Improved reweighted atomic norm-based high-resolution sparse ISAR imaging method

Abstract

The application discloses an improved reweighted atomic norm high-resolution sparse ISAR imaging method, which comprises the following steps: obtaining a sparse ISAR echo matrix of M distance units; based on the sparse ISAR echo matrix, using an MRAM-SDP method or an MRAM-ADMM method, and through multi-step iterative weighting, a recovered echo matrix S is obtained; obtaining the echo matrix B of all the distance units after recovery, and using an inverse fast Fourier transform to obtain a final ISAR image. The application has the effect that the number of iterative weightings of the reweighted atomic norm high-resolution sparse ISAR imaging method is reduced. By utilizing the sparse characteristics of scattering points in the imaging target in the Doppler domain, according to the continuous compressive sensing and low-rank matrix recovery theory, high-resolution imaging of a non-cooperative target under a sparse aperture signal is realized, the performance of the reweighted atomic norm high-resolution sparse ISAR imaging method is maintained, and the operation complexity is greatly reduced.

Description

Technical Field

[0001] This invention belongs to the field of radar signal processing technology, and in particular relates to an improved high-resolution sparse ISAR imaging method based on the reweighted atomic norm. Background Technology

[0002] Inverse Synthetic Aperture Radar (ISAR) imaging, as an all-weather, all-day microwave detection method, can achieve high-resolution imaging of air, space, and sea targets. ISAR systems achieve high range resolution by transmitting and receiving wideband signals, and high azimuth resolution by synthesizing multi-pulse echo signals within the coherent processing interval (CPI) using the change in viewing angle between the radar and the target. Typically, imaging can be achieved by using a fast inverse Fourier transform (FFT) on the received signal. However, in practice, radar systems operate under multi-tasking conditions and cannot obtain complete full aperture (FA) observation signals. Using a FFT on the missing sparse aperture (SA) signal results in severe main lobe broadening in the image.

[0003] With advancements in radar signal processing technology, sparse recovery techniques have become a popular method for signal reconstruction. Sparse ISAR imaging based on this technology leverages the sparse characteristics of target scattering centers in the spatial plane. Through appropriate sparse recovery algorithms, accurate reconstruction of sparse aperture signals can be achieved, thus enabling high-resolution sparse ISAR imaging. However, traditional sparse recovery-based ISAR imaging methods require discretization of the imaging scene, inevitably leading to grid mismatch and reduced ISAR imaging quality. While existing reweighted atomic norm (RAM) high-resolution sparse ISAR imaging methods can avoid grid mismatch in sparse recovery, these methods are complex and computationally time-consuming. Therefore, how to reduce the computational cost of reweighted atomic norm sparse ISAR imaging has become a pressing technical problem for those skilled in the art. Summary of the Invention

[0004] To address the aforementioned problems, the present invention aims to provide a high-resolution sparse ISAR imaging method based on the improved Modified Reweighted Atomic Norm (MRAM).

[0005] To achieve the above objectives, the present invention provides a high-resolution sparse ISAR imaging method based on an improved reweighted atomic norm, comprising the following steps performed in sequence:

[0006] 1) Obtain the sparse ISAR echo matrix of M range cells.

[0007] 2) Based on the above sparse ISAR echo matrix The recovered echo matrix S is obtained by using the MRAM-SDP method or the MRAM-ADMM method through multi-step iterative weighted solution.

[0008] 3) Repeat step 2) to obtain the echo matrix B of all recovered range cells and use inverse fast Fourier transform to obtain the final ISAR image.

