A Two-Dimensional Sparse ISAR Imaging Method Based on Weighted Matrix Completion

Abstract

The present invention relates to a two-dimensional sparse ISAR imaging method, and particularly to a two-dimensional sparse ISAR imaging method based on weighted matrix completion, which comprises the following steps: constructing an ISAR echo imaging model of random sampling into a matrix containing singular values; adopting a weighting scheme to protect small singular values and constrain large singular values, and using an iterative weighting method based on the alternating direction method of multipliers (ADMM) to solve the matrix; when all data are recovered, obtaining the final ISAR image by using two-dimensional inverse fast Fourier transform (2D FFT). By constructing a matrix containing singular values and using the method of the weighting scheme to solve the matrix to obtain the final ISAR image, the present invention enables the missing data in the random sampling mode to be recovered with high precision, effectively reducing the data computation amount while improving the recovery accuracy of the missing data in the random sampling mode.

Description

Technical Field

[0001] The present invention relates to a two-dimensional sparse ISAR imaging method, and particularly to a two-dimensional sparse ISAR imaging method based on weighted matrix completion. Background Art

[0002] Inverse synthetic aperture radar (ISAR) can achieve all-weather high-resolution observation of moving targets and obtain high-resolution target images, and has been widely used in military and civilian fields. Generally, a radar improves the range resolution by transmitting a broadband signal and obtains high resolution in the azimuth direction by observing a moving target for a long time. The stepped-frequency waveform synthesizes a large bandwidth at the receiving end by transmitting a series of sub-pulses with continuously jumping carrier frequencies. High azimuth resolution can be obtained by transmitting multiple pulse trains. Compared with the broadband LFM signal, it has the advantages of easy implementation and low hardware requirements. Therefore, the ISAR imaging method based on the stepped-frequency waveform has become the focus of current research and has received extensive attention from scholars.

[0003] In general, the final two-dimensional ISAR imaging results can be quickly obtained through traditional RD methods. However, due to the multi-mode operating mode of the radar in practice and the interference of the external environment, it is inevitable that some sub-pulse signals are not transmitted or unavailable due to interference. At this time, the traditional RD method will have problems with the degradation of imaging quality. Introducing the compressive sensing theory (CS) into the field of ISAR imaging, by utilizing the unique sparse characteristics of the target, the original signal can be accurately restored in the case of sparse echoes, and high-precision ISAR imaging results can be obtained. For example, the literature [L. Zhang, Z. Qiao, M. Xing, J. Sheng, R. Guo, and Z. Bao, “High-resolution ISAR imaging by exploiting sparse apertures,” IEEE Trans. Antennas Propag., vol. 60, no. 2, pp. 997-1008, Feb. 2012.] studied the method of synthesizing range profiles of sparse stepped-frequency waveforms. To achieve two-dimensional joint imaging in range / azimuth, the literature [H. Wang, Y. Quan, M. Xing, and S. Zhang, “ISAR imaging via sparse probing frequencies,” IEEE Geosci. Remote Sens. Lett., vol. 8, no. 3, pp. 451–455, May 2011.] converted the two-dimensional sparse reconstruction problem into a classical one-dimensional sparse reconstruction model through vectorization operations, and then performed matrix operations on the reconstruction results to obtain the final two-dimensional image. In fact, most of the above-mentioned CS-based sparse ISAR imaging methods consider the block-sparse method, that is, the relative positions of missing sub-pulses in different pulse trains are the same, that is, a row or a column of data in the echo matrix is completely missing. The literature [W. Qiu, H. Zhao, J. Zhou, and Q. Fu, “High-resolution fully polarimetric ISAR imaging based on compressive sensing,” IEEE Trans. Geosci. Remote Sens., vol. 52, no. 10, pp. 6119-6131, Oct. 2014.] calls it the separable sampling model. However, due to the real-time changes in the external environment and the multi-mode operating mode of the radar, the positions of missing sub-pulses in each pulse train are not the same, that is, the missing sub-pulses are often randomly distributed in the echo data matrix, which we call the random sampling model (Random sampling model).It should be noted that due to the random distribution of the sparse positions of sub-pulses in the random sampling mode, it is impossible to directly construct the corresponding matrix-based joint sparse reconstruction model. Although the vectorization processing model is applicable to the above random sampling mode, the computational complexity problem brought by the vectorization operation will be the primary problem that must be considered. When the data dimension is large, there is even a situation where an ordinary computer cannot handle it.

