A Ground Penetrating Radar Clutter Suppression Method Based on Low-Rank Sparse Decomposition and Total Variation

Abstract

This invention provides a ground-penetrating radar (GPR) clutter suppression method based on low-rank sparse decomposition and total variation. The low-rank sparse decomposition optimization problem is transformed into a joint optimization problem based on minimizing the sum of the rank function of the low-rank matrix, the sparsity of the sparse matrix, and the total variation regularization term of the sparse matrix. The total variation regularization term of the sparse matrix can further filter out clutter components. Furthermore, this invention does not require a priori values ​​of the rank of the low-rank background matrix. Even with non-uniform clutter, noise, and decreased target sparsity, it can effectively suppress clutter and noise in GPR data while preserving the target area to some extent. In addition, simulation and field experiments show that this invention can improve the signal-to-clutter ratio of GPR data, making it an effective GPR clutter suppression method.

Description

Technical Field

[0001] This invention belongs to the field of ground-penetrating radar data processing technology, and particularly relates to a ground-penetrating radar clutter suppression method based on low-rank sparse decomposition and total variation. Background Technology

[0002] Ground penetrating radar (GPR) is an effective electromagnetic geophysical method for detecting underground targets. Due to its non-destructive, high-precision, and efficient characteristics, it has been widely applied in civil engineering, archaeology, mineral exploration, and target detection. However, in practical applications, strong clutter caused by ground reflection, direct coupling between transmitting and receiving antennas, and scattering responses from non-target sources (such as underground gravel or non-uniform surfaces) often overwhelm the target response. Therefore, clutter suppression is crucial for GPR target detection and identification.

[0003] In recent years, some studies have proposed decomposing the ground-penetrating radar (GPR) data matrix into a low-rank background matrix and a sparse target matrix, thereby transforming the clutter suppression problem into a low-rank sparse decomposition (LRSD) optimization problem. The main solution methods for the LRSD optimization problem already applied to clutter suppression include robust principal component analysis (RPCA), Go decomposition (GoDec), robust nonnegative matrix factorization (RNMF), and factor group sparse regularization (FGSR). Among these, the RNMF method requires a prior value of the rank of the low-rank background matrix, and in complex scenarios, such as when the rank of the low-rank background matrix is ​​greater than 1, it cannot adaptively select the rank parameter. While the other three methods do not require a prior value of the rank of the low-rank background matrix, they cannot effectively suppress non-uniform clutter and noise while preserving as much target sparsity as possible, especially in situations involving non-uniform clutter, noise, and decreased target sparsity. Summary of the Invention

[0004] To address the aforementioned problems, this invention provides a ground-penetrating radar clutter suppression method based on low-rank sparse decomposition and total variation, which can effectively suppress clutter and noise in ground-penetrating radar data while preserving the target area to a certain extent.

[0005] A ground-penetrating radar clutter suppression method based on low-rank sparse decomposition and total variation includes the following steps:

[0006] S1: Model the ground-penetrating radar data matrix D as the sum of a low-rank data matrix L and a sparse data matrix S, where the low-rank data matrix L represents clutter data and the sparse data matrix S represents target data.

[0007] S2: The low-rank sparse decomposition optimization problem based on the sum of the low-rank data matrix L and the sparse data matrix S is transformed into a joint optimization problem based on minimizing the sum of the rank function of the low-rank matrix, the sparsity of the sparse matrix, and the total variational regularization term of the sparse matrix.

[0008] S3: The joint optimization problem is solved using the alternating direction multiplier method, and the sparse data matrix S obtained is the target data matrix after clutter suppression.

[0009] Furthermore, the joint optimization problem is:

[0010] min||L|| * +λ||S||1+β||S|| TVa

[0011] stD = L + S,

[0012] Among them, ||·|| * Let ||·||1 represent the kernel norm of the low-rank data matrix L, and let ||·||1 represent the l1 norm of the sparse data matrix S. TVa Let λ represent the anisotropic total variation of the sparse data matrix S, where λ is the set sparse regularization weight and β is the set total variation regularization weight.