[0009] In step 1), the sparse ISAR echo matrix of M range cells is obtained. The method is:

[0010] First, the complete echo vector of the m-th distance cell is represented as:

[0011]

[0012] in, This represents the complete echo vector of the m-th distance cell in an N×M dimension. Let t represent the complex space, where t = [0:N]. T Δt, [·] T This represents finding the transpose of a matrix, where Δt represents the pulse repetition time, N represents the number of pulses, K represents the number of scattering points, and f k σ represents the Doppler frequency at the k-th scattering point. k Indicates the amplitude of the scattering point;

[0013] Based on equation (1), the complete echo vectors of the M range cells are combined to obtain an N×M dimensional echo matrix:

[0014]

[0015] Meanwhile, echo matrix S 0 It can also be expressed as:

[0016]

[0017] in, σ k,m This represents the amplitude of the k-th scattering point, which appears in the m-th range cell, where m ≤ M;

[0018] Since the sparse ISAR echo matrix of M range cells can be regarded as the above echo matrix S 0 The random sampling matrix is ​​given by the sparse ISAR echo matrix, which can be represented as follows: in This represents the index set of N pulses, where N represents the number of random samples.

[0019] In step 2), the sparse ISAR echo matrix mentioned above... The method for obtaining the recovered echo matrix S using the MRAM-SDP or MRAM-ADMM method through multi-step iterative weighted solution is as follows:

[0020] Define weighted atom set for:

[0021]

[0022] Where w(f) represents the weight function of atom a(f); based on equation (4), the weighted atom l0 norm of the recovered echo matrix S can be obtained:

[0023]

[0024] Where inf{·} denotes solving for the infimum; since solving equation (5) is an NP-hard problem, equation (5) needs to be convexly relaxed to equation (6):

[0025]

[0026] When using the MRAM-SDP method, the problem of minimizing the weighted atomic norm of the recovered echo signal S can be transformed into an SDP problem, and the local optimum can be approximated through multi-step iterative weighting. Let u j Let J represent the variable optimized in the j-th iteration, based on the ISAR echo matrix obtained in step 1). The SDP optimization equation for the (j+1)th iteration is:

[0027]

[0028]

[0029] Where tr(·) denotes finding the trace of a matrix, W denotes the weighted matrix, X denotes the dual variable, T(u) denotes the Toeplitz matrix composed of the iterative optimization variable u, η denotes the regularization parameter, and ||·|| F Represents the Fourier norm;

[0030] The form of the weighting matrix W is determined by the introduced sparsity metric criterion;

[0031] A new sparsity metric is introduced to reduce the number of iterations required for RAM, namely:

[0032]

[0033] Therefore, according to equation (9), the weighted matrix W = ε / (T(u) + εI) 2 Then, the recovered echo signal S can be obtained by solving the SDP optimization equation in equation (7);

[0034] When using the ADMM method, the sparse ISAR echo matrix obtained based on equation (7) and step 1) is... Construct the Lagrange augmented equation:

[0035]

[0036] Where α and β are regularization parameters, and <,> denote inner product operations; matrix matrix They are respectively:

[0037]

[0038] in, If the initial matrices of Λ and Z are zero matrices, then the ADMM update solution formula is as follows:

[0039]

[0040]

[0041] Where D represents a diagonal matrix, and the diagonal elements N i =1,2,...,N;S Ωc This represents missing echo data; Ωc represents the complement of the index set Ω of N pulses; T · (·) indicates mapping a matrix to a vector, where the Nth element in the vector is... i The elements are the summation and average of the diagonal elements of the matrix; furthermore, the iterative update expression for matrix Z is as follows:

[0042]

[0043] Equation (17) is usually converted to a Hermitian matrix:

[0044]

[0045] Retain positive eigenvalues, that is:

[0046] Z i+1 =V i diag((d i ) + (v) i ) H (19)

[0047] Where d and v represent the eigenvalues ​​and eigenvectors of the Hermitian matrix, respectively. + This indicates that positive eigenvalues ​​are retained;

[0048] The recovered echo signal S is obtained by iteratively solving equations (12), (13), (14), (15), (16) and (19).

[0049] In step 3), the method of repeating step 2) to obtain the echo matrix B of all recovered range cells and using inverse fast Fourier transform to obtain the final ISAR image is as follows:

[0050] After integrating the echo matrices of the M recovered range cells, the echo matrix B of all range cells is obtained. Then, the final ISAR image is obtained using the inverse fast Fourier transform, as shown in the formula:

[0051] I = ifft(B) (20)

[0052] Here, ifft(·) represents the fast inverse Fourier transform operation.