[0004] To address the above problems, the present invention introduces the Matrix Completion (MC) theory into the field of two-dimensional ISAR imaging to solve the high-precision recovery of missing data in the random sampling mode. Summary of the Invention

[0005] Therefore, the present invention provides a two-dimensional sparse ISAR imaging method based on weighted matrix completion to overcome the problem that an ordinary computer cannot handle due to excessive data volume in the random sampling mode in the prior art.

[0006] To achieve the above object, the present invention provides a two-dimensional sparse ISAR imaging method based on weighted matrix completion, which is characterized by including:

[0007] Step S1, constructing the ISAR echo imaging model of random sampling into a matrix containing singular values;

[0008] Step S2, using a weighting scheme to protect small singular values and constrain large singular values, and solving the matrix based on the iterative weighting method of the alternating direction multiplier method;

[0009] Step S3, when all data is recovered, using the two-dimensional inverse fast Fourier transform to obtain the final ISAR image, thereby highly precisely recovering the missing data in the random sampling mode.

[0010] Further, the echo imaging model decomposes the motion of the target into two components: translational motion and rotational motion, and performs motion compensation on the translational motion component to avoid defocusing of the ISAR imaging result;

[0011] At time t, the distance between the radar and the target scattering point (x k , y k ) is approximated by Equation (1) as:

[0012]

[0013] where (x k , y k ) is the position of the scattering point in the target coordinate system, R o is the initial distance between the target center and the radar, ω is the equivalent rotational angular velocity of the target, and T r is the pulse repetition time;

[0014] For a stepped-frequency ISAR system, the transmitted waveform includes Na groups of pulse trains, and each group of pulse trains contains N sub-pulses. Assuming the bandwidth of each sub-pulse is Δf and the synthesized bandwidth is denoted as NΔf. At this time, the echo of the nth sub-pulse in the nth group of pulse trains can be approximated by Equation (2) as follows: a The echo of the nth sub-pulse in the nth group of pulse trains is approximated by Equation (2) as:

[0015]

[0016] δ k is the intensity of the kth scattering point, where δ k , k = 1, 2, 3,..., K, f0 is the center frequency, c is the speed of light. Substituting Equation (1) into Equation (2), Equation (3) can be obtained:

[0017]

[0018] According to the ISAR imaging resolution, the range resolution is Δy and the azimuth resolution is Δx. It is set that Δy = c / 2NΔf, Δx ≈ c / 2f0ωn a NT r ; At this time, the echo signal shown in Equation (3) is expressed by Equation (4) as:

[0019]

[0020] Writing Equation (4) in matrix form, Equation (5) can be obtained:

[0021] S = AXB T (5)

[0022] where is the echo data matrix, is the range dictionary matrix, is the azimuth dictionary matrix, is the two-dimensional imaging result.

[0023] Furthermore, in the random sampling mode, ⊙ is set as the Hadamard product, and a matrix containing only 0 and 1 elements is initially constructed. After the corresponding data random sampling rule is processed by vectorization, the data extraction process can be Equation (6):

[0024] Θ⊙S = Θ⊙(AXB T )(6)

[0025] After vectorization processing, Equation (6) can be Equation (7):

[0026] Θ⊙S = Θ⊙(AXB T ) → s = Φx(7)

[0027] where vec(·) is the vectorization operation, diag(·) is the diagonal matrix, is the Kronecker product,

[0028] P ≥ N, Q ≥ Na;

[0029] Based on the sparse prior information of the target, the target scenario x can be transformed into Equation (8):

[0030]

[0031] where ||x||0 is the L0 norm, || || F is the Frobenius norm, and ξ is a constant related to the noise level.

[0032] Furthermore, when the echo data matrix contains K scattering points in the target, Equation (3) can be Equation (9):

[0033]

[0034] where a n,k = exp(-j4πnΔfy k / c), b 2,k = exp(j2πf0x k ωn a NT r / c),

[0035] According to the characteristics of the low-rank matrix, the rank of the echo matrix satisfies Equation (10):

[0036]

[0037] For the sparse echo data, it can be fitted as a random sparse sampling of the full data S. Let Γ be the set of positions of the sampled data, then the sparse echo data Z can be Equation (11):

[0038]

[0039] If the matrix has the low-rank property, the recovery of the full data S can be simplified to Equation (12):

[0040]

[0041] where P Γ (·) is the element operation at the corresponding position of the matrix;

[0042] Relax Equation (12) to Equation (13):

[0043]

[0044] where is the nuclear norm of S, and σ i is the i-th eigenvalue of S, with σ1≥σ2≥...≥σ i ≥...≥0.