[0013] Furthermore, the joint optimization problem is solved using the alternating direction multiplier method as follows:

[0014] S31: The alternating update steps based on the alternating direction multiplier method are as follows:

[0015]

[0016]

[0017] Y k =Y k-1 +μ k-1 (DL k -S k )

[0018]

[0019] Among them, L k Let U represent the low-rank data matrix of the k-th iteration, and let U, Σ, and V represent the matrix... The left singular vector matrix, the singular value diagonal matrix, and the right singular vector matrix, [ ] H S represents the conjugate transpose of a matrix. k and S k-1 Y represents the sparse data matrices for the k-th and (k-1)-th iterations, respectively. k and Yk-1 Let μ represent the Lagrange multiplier matrices for the k-th and (k-1)-th iterations, respectively. k and μ k-1 Let represent the Lagrange multiplier matrix update parameters for the k-th and (k-1)-th iterations, respectively, and let B represent the auxiliary matrix, and have ρ is set as a constant, and ρ > 1; ε2 represents the first convergent residual; Ψ τ (·) is the element soft thresholding operator, defined as follows:

[0020]

[0021] Where u and τ are the input variables of the element soft thresholding operator;

[0022] S32: After initializing the Lagrange multiplier matrix, low-rank data matrix, and sparse data matrix to Y0 = L0 = S0 = 0 respectively, perform the alternating update steps. After each iteration, determine whether the iteration result meets the set convergence condition or whether the number of iterations reaches the set upper limit. If either condition is met, the final sparse data matrix is ​​obtained.

[0023] Furthermore, the convergence condition is set as follows:

[0024] ||DL k -S k || F / ||D|| F <ε1andμ k-1 ||S k -S k-1 || / ||D||<ε2

[0025] Among them, ||·|| F Let F denote the F norm of the matrix, and ε1 denote the second convergent residual.

[0026] Furthermore, Obtained by the fast gradient projection algorithm.

[0027] Furthermore, the ground-penetrating radar data matrix D is constructed from n channels of sampled data scanned from the depth direction. At the same time, the number of sampling points in each channel of sampled data is denoted as m, and m > n.

[0028] Beneficial effects:

[0029] 1. This invention provides a ground-penetrating radar clutter suppression method based on low-rank sparse decomposition and total variation. The low-rank sparse decomposition optimization problem is transformed into a joint optimization problem based on minimizing the sum of the rank function of the low-rank matrix, the sparsity of the sparse matrix, and the total variation regularization term of the sparse matrix. The total variation regularization term of the sparse matrix can further filter out clutter components. Moreover, this invention does not require the rank prior value of the low-rank background matrix. Under the conditions of non-uniform clutter, noise, and decreased target sparsity, it can effectively suppress clutter and noise in ground-penetrating radar data and retain the target area to a certain extent. In addition, simulation experiments and field experiments show that this invention can improve the signal-to-clutter ratio of ground-penetrating radar data and is an effective ground-penetrating radar clutter suppression method.

[0030] 2. This invention provides a ground-penetrating radar clutter suppression method based on low-rank sparse decomposition and total variation. It uses the alternating direction multiplier method to solve the joint optimization problem, which can guarantee the accuracy of the target data matrix after clutter suppression and has higher computational efficiency.

[0031] 3. This invention provides a ground-penetrating radar clutter suppression method based on low-rank sparse decomposition and total variation. It uses a fast gradient projection algorithm to obtain the auxiliary variables used in the alternating direction multiplier method, which can improve computational efficiency. Attached Figure Description

[0032] Figure 1 This is a flowchart of the method of the present invention;

[0033] Figure 2 This is a simulation scene diagram of the present invention;

[0034] Figure 3 Figures showing the results of simulation data processed by different methods under dry sandy soil conditions are shown below. (a) shows the original data, (b) shows the results of simulation data processed by the RPCA method under dry sandy soil conditions, (c) shows the results of simulation data processed by the RNMF method under dry sandy soil conditions, and (d) shows the results of simulation data processed by the method of this invention under dry sandy soil conditions.

[0035] Figure 4 This is a cross-sectional view of the actual test scenario of the present invention;

[0036] Figure 5 The figures show the results of processing the measured data using different methods. (a) shows the original data, (b) shows the results of processing the measured data using the RPCA method, (c) shows the results of processing the measured data using the RNMF method, and (d) shows the results of processing the measured data using the method of this invention. Detailed Implementation

[0037] To enable those skilled in the art to better understand the present application, the technical solutions in the embodiments of the present application will be clearly and completely described below with reference to the accompanying drawings.

[0038] A ground-penetrating radar clutter suppression method based on low-rank sparse decomposition and total variational regularization, the processing flow is as follows: Figure 1 As shown, the specific steps include

[0039] S1: Model the ground-penetrating radar data matrix D as the sum of a low-rank data matrix L and a sparse data matrix S, where the low-rank data matrix L represents clutter data and the sparse data matrix S represents target data.