[0053] The beneficial effects of this invention are:

[0054] This method can reduce the number of iterative weighting steps in the reweighted atomic norm high-resolution sparse ISAR imaging method. By utilizing the sparsity of scattering points in the Doppler domain of the imaging target, and based on continuous compressed sensing and low-rank matrix recovery theory, high-resolution imaging of non-cooperative targets under sparse aperture signals is achieved. This not only maintains the performance of the reweighted atomic norm high-resolution sparse ISAR imaging method, but also significantly reduces the computational complexity. Attached Figure Description

[0055] Figure 1 (a) and (b) show the actual full aperture data and imaging results of the Yak-42, respectively.

[0056] Figure 2 (a), (b), (c), and (d) are the imaging results of ANM, RAM, MRAM-SDP, and MRAM-ADMM methods, respectively, with a pulse reception pulse number of 50% and a signal-to-noise ratio of 5dB.

[0057] Figure 3 (a), (b), (c), and (d) are the imaging results of ANM, RAM, MRAM-SDP, and MRAM-ADMM methods, respectively, with a pulse reception pulse number of 30% and a signal-to-noise ratio of 7.5 dB.

[0058] Figure 4(a) and (b) show the reconstruction error under different signal-to-noise ratios with 50% pulse data and the reconstruction error under different pulse numbers with a signal-to-noise ratio of 5dB, respectively. Detailed Implementation

[0059] To enable those skilled in the art to better understand the technical solutions of the present invention, the present invention will be further described in detail below with reference to the accompanying drawings and preferred embodiments. All other embodiments obtained by those skilled in the art based on the embodiments of the invention without inventive effort are within the scope of protection of the invention.

[0060] The high-resolution sparse ISAR imaging method based on the improved reweighted atomic norm provided by this invention includes the following steps performed in sequence:

[0061] 1) Obtain the sparse ISAR echo matrix of M range cells.

[0062] First, the complete echo vector of the m-th distance cell is represented as:

[0063]

[0064] in, This represents the complete echo vector of the m-th distance cell in an N×M dimension. Let t represent the complex space, where t = [0:N]. T Δt, [·] T This represents finding the transpose of a matrix, where Δt represents the pulse repetition time, N represents the number of pulses, K represents the number of scattering points, and f k σ represents the Doppler frequency at the k-th scattering point. k Indicates the amplitude of the scattering point;

[0065] Based on equation (1), the complete echo vectors of the M range cells are combined to obtain an N×M dimensional echo matrix:

[0066]

[0067] Meanwhile, echo matrix S 0 It can also be expressed as:

[0068]

[0069] in, σ k,m This represents the amplitude of the k-th scattering point, which appears in the m-th range cell, where m ≤ M;

[0070] Since the sparse ISAR echo matrix of M range cells can be regarded as the above echo matrix S 0The random sampling matrix is ​​given by the sparse ISAR echo matrix, which can be represented as follows: in This represents the index set of N pulses. Represents the number of random samples.

[0071] 2) Based on the above sparse ISAR echo matrix The recovered echo matrix S is obtained by using the MRAM-SDP (reweighted atomic norm-semi-positive definite programming) method or the MRAM-ADMM (reweighted atomic norm-alternating direction multiplier method) method through multi-step iterative weighted solution.

