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

Abstract

The invention discloses a high-resolution sparse ISAR (inverse synthetic aperture radar) imaging method based on an improved reweighted atom norm. The method comprises the following steps: obtaining a sparse ISAR echo matrix # imgabs0 # of M distance units, based on the sparse ISAR echo matrix # imgabs1 #, using an MRAM-SDP method or an MRAM-ADMM method, and obtaining a recovered echo matrix S through multi-step iterative weighted solution; and obtaining a recovered echo matrix B of all distance units, and obtaining a final ISAR image by using inverse fast Fourier transform. The method has the advantages that the number of iteration weighting times of the reweighted atomic norm high-resolution sparse ISAR imaging method can be reduced. The sparse characteristic of scattering points in an imaging target in a Doppler domain is utilized, high-resolution imaging of a non-cooperative target under a sparse aperture signal is achieved according to continuous compressed sensing and a low-rank matrix recovery theory, the performance of a reweighted atomic norm high-resolution sparse ISAR imaging method is kept, and the operation complexity is greatly reduced.

Description

Technical Field

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

[0002] As a microwave detection method for all-day and all-weather, Inverse Synthetic Aperture Radar (ISAR) imaging can achieve high-resolution imaging of air, space and sea targets. The ISAR system achieves high resolution in range by transmitting and receiving large broadband signals, and uses the change in the viewing angle between the radar and the target to synthesize multi-pulse echo signals within the coherent processing interval (CPI) to achieve high resolution in azimuth. Usually, imaging can be achieved by using fast inverse Fourier transform on the received signal. However, in actual situations, the radar system is in a multi-task working state and cannot obtain a complete full aperture (FA) observation signal. If the fast inverse Fourier transform is used for the missing sparse aperture (SA) signal, the image will have a serious main lobe broadening problem.

[0003] With the technological progress in radar signal processing, sparse recovery technology has gradually become a popular method for reconstructing signals. Sparse ISAR imaging based on this technology utilizes the sparse characteristics of the target scattering center on the spatial plane. Through a suitable sparse recovery algorithm, it can achieve accurate reconstruction of sparse aperture signals, thereby completing high-resolution sparse ISAR imaging. At present, the traditional ISAR imaging method based on sparse recovery requires discretization of the imaging scene, which inevitably leads to the problem of grid mismatch, thereby reducing the quality of ISAR imaging. Although the existing reweighted atomic norm (RAM) high-resolution sparse ISAR imaging method can avoid the grid mismatch problem in sparse recovery, the method is highly complex and takes a long time to operate. In summary, how to reduce the reweighted atomic norm sparse ISAR imaging method has become a technical problem that needs to be solved urgently by those skilled in the art. Summary of the invention

[0004] In order to solve the above problems, the object of the present invention is to provide a high-resolution sparse ISAR imaging method based on modified reweighted atomic norm (MRAM).

[0005] In order to achieve the above object, the improved reweighted atomic norm high-resolution sparse ISAR imaging method provided by the present invention comprises the following steps performed in sequence:

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

[0007] 2) Based on the above sparse ISAR echo matrix Use the MRAM-SDP method or the MRAM-ADMM method to obtain the restored echo matrix S through multi-step iterative weighted solution;

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

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

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

[0011]

[0012] in, represents the complete echo vector of the mth distance unit in N×M dimensions, Represents the complex space, t = [0:N] T Δt, [·] T represents the transpose of the matrix, Δt represents the pulse repetition time, N represents the number of pulses, K represents the number of scattering points, and f k represents the Doppler frequency of the kth scattering point, σ k represents the amplitude of the scattering point;

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

[0014]

[0015] At the same time, the echo matrix S 0 It can also be specifically expressed as:

[0016]

[0017] in, σ k,m Represents the amplitude of the kth scattering point, and appears in the mth distance unit, 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 of , so the sparse ISAR echo matrix is ​​expressed as in represents the index set of N pulses, N represents the number of random samples,

[0019] In step 2), the sparse ISAR echo matrix Using the MRAM-SDP method or the MRAM-ADMM method, through multi-step iterative weighted solution, the method to obtain the restored echo matrix S is:

[0020] Defining weighted atom sets for:

[0021]

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

[0023]

[0024] Where inf{·} means solving the infimum. Since solving equation (5) is an NP-hard problem, it is necessary to convexly relax equation (5) into equation (6):

[0025]