[0040] For example, consider a two-dimensional scanning data matrix of a ground-penetrating radar. Where m is the number of data sampling points in one scan along the depth direction, and n is the total number of scan channels, generally m > n. According to Low-Rank Sparse Decomposition (LRSD), the ground-penetrating radar data matrix D can be modeled as:

[0041] D = L + S

[0042] in, These represent the low-rank clutter component and the target component, respectively.

[0043] S2: The low-rank sparse decomposition optimization problem based on the sum of the low-rank data matrix L and the sparse data matrix S is transformed into a joint optimization problem based on minimizing the sum of the rank function of the low-rank matrix, the sparsity of the sparse matrix, and the total variational regularization term of the sparse matrix.

[0044] In other words, after completing the modeling of the ground-penetrating radar data matrix, in order to both recover L and S from the ground-penetrating radar data matrix D and suppress non-uniform clutter and noise while preserving the target area, the following optimization problem is first established:

[0045]

[0046] stD = L + S,

[0047] The first term minimizes the rank function of the low-rank matrix, the second term represents the sparsity of the sparse matrix, the third term is the total variational regularization term of the sparse matrix, rank(·) denotes the rank of the matrix, ||·||0 denotes the l0 norm of the matrix, and ||·|| TV Let represent the discrete total variation (TV) seminorm of the matrix, and λ and β be the sparse regularization weight parameter and the total variation regularization weight parameter, respectively. Since the first two terms are non-convex, the above optimization problem is NP-hard. To solve this problem, the first two terms need to be relaxed to be convex, that is, the above optimization problem is transformed into the following optimization problem:

[0048]

[0049] stD = L + S

[0050] Among them, ||·|| * The nuclear norm of a matrix is ​​defined as the sum of its singular values:

[0051]

[0052] ||·||1 represents the l1 norm of a matrix, defined as the sum of the absolute values ​​of its elements:

[0053]

[0054] ||·|| TVa The anisotropic total variation (TV) of a matrix is ​​defined as follows:

[0055]

[0056] It should be noted that the Discrete Total Variation (TV) seminorm of this invention represents the sum of the absolute values ​​of all elements in the horizontal and vertical difference matrices of the sparse data matrix S after obtaining the horizontal and vertical difference matrices. The smaller the sum, the less clutter component there is in the sparse data matrix S.

[0057] S3: The joint optimization problem is solved using the alternating direction multiplier method, and the sparse data matrix S obtained is the target data matrix after clutter suppression.

[0058] Furthermore, to solve the joint optimization problem, we first construct the augmented Lagrangian function.

[0059]

[0060] in, Let ||·|| denote the Lagrange multiplier matrix, μ > 0 as a sufficiently small parameter, and 〈·, ·〉 denote the matrix inner product, ||·|| F Let F be the norm of the matrix. According to the alternating direction multiplier method, the alternating update steps are as follows:

[0061]

[0062] Where k represents the current iteration number, and ε2 represents the convergence residual. Next, the alternating update operator is derived in detail to update the low-rank data matrix L and the sparse data matrix S.

[0063] First, update the low-rank data matrix L by solving the following problem.

[0064]

[0065] Among them, the above L k The solution is:

[0066] L k =UΨ 1 / μ (Σ)V H

[0067] U, Σ, and V are obtained from the following singular value decomposition.

[0068]

[0069] svd() represents singular value decomposition, where U, Σ, and V represent the matrix... The left singular vector matrix, the singular value diagonal matrix, and the right singular vector matrix, [ ] H Ψ represents the conjugate transpose of a matrix. τ (·) is the element soft thresholding operator, defined as follows:

[0070]

[0071] Where u and τ are the input variables of the element soft thresholding operator, specifically, in this invention, τ is 1 / μ k or λ / μ k-1 ,u is Σ or

[0072] Secondly, the sparse data matrix S is updated by solving the following problem.

[0073]

[0074] Among them, the above S k The solution is:

[0075]

[0076] in, The solution to the following optimization problem is, i.e.

[0077]

[0078] Furthermore, It can be solved by the Fast Gradient Projection (FGP) algorithm, as shown in Algorithm 1.

[0079]

[0080] Among them, S tvs That is Defined as S tvs The dual variables of the longitudinal gradient matrix and the transverse gradient matrix, where, Defined as calculating the vertical gradient matrix and the horizontal gradient matrix of the input matrix. for The adjoint operator is defined as the divergence of the input gradient. For anisotropic total variation, Defined as the projection (r,s) of (z,w) onto set C = P C (z, w), with respect to the definition of set C, has:

[0081]

[0082] Projection operation (r,s) = P C (z,w) is given by the following two equations.