[0045] Furthermore, the nuclear norm of S can be regarded as the norm of the L0 singular values of the data matrix S. By setting different eigenvalues with different thresholds, the weighted nuclear norm optimization model of Equation (14) is obtained:

[0046]

[0047] where ω i is the weight corresponding to the singular value σ i . Let i = 1, 2, 3,..., L and 0≤ω1≤...≤ω i ≤...≤ω L . When setting the weights, if 0.3σ1≥σ i , then the weight ω i satisfies ω i ≥0.7ω L . If 0.7σ1≤σ i , then the weight ω i satisfies ω i ≤0.3ω L . ω i has no linear relationship with σ i , so that while reducing the over-penalty for large singular values, small singular values can be better protected;

[0048] Considering the influence of noise, Equation (14) can be relaxed to Equation (15):

[0049]

[0050] where ||g|| F is the F norm, E is the noise matrix, and λ is the regularization parameter;

[0051] For the convenience of solving, an auxiliary variable X is introduced, and then Equation (15) can be equivalently written as Equation (16):

[0052]

[0053] For Equation (16), the augmented Lagrangian function is constructed as Equation (17):

[0054]

[0055] where is the Lagrange multiplier and μ is the penalty parameter.

[0056] Furthermore, for the data matrix S, the formula (17) can be transformed into formula (18):

[0057]

[0058] where H = X + Y2 / μ, and the optimal solution of S is formula (19):

[0059]

[0060] where H = U∑V H is the singular value decomposition of matrix H, Diag{g} is the symbol of the diagonal matrix, and (g) + is to take the elements greater than zero.

[0061] Furthermore, the noise matrix E can be transformed into formula (20):

[0062]

[0063] The optimal solution of E in formula (20) is formula (21):

[0064]

[0065] Furthermore, the auxiliary variable X is equivalent to formula (22):

[0066]

[0067] In formula (22), there are sampling values and sparse values, which need to be updated separately, and the update formula can be written as formula (23):

[0068]

[0069] where Γ′ is the complement of the set Γ.

[0070] Furthermore, the weight vector W = [ω1, ω2,..., ω I can be written as formula (24):

[0071]

[0072] where α is a positive constant, is a very small positive number.

[0073] Furthermore, the iterative optimization includes the following process:

[0074] Input: Γ,

[0075] Initialization quantity: i = 1, 2, 3, ..., I, λ > 0, μ > 0, l = 1, ρ > 0. The iterative process is as follows.

[0076] Step T1

[0077] Step T2

[0078] Step T3

[0079] Step T4

[0080] Step T5

[0081] Step T6

[0082] Step T7, Y1 l+1 = Y1 l + μ(P Γ (X l+1 ) + E l+1 - Z)(31);

[0083] Step T8

[0084] Step T9, μ l+1 = ρυ l (33);

[0085] Step T10, l = l + 1 (34);

[0086] The iteration ends.

[0087] Output the results: S, X.

[0088] Compared with the prior art, the beneficial effects of the present invention are as follows. The present invention provides a two-dimensional sparse ISAR imaging method based on weighted matrix completion. By constructing the randomly sampled ISAR echo imaging model into a matrix containing singular values and using a weighted scheme based on the iterative weighted method of the alternating direction method of multipliers (ADMM) to solve the matrix, and adopting the traditional two-dimensional inverse fast Fourier transform (2D FFT) to obtain the final ISAR image, the missing data in the random sampling mode is accurately restored. While effectively reducing the data operation amount, the restoration accuracy of the missing data in the random sampling mode is improved.

[0089] Furthermore, the present invention sets weights for the singular values and uses the weights to protect the singular values, so as to reduce the over-penalty of the large singular values while better protecting the small singular values, and further improves the accuracy of data restoration.