[0026] When using the MRAM-SDP method, the weighted atomic norm minimization problem of the recovered echo signal S can be transformed into an SDP problem, and the local optimal solution is approached through multi-step iterative weighting. Let u j represents the jth iteration optimization variable, based on the ISAR echo matrix obtained in step 1). Then the SDP optimization equation for the j+1th iteration is:

[0027]

[0028]

[0029] Where tr(·) means finding the trace of the matrix, W means the weighting matrix, X means the dual variable, T(u) means the Toeplitz matrix composed of iterative optimization variables u, η means the regularization parameter, ||·|| F represents the Fourier norm;

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

[0031] A new sparse metric criterion is introduced to reduce the number of iterations required by RAM, namely:

[0032]

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

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

[0035]

[0036] Among them, α and β are regularization parameters, <,> represents the inner product operation; the matrix matrix They are:

[0037]

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

[0039]

[0040]

[0041] Where D is a diagonal matrix with diagonal elements , N i =1,2,...,N;S Ωc represents the missing echo data, Ωc represents the complement of the index set Ω of N pulses; T · (·) means mapping the matrix to a vector. The Nth i The diagonal elements of the matrix are summed and averaged; in addition, the iterative update expression of the matrix Z is as follows:

[0042]

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

[0044]

[0045] Keep the positive eigenvalues, that is:

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

[0047] where d and v represent the eigenvalue and eigenvector of the Hermitian matrix, respectively, (·) + Indicates retaining positive eigenvalues;

[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 restored echo matrix B of all range units and using inverse fast Fourier transform to obtain the final ISAR image is:

[0050] After integrating the echo matrices of multiple groups of M range units after recovery, the echo matrix B of all range units is obtained and the final ISAR image is obtained by using the inverse fast Fourier transform. The formula is:

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

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

[0053] The beneficial effects of the present invention are:

[0054] The number of iterative weightings of the reweighted atomic norm high-resolution sparse ISAR imaging method can be reduced. By utilizing the sparse characteristics of the scattering points in the imaging target in the Doppler domain, based on continuous compressed sensing and low-rank matrix recovery theory, high-resolution imaging of non-cooperative targets under sparse aperture signals is achieved, which not only maintains the performance of the reweighted atomic norm high-resolution sparse ISAR imaging method, but also greatly reduces the computational complexity. BRIEF DESCRIPTION OF THE DRAWINGS

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

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

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

[0058] Figure 4(a) and (b) are the comparison results of reconstruction error at different signal-to-noise ratios under 50% pulse data and reconstruction error at different pulse numbers under a signal-to-noise ratio of 5dB. DETAILED DESCRIPTION

[0059] In order to enable those skilled in the art to better understand the technical solution of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and the best embodiment. Based on the embodiments in the invention, all other embodiments obtained by ordinary technicians in the field without making creative work are within the scope of protection of the invention.

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

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

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

[0063]

[0064] in, represents the complete echo vector of the mth distance unit in N×M dimensions, Represents the complex space, t = [0:N] T Δt, [·] T represents the transpose of the matrix, Δt represents the pulse repetition time, N represents the number of pulses, K represents the number of scattering points, and f k represents the Doppler frequency of the kth scattering point, σ k represents the amplitude of the scattering point;

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

[0066]

[0067] At the same time, the echo matrix S 0 It can also be specifically expressed as:

[0068]

[0069] in, σ k,m Represents the amplitude of the kth scattering point, and appears in the mth distance unit, 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 of , so the sparse ISAR echo matrix is ​​expressed as in represents the index set of N pulses, represents the number of random samples,

[0071] 2) Based on the above sparse ISAR echo matrix Using the MRAM-SDP (reweighted atomic norm-semidefinite programming) method or the MRAM-ADMM (reweighted atomic norm-alternating direction multiplier method) method, a multi-step iterative weighted solution is performed to obtain the restored echo matrix S;

[0072] Defining weighted atom sets for:

[0073]

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

[0075]

[0076] Where inf{·} means solving the infimum. Since solving equation (5) is an NP-hard problem, it is necessary to convexly relax equation (5) into equation (6):

[0077]

[0078] When using the MRAM-SDP (reweighted atomic norm-semidefinite programming) method, the weighted atomic norm minimization problem of the restored echo signal S can be transformed into an SDP problem, and the local optimal solution is approached through multi-step iterative weighting. Let u j represents the jth iteration optimization variable, based on the ISAR echo matrix obtained in step 1). Then the SDP optimization equation for the j+1th iteration is:

[0079]

[0080]

[0081] Where tr(·) means finding the trace of the matrix, W means the weighting matrix, X means the dual variable, T(u) means the Toeplitz matrix composed of iterative optimization variables u, η means the regularization parameter, ||·|| F represents the Fourier norm;

[0082] The form of the weighted matrix W is determined by the introduced sparse metric criterion. The conventional RAM method generally adopts the following sparse metric criterion:

[0083]

[0084]

[0085] Among them, S H represents the conjugate transpose of the recovered echo matrix S, ε represents the sparse regularization parameter, when the sparse regularization parameter ε tends to positive infinity, the sparse metric represented by the sparse metric function is equivalent to the atomic norm, when the sparse regularization parameter ε tends to zero, the sparse metric is equivalent to the atomic l0 norm.