[0083]

[0084]

[0085] In summary, the solution algorithm for ground-penetrating radar clutter suppression based on low-rank sparse decomposition and total variational regularization can be summarized as shown in Algorithm 2 below.

[0086]

[0087]

[0088] Convergence condition is defined as

[0089] ||DL k -S k || F / ||D|| F <ε1andμ k-1 ||S k -S k-1 || / ||D||<ε2

[0090] It should be noted that if the convergence condition is not met even after the set upper limit is reached, the result of the last iteration will be used as the final sparse data matrix S.

[0091] Furthermore, to verify the ground-penetrating radar clutter suppression method based on low-rank sparse decomposition and total variational regularization proposed in this invention, simulation data and measured data were used for verification.

[0092] The positive-to-negativity ratio improvement factor used for quantitative analysis is defined as follows:

[0093] IF(dB) = 10log(SCR) a / SCR b )

[0094] SCR is defined as the signal-to-clutter ratio of ground-penetrating radar data.a The signal-to-noise ratio (SCR) after processing is given by the method. b The signal-to-noise ratio before the method processing

[0095]

[0096] In the above formula, N T and N C Representing the target region R respectively T and clutter region R C The amount of data, I(p) is the data value at position p.

[0097] The simulation data was generated by the simulation software gprMax, and the simulation scene is as follows: Figure 2 As shown, the simulation scene is a two-dimensional scene with dimensions of 2.2m × 0.005m × 1.0m. The ground surface is rough, with an elevation change of ±λ / 20, where λ is the center frequency wavelength. The target is a regularly shaped, hollow target, with its position and size as shown. Figure 2 As shown in Table 1, the transmitted waveform used a Ricker wavelet with a center frequency of 1.5 GHz, a time window of 15 ns, and a co-located transmitting and receiving antenna close to the ground surface with a step size of 2 cm. The resulting ground-penetrating radar raw data matrix was 1697 × 10¹. Simulation parameters for different soil media are shown in Table 1.

[0098] Table 1 Simulation parameters for different soil media

[0099]

[0100] Table 2 shows the improvement factor results of the method of the present invention and the other two methods for processing five sets of simulation data. As can be seen from Table 2, the method of the present invention has the best clutter suppression effect on ground penetrating radar data. Figure 3 The images show the results of simulation data processed by different methods under dry sandy soil conditions, for comparison. Figure 3 As can be seen from (a), (b), (c) and (d), the method of the present invention, compared with the other two methods, not only improves the clutter suppression effect, but also preserves the target area to a certain extent.

[0101] Table 2 Results of the Improvement Factor (IF) for Processing Simulation Data

[0102]

[0103] The measured data was obtained by the LTD-2600 ground penetrating radar, and the cross-sectional diagram of the measured scene is shown below. Figure 4 As shown, the cross-sectional dimensions of the measured scene area are 1.8m × 0.9m. The target is a rectangular cardboard box with an internal filling of 50cm × 25cm × 14cm, and its location is as follows. Figure 4As shown. The transmitted waveform uses the Ricker subwave with a center frequency of 1.5 GHz, a time window of 10 ns, 512 sampling points, and the transmit and receive antennas are placed together, close to the ground surface, with a step of 2.5 cm. The resulting ground penetrating radar raw data matrix is ​​512×73.

[0104] Table 3 shows the improvement factor results of the method of the present invention and the other two methods for processing the measured data. As can be seen from Table 3, the clutter suppression effect of the method of the present invention is better than RPCA and comparable to RNMF. Figure 5 Figures (a), (b), (c), and (d) in the figure show the results of processing measured data using different methods, for comparison. Figure 3 As can be seen from (a), (b), (c) and (d) in the figure, the method of the present invention has a better clutter suppression effect than RPCA; in addition, as can be seen from the data in Table 3, the method of the present invention can not only achieve an output improvement factor comparable to RNMF, but also retain more target area to a certain extent compared with the RNMF method.

