 2 m area. Spectral data points of 250 × 250 × 300 were collected at a height of 2 m within this 1.25 km × 1.25 km area. x , y Represents the spatial grid dimension, z This indicates that the received signal strength data in a dynamic scene is a time-dimension raster, with numerical values ​​represented in... dBm Electromagnetic spectrum tensor completion is performed on the observation of three-dimensional random sampling, using units as the unit.

[0061] Table 1 Simulation Parameters

[0062] The spectral completion method based on third-dimensional tensor cooperative sparse-low-rank transform learning proposed in this embodiment includes the following steps: Step S1: Obtain the 3D Spectral Map Tensor and observation index set Determine the size of the tensor to be recovered. In this example The first and second dimensions correspond to space, and the third dimension corresponds to time. A tensor completion optimization method based on third-dimensional collaborative sparsity and low-rank constraints is constructed: the relationship between low-rank and sparse branches is established and complementary expressions are achieved collaboratively; the nuclear norm is introduced into the objective function. Norm regularization terms are introduced to promote low-rank spatial slices and sparsity of third-dimensional fibers, respectively, by introducing penalty parameters and inertial smoothing coefficients. A divisible near-terminal problem is formed and solved by combining the observation consistency condition.

[0063] The objective function jointly optimizes the low-rank branch, the sparse branch, and their corresponding transformation matrices, introducing a proximal penalty term to constrain the update magnitude of each variable in adjacent iterations. The expression is as follows:

[0064] In this example, take and , For sparse regularization coefficients, in this example, we take... ; Step S2: Initialize and fill the data at unobserved locations in the 3D spectrum map tensor to obtain the initialized recovery tensor; initialize the low-rank branch variables, sparse branch variables, and the low-rank transformation matrix and sparse transformation matrix of the third dimension.

[0065] Step S3: Starting with the initial values ​​of each variable in Step S2, iteratively optimize each variable using a proximal iteration method with alternating updates of multiple variables according to the objective function established in Step S1. In each iteration, update the low-rank branch variable, the sparse branch variable, and their corresponding transformation matrix respectively. Perform weighted fusion on the updated low-rank branch and sparse branch, update the recovery tensor, and correct the recovery tensor according to the observation consistency constraint. Use the corrected recovery tensor as the input for the next iteration optimization, and calculate the peak signal-to-noise ratio to evaluate the completion effect. In this example, the number of preheating stages is set to be... The number of rounds in the dictionary preheating phase is .forward Fixed Tensor to be Completed and in fixed , Under the condition of low-rank branch variables With sparse branch variables Perform a proximal update to obtain a stable initial representation in the third-dimensional transform domain. Then... During the round, tensors to be completed are allowed. Participate in the fusion update, while updating the third-dimensional transformation matrix at a lower frequency or under stable conditions. , The observation consistency constraint is continuously applied. After the warm-up is complete, the main iteration step begins.

[0066] After each main iteration, to quantitatively evaluate the closeness between the completed result and the true tensor, this example uses peak signal-to-noise ratio. As an evaluation metric for the completion effect, the mean square error is first calculated. :

[0067] Peak signal-to-noise ratio is defined as:

[0068] in, This represents the peak value of the three-dimensional spectrum map tensor. The larger the value, the closer the completion result is to the true tensor, and the higher the completion accuracy.

[0069] Step S4: Repeat the iterative optimization until the convergence threshold or the maximum number of iterations is reached, and output the final completed spectrum map tensor.

[0070] In this example, a convergence threshold is set. Maximum number of iterations At observed locations, the observed values ​​are directly used for replacement; at unobserved locations, the fused results are used for completion.

[0071] The result obtained in this example is as follows: Figure 2 and Figure 3 As shown, the specific data obtained in the simulation experiment will be further explained.

[0072] like Figure 2 As shown, through quantitative and visual analysis of the completion results of different reconstruction methods at different sampling rates, it can be observed that: as the sampling rate increases, the completion accuracy steadily improves; and compared with traditional completion methods, this method has the highest completion accuracy at different sampling rates. Figure 3 As shown in the figure, the two-dimensional slice visualizations of the electromagnetic spectrum map completion results of different reconstruction methods are displayed at a sampling rate of 30%. Our method can still recover the main continuous power distribution pattern and significantly improves local details and boundary transitions when the sampling rate is further increased, verifying the effectiveness of the third-dimensional cooperative sparse-low-rank prior in the electromagnetic spectrum map completion task.