[0086] The present invention introduces a new sparse metric criterion to reduce the number of iterations required by RAM, namely:

[0087]

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

[0089] The ADMM method can further improve the computational efficiency. Based on equation (7) and step 1), the sparse ISAR echo matrix is ​​obtained: Construct the Lagrange augmented equation:

[0090]

[0091] Among them, α and β are regularization parameters, <,> represents the inner product operation; the matrix matrix They are:

[0092]

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

[0094]

[0095]

[0096] Where D is a diagonal matrix with diagonal elements N i =1,2,...,N;S Ωc represents the missing echo data, Ωc represents the complement of the index set Ω of N pulses; T · (·) means mapping the matrix to a vector. The Nth iThe diagonal elements of the matrix are summed and averaged; in addition, the iterative update expression of the matrix Z is as follows:

[0097]

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

[0099]

[0100] Keep the positive eigenvalues, that is:

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

[0102] where d and v represent the eigenvalue and eigenvector of the Hermitian matrix, respectively, (·) + Indicates retaining positive eigenvalues;

[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 restored echo matrix B of all range units and use inverse fast Fourier transform to obtain the final ISAR image.

[0105] After integrating the echo matrices of multiple groups of M range units after recovery, the echo matrix B of all range units is obtained and the final ISAR image is obtained by using the inverse fast Fourier transform. The formula is:

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

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

[0108] In order to verify the effect of the method of the present invention, the inventors conducted the following characterization experiments:

[0109] The experiment used Figure 1The real Yak-42 full-aperture data shown in the figure has the following radar parameters: the radar operating frequency is 5.52 GHz, the pulse repetition frequency is 100 Hz, the signal bandwidth is 400 MHz, the number of full-aperture data pulses used in the experiment is 128, and the number of sparse apertures is 64. In the experiment, the initialization sparse regularization parameter ε is set to 1, and each iteration is half-decayed, the number of iterations of the RAM method is set to 6, and the number of iterations of the method of the present invention is set to 4. The simulation experiment compares the ANM method and the RAM method. The sparse ISAR imaging results under low signal-to-noise ratio and low pulse number are compared.

[0110] Experiment 1: Low signal-to-noise ratio experimental verification. Figure 2 The imaging results of different methods are shown when the signal-to-noise ratio is set to 5dB using 50% pulse data. Although the ANM method considers the imaging target in the continuous domain, its imaging performance is not as good as the RAM and MRAM methods due to the influence of convex relaxation. The non-convex surrogate function makes up for the gap with the norm, and the imaging performance of the RAM and MRAM methods is significantly better than that of the ANM method for the weaker scattering center at the head of the aircraft. In addition, the fast solution method based on ADMM does not result in any loss in imaging performance.

[0111] Experiment 2: Experimental verification with low pulse number. Figure 3 The imaging results of different methods are shown when the signal-to-noise ratio is set to 7.5dB using 30% pulse data. It is obvious that when the number of apertures is limited, the influence of convex relaxation is more obvious. There are a large number of erroneous scattering points in the imaging results of the ANM method, and the imaging quality is low. The RAM and MRAM methods based on the reweighted mode effectively enhance the sparsity of ANM, and the imaging results have good focusing. The image quality obtained is significantly better than that of 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 0 || F / ||B 0 || F The imaging performance of different methods is measured by the computing time, where B 0 represents the original full-aperture data, and B represents the full-aperture data reconstructed using SA data.

[0113] Figure 4 (a) shows the reconstruction error at different signal-to-noise ratios under 50% pulse data. Figure 4 (b) shows the reconstruction error for different pulse numbers at a signal-to-noise ratio of 5 dB. It can be seen that the mean square errors of the RAM and MRAM methods are lower than that of ANM when the signal-to-noise ratio and the number of apertures change, and the mean square error curves of the 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. From the numerical comparison, it can be seen that the computational efficiency of the SDP method is effectively improved compared with the RAM method, and the ADMM method only requires half the computational time of the SDP method.