[0090] Furthermore, the present invention effectively reduces the computational workload of the computer by optimizing the echo data matrix S, the noise matrix E, and setting the auxiliary variable X. Thus, while effectively reducing the data computational workload, the recovery accuracy of the missing data in the random sampling mode is further improved. BRIEF DESCRIPTION OF THE DRAWINGS

[0091] Figure 1 is a flowchart of the method of the present invention;

[0092] Figure 2 is a schematic diagram of the ISAR imaging turntable model of the method of the present invention;

[0093] Figure 3 is a schematic diagram of the sparse measurement matrix of the method of the present invention;

[0094] Figure 4 is an imaging result diagram obtained by a conventional method for the echo data matrix of the method of the present invention;

[0095] Figure 5 is a comparison of imaging results of different algorithms under different SPR conditions for the method of the present invention;

[0096] Figure 6 is a performance comparison of different methods under different SPR conditions for the method of the present invention;

[0097] Figure 7 is a comparison of imaging results of different algorithms under different signal-to-noise ratio conditions for the method of the present invention;

[0098] Figure 8 is a performance comparison of different methods under different signal-to-noise ratio conditions for the method of the present invention.

[0099] SPECIFIC EMBODIMENTS

[0100] In order to make the objectives and advantages of the present invention more clear, the present invention will be further described below in conjunction with embodiments; it should be understood that the specific embodiments described herein are only used to explain the present invention and are not used to limit the present invention.

[0101] The preferred embodiments of the present invention will be described below with reference to the accompanying drawings. Those skilled in the art should understand that these embodiments are only used to explain the technical principles of the present invention and do not limit the protection scope of the present invention.

[0102] First, the professional terms used in the embodiments are explained.

[0103] ISAR: The full Chinese name is: Inverse Synthetic Aperture Radar, and the full English name is: Inverse Synthetic Aperture Radar. The inverse synthetic aperture radar can perform high-resolution imaging on distant targets, detect targets with a large bandwidth transmitted signal, and high range resolution can provide more detailed target features.

[0104] CS: The full Chinese name is: Compressive Sensing, and the full English name is: Compressive Sensing. The compressive sensing method abandons the redundant information in current signal sampling. CS directly obtains compressed samples from continuous-time signals and then uses optimization methods to process the compressed samples in digital signal processing.

[0105] SNR: The full Chinese name is: Signal Noise Ratio, and the full English name is: Signal Noise Ratio.

[0106] SVT: The full Chinese name is: Singular Value Thresholding Algorithm, and the full English name is: Singular Value Thresholding Algorithm

[0107] IALM: The full Chinese name is: Inexact Augmented Lagrange Multiplier Method, and the full English name is: Inexact Augmented lagrange multiplier

[0108] IRNN: The full Chinese name is: Recurrent Neural Network Method, and the full English name is: infinitely recurrent neural network

[0109] Please refer to Figure 1 as shown, which is the flow chart of the method of the present invention, including:

[0110] Step S1, constructing the randomly sampled ISAR echo imaging model into a matrix containing singular values;

[0111] Step S2, using a weighting scheme to protect small singular values and constrain large singular values, and solving the matrix based on the iterative weighted method of the alternating direction multiplier method;

[0112] Step S3, when all data is recovered, using two-dimensional inverse fast Fourier transform to obtain the final ISAR image, so as to accurately recover the missing data in the random sampling mode.

[0113] Please refer to Figure 2 as shown, in this embodiment, the echo imaging model decomposes the movement of the target into two components: translational and rotational. Among them, the rotational component of the target is the key to azimuth focusing, while the translational component will cause defocusing of the ISAR imaging result and motion compensation is required;

[0114] At time t, the distance between the radar and the target scattering point (x k , y k ) can be approximated as:

[0115]

[0116] where (x k , y k ) is the position of the scattering point in the target coordinate system, R o is the initial distance between the target center and the radar, ω is the equivalent rotational angular velocity of the target, and T r is the pulse repetition time;

[0117] For a stepped-frequency ISAR system, its transmitted waveform generally consists of Na groups of pulse trains, and each group of pulse trains contains N sub-pulses. Assuming the bandwidth of each sub-pulse is Δf, the final synthesized bandwidth is NΔf. At this time, the echo of the n a th sub-pulse in the nth group of pulse trains can be approximated as:

[0118]

[0119] where δ k , k = 1, 2, 3,..., K is the intensity of the kth scattering point, f0 is the center frequency, c is the speed of light. Substituting the formula (1) into the formula (2), we can obtain:

[0120]

[0121] According to the ISAR imaging resolution, the range resolution and azimuth resolution can be Δy = c / 2NΔf and Δx ≈ c / 2f0ωn a NT r . At this time, the echo signal shown in the formula (3) can be:

[0122]

[0123] Writing the above echo model in matrix form gives:

[0124] S = AXB T (5)

[0125] where is the echo data matrix, are the range and azimuth dictionary matrices respectively, is the two-dimensional imaging result.