[0072] Define weighted atom set for:

[0073]

[0074] Where w(f) represents the weight function of atom a(f); based on equation (4), the weighted atom l0 norm of the recovered echo matrix S can be obtained:

[0075]

[0076] Where inf{·} denotes solving for the infimum; since solving equation (5) is an NP-hard problem, equation (5) needs to be convexly relaxed to equation (6):

[0077]

[0078] When using the MRAM-SDP (Reweighted Atom Norm-Semidefinite Programming) method, the problem of minimizing the weighted atom norm of the recovered echo signal S can be transformed into an SDP problem, and a local optimum can be approximated through multi-step iterative weighting. Let u j Let J represent the variable optimized in the j-th iteration, based on the ISAR echo matrix obtained in step 1). The SDP optimization equation for the (j+1)th iteration is:

[0079]

[0080]

[0081] Where tr(·) denotes finding the trace of a matrix, W denotes the weighted matrix, X denotes the dual variable, T(u) denotes the Toeplitz matrix composed of the iterative optimization variable u, η denotes the regularization parameter, and ||·|| F Represents the Fourier norm;

[0082] The form of the weighting matrix W is determined by the introduced sparsity metric. Conventional RAM methods generally use the following sparsity metric:

[0083]

[0084]

[0085] Among them, S H Let ε represent the conjugate transpose of the recovered echo matrix S, and let ε represent the sparsity regularization parameter. When the sparsity regularization parameter ε approaches positive infinity, the sparsity metric represented by the sparsity metric function is equivalent to the atomic norm. When the sparsity regularization parameter ε approaches zero, the sparsity metric is equivalent to the atomic l0 norm.

[0086] This invention introduces a novel sparsity metric to reduce the number of iterations required for RAM, namely:

[0087]

[0088] Therefore, according to equation (9), the weighted matrix W = ε / (T(u) + εI) 2 Then, the recovered echo signal S can be obtained by solving the SDP optimization equation in equation (7);

[0089] The computational efficiency can be further improved by using the ADMM method, based on the sparse ISAR echo matrix obtained by equation (7) and step 1). Construct the Lagrange augmented equation:

[0090]

[0091] Where α and β are regularization parameters, and <,> denote inner product operations; matrix matrix They are respectively:

[0092]

[0093] in, If the initial matrices of Λ and Z are zero matrices, then the ADMM update solution formula is as follows:

[0094]

[0095]

[0096] Where D represents a diagonal matrix, and the diagonal elements N i =1,2,...,N;S Ωc This represents missing echo data; Ωc represents the complement of the index set Ω of N pulses; T · (·) indicates mapping a matrix to a vector, where the Nth element in the vector is... iThe elements are the summation and average of the diagonal elements of the matrix; furthermore, the iterative update expression for matrix Z is as follows:

[0097]

[0098] Equation (17) is usually converted to a Hermitian matrix:

[0099]

[0100] Retain positive eigenvalues, that is:

[0101] Z i+1 =V i diag((d i ) + (v) i ) H (19)

[0102] Where d and v represent the eigenvalues ​​and eigenvectors of the Hermitian matrix, respectively. + This indicates that positive eigenvalues ​​are retained;

[0103] The recovered echo signal S is obtained by iteratively solving equations (12), (13), (14), (15), (16) and (19).

[0104] 3) Repeat step 2) to obtain the echo matrix B of all recovered range cells and use inverse fast Fourier transform to obtain the final ISAR image.

[0105] After integrating the echo matrices of the M recovered range cells, the echo matrix B of all range cells is obtained. Then, the final ISAR image is obtained using the inverse fast Fourier transform, as shown in the formula:

[0106] I = ifft(B) (20)

[0107] Here, ifft(·) represents the fast inverse Fourier transform operation.

[0108] To verify the effectiveness of the method of the present invention, the inventors conducted the following characterization experiments:

[0109] Experimental use such as Figure 1The actual Yak-42 full-aperture data shown has the following radar parameters: radar operating frequency of 5.52 GHz, pulse repetition frequency of 100 Hz, signal bandwidth of 400 MHz, and the full-aperture data used in the experiment has 128 pulses and 64 sparse apertures. In the experiment, the initial sparse regularization parameter ε was set to 1, and the attenuation was halved in each iteration. The number of iterations for the RAM method was set to 6, and the number of iterations for the method of this invention was set to 4. Simulation experiments compared the ANM method and the RAM method. The sparse ISAR imaging results under low signal-to-noise ratio and low pulse number conditions were compared.