[0105] Table 3. Results of the Improvement Factor (IF) for Processing Measured Data

[0106]

[0107] Therefore, the ground-penetrating radar clutter suppression method proposed in this invention, based on low-rank sparse decomposition and total variational regularization, first models the ground-penetrating radar data matrix as the sum of a low-rank data matrix and a sparse data matrix. The low-rank data matrix represents the main clutter component, and the sparse data matrix represents the target. Next, the low-rank sparse decomposition optimization problem is transformed into an optimization problem that minimizes the sum of the rank function of the low-rank matrix, the sparsity of the sparse matrix, and the total variational regularization term of the sparse matrix. Finally, the above optimization problem is solved using the Alternating Direction Multiplier Method (ADMM), and the resulting sparse data matrix is ​​the target data matrix after clutter suppression. Compared with existing technologies, this invention does not require a priori rank value of the low-rank background matrix. Even with non-uniform clutter, noise, and reduced target sparsity, it can effectively suppress clutter and noise in ground-penetrating radar data while preserving the target area to some extent. Experiments show that the signal-to-clutter ratio of ground-penetrating radar data processed by this invention is effectively improved.

[0108] Of course, the present invention may have other various embodiments. Without departing from the spirit and essence of the present invention, those skilled in the art can make various corresponding changes and modifications according to the present invention, but these corresponding changes and modifications should all fall within the protection scope of the appended claims.

Claims

1. A ground-penetrating radar clutter suppression method based on low-rank sparse decomposition and total variation, characterized in that, Includes the following steps: S1: Model the ground-penetrating radar data matrix D as the sum of a low-rank data matrix L and a sparse data matrix S, where the low-rank data matrix L represents clutter data and the sparse data matrix S represents target data. S2: The low-rank sparse decomposition optimization problem based on the sum of the low-rank data matrix L and the sparse data matrix S is transformed into a joint optimization problem based on minimizing the sum of the rank function of the low-rank matrix, the sparsity of the sparse matrix, and the total variational regularization term of the sparse matrix. S3: The joint optimization problem is solved using the alternating direction multiplier method, and the sparse data matrix S obtained is the target data matrix after clutter suppression. The joint optimization problem is: min||L|| * +λ||S||1+β||S|| TVa stD = L + S, Among them, ||·|| * Let ||·||1 represent the kernel norm of the low-rank data matrix L, and let ||·||1 represent the l1 norm of the sparse data matrix S. TVa Let λ represent the anisotropic total variation of the sparse data matrix S, where λ is the set sparse regularization weight and β is the set total variation regularization weight. The joint optimization problem is solved using the alternating direction multiplier method as follows: S31: The alternating update steps based on the alternating direction multiplier method are as follows: Y k =Y k-1 +μ k-1 (DL k -S k ) Among them, L k Let U represent the low-rank data matrix of the k-th iteration, and let U, Σ, and V represent the matrix... The left singular vector matrix, the singular value diagonal matrix, and the right singular vector matrix, [ ] H S represents the conjugate transpose of a matrix. k and S k-1 Y represents the sparse data matrices for the k-th and (k-1)-th iterations, respectively. k and Y k-1 Let μ represent the Lagrange multiplier matrices for the k-th and (k-1)-th iterations, respectively. k and μ k-1 Let represent the Lagrange multiplier matrix update parameters for the k-th and (k-1)-th iterations, respectively, and let B represent the auxiliary matrix, and have ρ is set as a constant, and ρ > 1; ε2 represents the first convergent residual; Ψ τ (·) is the element soft thresholding operator, defined as follows: Where u and τ are the input variables of the element soft thresholding operator; τ is 1 / μ k or λ / μ k-1 u is Σ or S32: After initializing the Lagrange multiplier matrix, low-rank data matrix and sparse data matrix to Y0 = L0 = S0 = 0 respectively, iterate according to the alternating update steps. After each iteration, determine whether the iteration result meets the set convergence condition or whether the number of iterations reaches the set upper limit. If either condition is met, the final sparse data matrix is ​​obtained. The convergence condition is set as follows: ||D-L k -S k || F / ||D|| F <ε1andμ k-1 ||S k -S k-1 || / ||D||<ε2 Among them, ||·|| F Let F denote the F norm of the matrix, and ε1 denote the second convergent residual.

2. The ground-penetrating radar clutter suppression method based on low-rank sparse decomposition and total variation as described in claim 1, characterized in that, Obtained by the fast gradient projection algorithm.

3. A ground-penetrating radar clutter suppression method based on low-rank sparse decomposition and total variation as described in any one of claims 1 to 2, characterized in that, The ground-penetrating radar data matrix D is constructed from n channels of sampled data scanned from the depth direction. At the same time, the number of sampling points in each channel of sampled data is denoted as m, and m > n.

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