[0073] In another embodiment of the present invention, a spectral data completion system based on third-dimensional tensor cooperative sparsity and low-rank transform learning is proposed, comprising: The model building module is used to obtain the 3D spectral map tensor and its observation index set of incomplete observations, and determine the size of the tensor to be completed. The first and second dimensions of the 3D spectral map tensor correspond to spatial location, and the third dimension corresponds to time series. A tensor completion optimization model based on third-dimensional cooperative sparsity and low-rank constraints is constructed. The tensor completion optimization model includes low-rank branches and sparse branches. By introducing regularization terms to constrain the low-rank property of spatial slices and the sparsity of third-dimensional fibers, combined with the observation consistency condition, the objective function to be optimized is established. The initialization module is used to initialize and fill the data at unobserved locations in the 3D spectrum map tensor to obtain the initialized tensor to be completed; it initializes the low-rank branch variables, sparse branch variables, and the low-rank transformation matrix and sparse transformation matrix of the third dimension. The iterative optimization module starts with the initial values ​​of each variable in the initialization module and iteratively optimizes each variable using a proximal iterative approach with alternating updates of multiple variables based on the objective function established in the model building module. In each iteration, it updates the low-rank branch variables, sparse branch variables, and their corresponding transformation matrices. The updated low-rank and sparse branches are then weighted and fused to update the tensor to be completed. The tensor to be completed is then corrected according to the observation consistency constraint. The corrected tensor to be completed is used as the input for the next iterative optimization, and the peak signal-to-noise ratio is calculated to evaluate the completion effect. The output module is used to iterate and optimize until the convergence threshold or the maximum number of iterations is reached, and outputs the final completed spectrum map tensor.

[0074] In another embodiment of the present invention, an electronic device is proposed, comprising: a memory, a processor, and a computer program stored in the memory and executable on the processor. When the processor executes the computer program, it implements a spectral data completion method based on third-dimensional tensor cooperative sparsity and low-rank transform learning as described above.

[0075] In another embodiment of the present invention, a computer-readable storage medium is proposed, storing a computer program that enables a computer to execute a spectral data completion method based on third-dimensional tensor cooperative sparsity and low-rank transform learning as described above.

[0076] In the embodiments disclosed in this application, a computer storage medium may be a tangible medium that may contain or store programs for use by or in conjunction with an instruction execution system, apparatus, or device. The computer storage medium may include, but is not limited to, electronic, magnetic, optical, electromagnetic, infrared, or semiconductor systems, apparatus, or devices, or any suitable combination of the foregoing. More specific examples of computer storage media include electrical connections based on one or more wires, portable computer disks, hard disks, random access memory (RAM), read-only memory (ROM), erasable programmable read-only memory (EPROM or flash memory), optical fibers, portable compact disk read-only memory (CD-ROM), optical storage devices, magnetic storage devices, or any suitable combination of the foregoing.

[0077] The above description is merely a preferred embodiment of the present invention and is not intended to limit the present invention in any way. Any simple modifications, equivalent substitutions, and improvements made by those skilled in the art to the above embodiments without departing from the scope of the technical solution of the present invention, based on the technical essence of the present invention, shall still fall within the protection scope of the technical solution of the present invention.