[0126] Please refer to Figure 3As shown, in the random sampling mode, a matrix containing only 0 and 1 elements is initially constructed. Corresponding to the random sampling law of the data and after vectorization processing, let ⊙ be the Hadamard product. The extraction process of this data can be as follows:

[0127] Θ⊙S = Θ⊙(AXB T )(6)

[0128] After vectorization processing, Equation (6) can be:

[0129] Θ⊙S = Θ⊙(AXB T )→s = Φx(7)

[0130] vec(·) is the vectorization operation, P≥N, Q≥Na, diag(·) is the diagonal matrix, is the Kronecker product,

[0131] Based on the sparse prior information of the target, the target scene x can be transformed into the following sparse optimization problem for solution:

[0132]

[0133] where ||x||0 is the L0 norm, || || F is the Frobenius norm, and ξ is a constant related to the noise level.

[0134] Specifically, when the target contains K scattering points in the echo data matrix, Equation (3) can be,

[0135]

[0136] where a n,k = exp(-j4πnΔfy k / c), b 2,k = exp(j2πf0x k ωn a NT r / c),

[0137] According to the characteristics of the low-rank matrix, the rank of the echo matrix satisfies,

[0138]

[0139] Obviously, the above inequality shows that the rank of the echo matrix is less than the number of target scattering points K, which also verifies that the echo data has the low-rank characteristic;

[0140] For sparse echo data, it can be regarded as random sparse sampling of the full data S; assuming that Γ is the set of positions of the sampled data, the sparse echo data Z can be:

[0141]

[0142] If the matrix has the low-rank property, the full data recovery problem can be the following low-rank recovery problem:

[0143]

[0144] where P Γ (·) is the operation of taking the elements at the corresponding positions of the matrix;

[0145] The low-rank problem shown in the above formula (12) is difficult to solve in practice. Therefore, in general, the formula (12) is relaxed into the following nuclear norm minimization problem for solution.

[0146]

[0147] where is the nuclear norm of S, σ i is the i-th eigenvalue of S, and σ1≥σ2≥...≥σ i ≥...≥0.

[0148] For the above nuclear norm minimization model, the SVT (Singular Value Thresholding) method can be used for solution. When the full data echo matrix is obtained, the final ISAR imaging result can be quickly obtained by using the traditional RD algorithm.

[0149] Specifically, the nuclear norm of S can be regarded as the L0 norm of the singular values of the data matrix S. However, the SVT method treats the singular values with the same threshold, which is disadvantageous for small singular values; by setting different thresholds for different eigenvalues, the following weighted nuclear norm optimization model is obtained:

[0150]

[0151] where 0≤ω1≤...≤ω i ≤...≤ω L is the weight corresponding to the singular value σ i When setting the weights, for large singular values, small weights are selected, and for small singular values, large weights are selected; in this way, while reducing the excessive penalty for large singular values, small singular values can be better protected.

[0152] By setting weights for the singular values and using the weights to protect the singular values, while reducing the transition penalty for large singular values, it better protects small singular values and further improves the accuracy of data recovery.

[0153] Considering the influence of noise, Equation (14) can be relaxed into the following nuclear norm minimization problem for solution.

[0154]

[0155] where ||g|| F is the Frobenius norm, E is the noise matrix, and λ is the regularization parameter, which is used to balance the proportion of low rank and noise in the optimization problem;

[0156] For the above optimization problem, the alternating direction method of multipliers (ADMM) can be used to solve it. The idea of ADMM for optimizing the above problem is: under the condition of keeping other variables unchanged, iteratively optimize the variables to be updated, and then use the optimized results to optimize and update other variables. For the convenience of solution, an auxiliary variable X is introduced, then the above optimization model can be equivalently written as

[0157]

[0158] For Equation (16), the augmented Lagrangian function is constructed as

[0159]

[0160] where is the Lagrange multiplier and μ is the penalty parameter.