[0115] The present 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 greatly reduces the computational complexity while obtaining the high-precision reconstruction performance of the RAM imaging method.

[0116] The above is only a preferred embodiment of the present invention. It should be pointed out that for ordinary technicians in this technical field, several improvements and modifications can be made without departing from the principle of the present invention. These improvements and modifications should also be regarded as 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 high-resolution sparse ISAR imaging method comprises the following steps performed in sequence: 1) Obtain a sparse ISAR echo matrix of M range cells 2) Based on the above sparse ISAR echo matrix Use the MRAM-SDP method or the MRAM-ADMM method to obtain the restored echo matrix S through multi-step iterative weighted solution; 3) Repeat step 2) to obtain the restored echo matrix B of all range units and use inverse fast Fourier transform to obtain the final ISAR image.

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 units is obtained. The method is: First, the complete echo vector of the m-th distance unit is expressed as: in, represents the complete echo vector of the mth distance unit in N×M dimensions, Represents the complex space, t = [0:N] T Δt, [·] T represents the transpose of the matrix, Δt represents the pulse repetition time, N represents the number of pulses, K represents the number of scattering points, and f k represents the Doppler frequency of the kth scattering point, σ k represents the amplitude of the scattering point; Based on formula (1), the complete echo vectors of M range units are combined together to obtain an N×M dimensional echo matrix: At the same time, the echo matrix S 0 It can also be specifically expressed as: in, σ k,m Represents the amplitude of the kth scattering point, and appears in the mth distance unit, 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 of , so the sparse ISAR echo matrix is ​​expressed as in 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 according to claim 1, characterized in that: In step 2), the sparse ISAR echo matrix Using the MRAM-SDP method or the MRAM-ADMM method, through multi-step iterative weighted solution, the method to obtain the restored echo matrix S is: Defining weighted atom sets for: Where w(f) represents the weight function of atom a(f); based on formula (4), the weighted atomic l0 norm of the restored echo matrix S can be obtained: Where inf{·} means solving the infimum. Since solving equation (5) is an NP-hard problem, it is necessary to convexly relax equation (5) into equation (6): When using the MRAM-SDP method, the weighted atomic norm minimization problem of the recovered echo signal S can be transformed into an SDP problem, and the local optimal solution is approached through multi-step iterative weighting. Let u j represents the jth iteration optimization variable, based on the ISAR echo matrix obtained in step 1). Then the SDP optimization equation for the j+1th iteration is: Where tr(·) means finding the trace of the matrix, W means the weighting matrix, X means the dual variable, T(u) means the Toeplitz matrix composed of iterative optimization variables u, η means the regularization parameter, ||·|| F represents the Fourier norm; The form of the weight matrix W is determined by the introduced sparse metric criterion; A new sparse metric criterion is introduced to reduce the number of iterations required by RAM, namely: Then, according to formula (9), we can get the weight matrix W = ε / (T(u)+εI) 2 , and then the recovered echo signal S can be obtained by solving the SDP optimization equation in equation (7); When the ADMM method is used, the sparse ISAR echo matrix obtained based on equation (7) and step 1) is Construct the Lagrange augmented equation: Among them, α and β are regularization parameters, <,> represents the inner product operation; the matrix matrix They are: in, The initial matrices of Λ and Z are zero matrices, and the ADMM update solution formula is as follows: Where D is a diagonal matrix with diagonal elements N i =1,2,…,N;S Ωc represents the missing echo data, Ωc represents the complement of the index set Ω of N pulses; T · (·) means mapping the matrix to a vector. The Nth i The diagonal elements of the matrix are summed and averaged; in addition, the iterative update expression of the matrix Z is as follows: Equation (17) is usually converted to a Hermitian matrix: Keep the positive eigenvalues, that is: where d and v represent the eigenvalue and eigenvector of the Hermitian matrix, respectively, (·) + Indicates retaining positive eigenvalues; The recovered echo signal S is obtained by iteratively solving equations (12), (13), (14), (15), (16) and (19).

4. The improved reweighted atomic norm high-resolution sparse ISAR imaging method according to claim 1, characterized in that: In step 3), the method of repeating step 2) to obtain the restored echo matrix B of all range units and using inverse fast Fourier transform to obtain the final ISAR image is: After integrating the echo matrices of multiple groups of M range units after recovery, the echo matrix B of all range units is obtained and the final ISAR image is obtained by using the inverse fast Fourier transform. The formula is: I=ifft(B) (20) where ifft(·) represents the fast inverse Fourier transform operation.

Images