Claims

1. A spectral data completion method based on third-dimensional tensor cooperative sparsity and low-rank transform learning, characterized in that, Includes the following steps: S1: Obtain the 3D spectral map tensor and its observation index set for incomplete observations, and determine the size of the tensor to be completed; wherein, the first and second dimensions of the 3D spectral map tensor correspond to spatial location, and the third dimension corresponds to time series; construct a tensor completion optimization model based on third-dimensional cooperative sparsity and low-rank constraints. The tensor completion optimization model includes low-rank branches and sparse branches. By introducing regularization terms to constrain the low-rank property of spatial slices and the sparsity of third-dimensional fibers, and combining the observation consistency condition, establish the objective function to be optimized. S2: Initialize and fill the data at unobserved locations in the 3D spectrum map tensor to obtain the initialized tensor to be completed; initialize the low-rank branch variables, sparse branch variables, and the low-rank transformation matrix and sparse transformation matrix of the third dimension; S3: Starting with the initial values ​​of each variable in step S2, iteratively optimize each variable using a proximal iteration method with alternating updates of multiple variables according to the objective function established in step S1. In each iteration, update the low-rank branch variable, the sparse branch variable and their corresponding transformation matrix respectively; perform weighted fusion on the updated low-rank branch and sparse branch, update the tensor to be completed, and correct the tensor to be completed according to the observation consistency constraint. Use the corrected tensor to be completed as the input for the next iteration optimization, and calculate the peak signal-to-noise ratio to evaluate the completion effect. S4: Repeat the iterative optimization until the convergence threshold or the maximum number of iterations is reached, and output the final completed spectrum map tensor.

2. The spectral data completion method based on third-dimensional tensor cooperative sparsity and low-rank transform learning according to claim 1, characterized in that: In step S1, the objective function of the tensor completion optimization model jointly optimizes the low-rank branch, the sparse branch, and their corresponding transformation matrices, introducing a proximal penalty term to constrain the update magnitude of each variable in adjacent iterations. The expression is: in, For block diagonal operators; and These represent the reconstruction weights of the low-rank branch variable and the sparse branch variable, respectively. Indicates low-rank branch variables; Represents sparse branch variables; The tensor to be completed represents incomplete observations; Represents a three-dimensional spectrum map tensor; Indicates the observation index set A projection operator that takes a value at one position and sets it to zero at the other positions; symbol Represents the third modulo product; For nuclear norm; for Norm; Denotes the Frobenius norm; These are sparse regularization coefficients; , They represent , The List; Represents a low-rank transformation matrix; Represents a sparse transformation matrix; This is the penalty coefficient; The inertial smoothing coefficient; Indicates the first The low-rank branch variables after the round of updates; Indicates the first The sparse branch variables after each round of updates; Indicates the first The low-rank transformation matrix after the round of updates; Indicates the first The sparse transformation matrix after the round of updates; Indicates the first The tensor to be completed after the round of updates.

3. The spectral data completion method based on third-dimensional tensor cooperative sparsity and low-rank transform learning according to claim 1, characterized in that: In step S2, the initial filling of data at unobserved locations in the three-dimensional spectrum map tensor is specifically performed as follows: the boundary of the three-dimensional spectrum map tensor is extended along the spatial dimension, spatial interpolation is performed on the unobserved locations in the three-dimensional spectrum map tensor on each third-dimensional slice, the interpolation results are weighted and fused with the observed values, and the observed values ​​are forcibly backfilled at the observed locations in the three-dimensional spectrum map tensor to obtain the initial tensor to be filled.

4. The spectral data completion method based on third-dimensional tensor cooperative sparsity and low-rank transform learning according to claim 1, characterized in that: In step S3, the low-rank branch variable and the sparse branch variable are updated, specifically as follows: The expression for updating the low-rank branch variable is: The expression for updating the sparse branch variable is: in, Indicates the first The low-rank transformation matrix slice variables of the wheel; Indicates the first Slice variables of the sparse transformation matrix of the wheel; For the proximal operator corresponding to the nuclear norm regularization term; Low-rank transformation matrix The Column vector; Low-rank transformation matrix The Column vector; and These represent the reconstruction weights of the low-rank branch variable and the sparse branch variable, respectively. This is the penalty coefficient; for The proximal operator corresponding to the regularization term; These represent the current tensor to be completed along the first rank, with the remaining low-rank components or sparse components fixed. The third-dimensional residual matrix of the low-rank transformation direction and the sparse transformation direction; Indicates the first The low-rank transformation matrix slice variables of the wheel; Indicates the first Slice variables of the sparse transformation matrix of the wheel; These are sparse regularization coefficients; Represents the vector reconstruction operator.