[0110] Experiment 1: Low signal-to-noise ratio experimental verification. Figure 2 Imaging results of different methods are presented using 50% pulse data with a signal-to-noise ratio of 5 dB. Although the ANM method considers the imaging target in the continuous domain, its imaging performance is inferior to the RAM and MRAM methods due to the effect of convex relaxation. The non-convex substitution function compensates for the gap with the norm, and for the weak scattering center of the aircraft nose, the imaging performance of the RAM and MRAM methods is significantly better than that of the ANM method. Furthermore, the fast solution method based on ADMM does not lead to any loss in imaging performance.

[0111] Experiment 2: Low pulse number experiment verification. Figure 3 Imaging results of different methods are presented using 30% pulse data with a signal-to-noise ratio of 7.5 dB. It is evident that when the number of apertures is limited, the effect of convex relaxation is significant, resulting in numerous erroneous scattering points and low image quality in the ANM method's imaging results. The RAM and MRAM methods based on reweighted modes effectively enhance the sparsity of ANM, producing imaging results with good focusing properties and significantly superior image quality compared to the ANM method.

[0112] Experiment 3: Monte Carlo Experiment Verification. To better illustrate the superiority of the algorithm, the root mean square error (RMSE) ||BB is used. 0 || F / ||B 0 || F The imaging performance of different methods is measured by computation time, where B... 0 B represents the original full pore size data, and B represents the full pore size data reconstructed using SA data.

[0113] Figure 4 (a) The reconstruction error is given for different signal-to-noise ratios under 50% pulse data. Figure 4 (b) The reconstruction error is given for different pulse numbers at a signal-to-noise ratio of 5dB. It can be seen that the mean square error of both RAM and MRAM methods is lower than that of ANM when the signal-to-noise ratio and the number of apertures change, and the mean square error curves of RAM and MRAM methods are highly similar.

[0114] In Experiment 1, the imaging time required by the ANM method was 213 seconds, the RAM method was 1253 seconds, the MRAM-SDP method was 827 seconds, and the MRAM-ADMM method was 313 seconds. In Experiment 2, the imaging time required by the ANM method was 204 seconds, the RAM method was 1153 seconds, the MRAM-SDP method was 769 seconds, and the MRAM-ADMM method was 354 seconds. The numerical comparison shows that the SDP method significantly improves computational efficiency compared to the RAM method, and the ADMM method requires only half the computation time of the SDP method.

[0115] This invention utilizes the sparse characteristics of scattering points in the imaging target in the Doppler domain and realizes the recovery of missing aperture data based on continuous compressed sensing and low-rank matrix recovery theory. This method not only effectively avoids the dictionary mismatch problem in sparse recovery, but also achieves the high-precision reconstruction performance of RAM imaging method while greatly reducing computational complexity.

[0116] The above description is only a preferred embodiment of the present invention. It should be noted that for those skilled in the art, several improvements and modifications can be made without departing from the principle of the present invention, and these improvements and modifications should also be considered within the scope of protection of the present invention.