5. The spectral data completion method based on third-dimensional tensor cooperative sparsity and low-rank transform learning according to claim 4, characterized in that: In step S3, the transformation matrices corresponding to the low-rank branch variables and sparse branch variables are updated, specifically as follows: Based on the current low-rank branch variables and sparse branch variables, the columns of the matrix are optimized by minimizing the reconstruction error; after each update, the resulting matrix column vectors are scaled to eliminate the scale uncertainty between the transformation matrix and the representation coefficients.

6. The spectral data completion method based on third-dimensional tensor cooperative sparsity and low-rank transform learning according to claim 1, characterized in that: In step S3, the updated low-rank branch and sparse branch are merged using a weighted summation method, and an inertial smoothing coefficient is introduced to enhance iteration stability, specifically as follows: in, The low-rank representation indicates the tensor's rank in the third dimension. A two-dimensional slice The low-rank coefficient vector obtained by vectorization; and These represent the reconstruction weights of the low-rank branch variable and the sparse branch variable, respectively. The inertial smoothing coefficient; Indicates the tensor to be completed; For the sparse tensor in the third dimension A two-dimensional slice The sparse coefficient vector obtained by vectorization; Indicates the first The low-rank transformation matrix obtained during the iteration; Indicates the first The sparse transformation matrix is ​​obtained through iterative updates.

7. The spectral data completion method based on third-dimensional tensor cooperative sparsity and low-rank transform learning according to claim 1, characterized in that: In step S3, the observation consistency constraint is implemented in each iteration by forcibly replacing the value of the tensor to be completed at the observation location with the original observation value. Specifically: in, Represents the set of unobserved locations; Represents the set of observation indices; Indicates the first The final tensor to be supplemented in the round; Indicates the tensor to be completed In the set of unobserved locations The element value on; Represents a three-dimensional spectrum map tensor In the observation index set The observed value at that location. The corrected tensor to be completed is used as the input for the next iteration of optimization, and the peak signal-to-noise ratio is calculated to evaluate the completion effect; wherein, a preheating process is performed before the formal iteration; the preheating process includes a coefficient preheating stage and a dictionary preheating stage.

8. A spectrum data completion system based on third-dimensional tensor cooperative sparsity and low-rank transform learning, characterized in that, include: The model building module is used to obtain the 3D spectral map tensor and its observation index set of incomplete observations, and determine the size of the tensor to be completed. The first and second dimensions of the 3D spectral map tensor correspond to spatial location, and the third dimension corresponds to time series. A tensor completion optimization model based on third-dimensional cooperative sparsity and low-rank constraints is constructed. The tensor completion optimization model includes low-rank branches and sparse branches. By introducing regularization terms to constrain the low-rank property of spatial slices and the sparsity of third-dimensional fibers, combined with the observation consistency condition, the objective function to be optimized is established. The initialization module is used to initialize and fill the data at unobserved locations in the 3D spectrum map tensor to obtain the initialized tensor to be completed; it initializes the low-rank branch variables, sparse branch variables, and the low-rank transformation matrix and sparse transformation matrix of the third dimension. The iterative optimization module starts with the initial values ​​of each variable in the initialization module and iteratively optimizes each variable using a proximal iterative approach with alternating updates of multiple variables based on the objective function established in the model building module. In each iteration, it updates the low-rank branch variables, sparse branch variables, and their corresponding transformation matrices. The updated low-rank and sparse branches are then weighted and fused to update the tensor to be completed. The tensor to be completed is then corrected according to the observation consistency constraint. The corrected tensor to be completed is used as the input for the next iterative optimization, and the peak signal-to-noise ratio is calculated to evaluate the completion effect. The output module is used to iterate and optimize until the convergence threshold or the maximum number of iterations is reached, and outputs the final completed spectrum map tensor.

9. An electronic device, characterized in that, include: The memory, the processor, and the computer program stored in the memory and executable on the processor, wherein when the processor executes the computer program, it implements the spectral data completion method based on third-dimensional tensor cooperative sparsity and low-rank transform learning as described in any one of claims 1 to 7.

10. A computer-readable storage medium having a computer program stored thereon, characterized in that: The computer program causes the computer to execute a spectral data completion method based on third-dimensional tensor cooperative sparsity and low-rank transform learning as described in any one of claims 1 to 7.

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