Claims

1. A high-resolution sparse ISAR imaging method based on an improved reweighted atomic norm, characterized in that: The improved reweighted atomic norm-based high-resolution sparse ISAR imaging method comprises the following steps performed sequentially: 1) Obtain the sparse ISAR echo matrix of M range cells. 2) Based on the above sparse ISAR echo matrix The recovered echo matrix S is obtained by using the MRAM-SDP method or the MRAM-ADMM method through multi-step iterative weighted solution. 3) Repeat step 2) to obtain the echo matrix B of all recovered range cells and use inverse fast Fourier transform to obtain the final ISAR image; In step 2), the sparse ISAR echo matrix mentioned above... The method for obtaining the recovered echo matrix S using the MRAM-SDP or MRAM-ADMM method through multi-step iterative weighted solution is as follows: Define weighted atom set for: Where w(f) represents the weight function of atom a(f); based on equation (4), the weighted atom l0 norm of the recovered echo matrix S can be obtained: Where inf{·} denotes solving for the infimum; f k σ represents the Doppler frequency at the k-th scattering point. k Let represent the amplitude of the scattering point; since solving equation (5) is an NP-hard problem, it is necessary to relax equation (5) into equation (6): When using the MRAM-SDP method, the problem of minimizing the weighted atomic norm of the recovered echo signal S can be transformed into an SDP problem, and the local optimum can be approximated through multi-step iterative weighting. Let u j Let J represent the variable optimized in the j-th iteration, based on the ISAR echo matrix obtained in step 1). The SDP optimization equation for the (j+1)th iteration is: Where tr(·) denotes finding the trace of a matrix, W denotes the weighted matrix, X denotes the dual variable, T(u) denotes the Toeplitz matrix composed of the iterative optimization variable u, η denotes the regularization parameter, and ||·|| F Represents the Fourier norm; The form of the weighting matrix W is determined by the introduced sparsity metric criterion; A new sparsity metric is introduced to reduce the number of iterations required for RAM, namely: Therefore, according to equation (9), the weighted matrix W = ε / (T(u) + εI) 2 Then, the recovered echo signal S can be obtained by solving the SDP optimization equation in equation (7); When using the ADMM method, the sparse ISAR echo matrix obtained based on equation (7) and step 1) is... Construct the Lagrange augmented equation: Where α and β are regularization parameters, and <,> denote inner product operations; matrix matrix They are respectively: in, If the initial matrices of Λ and Z are zero matrices, then the ADMM update solution formula is as follows: Where D represents a diagonal matrix, and the diagonal elements S Ωc This represents missing echo data; Ωc represents the complement of the index set Ω of N pulses; T·(·) represents mapping the matrix to a vector, where the Nth pulse in the vector... i The elements are the summation and average of the diagonal elements of the matrix; furthermore, the iterative update expression for matrix Z is as follows: Equation (17) is usually converted to a Hermitian matrix: Retain positive eigenvalues, that is: Z i+1 =V i diag((d i ) + (v) i ) H (19) Where d and v represent the eigenvalues ​​and eigenvectors of the Hermitian matrix, respectively. + This indicates that positive eigenvalues ​​are retained; The recovered echo signal S is obtained by iteratively solving equations (12), (13), (14), (15), (16) and (19).

2. The high-resolution sparse ISAR imaging method based on the improved reweighted atomic norm according to claim 1, characterized in that: In step 1), the sparse ISAR echo matrix of M range cells is obtained. The method is: First, the complete echo vector of the m-th distance cell is represented as: in, This represents the complete echo vector of the m-th distance cell in an N×M dimension. Let t represent the complex space, where t = [0:N]. T Δt, [·] T This represents finding the transpose of a matrix, where Δt represents the pulse repetition time, N represents the number of pulses, K represents the number of scattering points, and f k σ represents the Doppler frequency at the k-th scattering point. k Indicates the amplitude of the scattering point; Based on equation (1), the complete echo vectors of the M range cells are combined to obtain an N×M dimensional echo matrix: Meanwhile, echo matrix S 0 It can also be expressed as: in, σ k,m This represents the amplitude of the k-th scattering point, which appears in the m-th range cell, where m ≤ M; Since the sparse ISAR echo matrix of M range cells can be regarded as the above echo matrix S 0 The random sampling matrix is ​​given by the sparse ISAR echo matrix, which can be represented as follows: in This represents the index set of N pulses. Represents the number of random samples.

3. The high-resolution sparse ISAR imaging method based on the improved reweighted atomic norm as described in claim 1, characterized in that: In step 3), the method of repeating step 2) to obtain the echo matrix B of all recovered range cells and using inverse fast Fourier transform to obtain the final ISAR image is as follows: After integrating the echo matrices of the M recovered range cells, the echo matrix B of all range cells is obtained. Then, the final ISAR image is obtained using the inverse fast Fourier transform, as shown in the formula: I = ifft(B) (20) Here, ifft(·) represents the fast inverse Fourier transform operation.

Images