[0161] Specifically, for Equation (17), for the data matrix S, it can be transformed into the following problem for update:

[0162]

[0163] where H = X + Y2 / μ. The above optimization problem can actually be regarded as a weighted Frobenius norm minimization problem, and the optimal solution of S can be directly obtained as:

[0164]

[0165] where H = U∑V H is the singular value decomposition of matrix H, Diag{g} is the symbol of the diagonal matrix, and (g) + takes the elements greater than zero.

[0166] Specifically, the noise matrix E can be solved as follows, that is:

[0167]

[0168] The optimal solution of E in the formula (20) is as follows:

[0169]

[0170] Specifically, the auxiliary variable X can be equivalent to:

[0171]

[0172] In the formula (22), since there are two parts: the sampled value and the sparse value, they need to be updated separately, and the corresponding update formula can be written as:

[0173]

[0174] where Γ′ is the complement set of Γ, that is, the position set of the sparse echo signal.

[0175] Specifically, the weight vector W = [ω1, ω2,..., ω I can be calculated by the following formula:

[0176]

[0177] where α is a positive constant, is a very small positive number used to avoid the appearance of extremely large weights.

[0178] By optimizing the echo data matrix S, the noise matrix E, and setting the auxiliary variable X, the computational load of the computer is effectively reduced. Thus, while effectively reducing the data computational load, the recovery accuracy of the missing data in the random sampling mode is further improved.

[0179] Specifically, the iterative optimization includes the following process:

[0180] Input: Γ,

[0181] Initialization: i = 1, 2, 3,..., I, λ > 0, μ > 0, l = 1, ρ > 0, and the iterative process is as follows:

[0182] Step T1,

[0183] Step T2,

[0184] Step T3,

[0185] Step T4,

[0186] Step T5,

[0187] Step T6,

[0188] Step T7, Y1 l+1 = Y1 l + μ(P Γ (X l+1 ) + E l+1 - Z)(31);

[0189] Step T8,

[0190] Step T9, μ l+1 = ρυ(33); l (33);

[0191] Step T10, l = l + 1(34);

[0192] Iteration ends,

[0193] Output result: S, X.

[0194] The present invention provides a two - dimensional sparse ISAR imaging method based on weighted matrix completion. By constructing the randomly sampled ISAR echo imaging model into a matrix containing singular values, and using a weighted scheme based on the iterative weighted method of the alternating direction method of multipliers (ADMM) to solve the matrix, and adopting the traditional two - dimensional inverse fast Fourier transform (2D FFT) to obtain the final ISAR image, the missing data in the random sampling mode is accurately restored with high precision. While effectively reducing the data operation volume, the restoration accuracy of the missing data in the random sampling mode is improved.

[0195] Please refer to Figures 4-5 As shown, this embodiment uses measured data to verify the effectiveness of the method. The data is obtained by a radar with a bandwidth of 400 MHz and a pulse repetition period of 0.01 s. It contains 256 pulse trains, and each pulse train contains 256 sub - pulses. For such actual data, the dimension of the dictionary matrix corresponding to the traditional vectorized CS method is as high as 2562×2562, and the length of the signal vector is 2562. Such data dimensions are difficult to solve on an ordinary computer. Therefore, the results obtained by the vectorized CS method will not be shown in the following experiments. For comparison, Figure 3 the imaging results obtained by the conventional 2D FFT are given, where the data adopts the random sampling mode and zero - filling operation is performed before the sparse data imaging process. It can be seen from the imaging results that the image is submerged in the background of false reconstruction, which means the low - resolution performance of the 2D FFT method under sparse data conditions.

[0196] Specifically, we present the imaging results of the proposed method under different sampling rates (SPR) and signal-to-noise ratios (SNR). SPR is defined as SPR = M / NNa, where M is the measurement data. In addition, the mean square error (MSE) is used to measure the performance of the imaging results, and the calculation method is where is the recovered complete data. In addition, the SVT method [J.F. Cai, J.C. Emmanuel, and Z.W Shen, “A singular value thresholding algorithm for matrix completion,” SIAM J. OPTIM., vol. 20, no. 4, pp. 1956–1982, Mar. 2010.], the IALM method [Z.C. Lin, M.M. Chen, L.Q. Wu, and Y. Ma, “The augmented lagrange multiplier method for exact recovery of corrupted low-rank matrices,” [Online]. Available: https: / / arxiv.org / abs / 1009.5055.] and the IRNN method [C.Y. Lu, J.H. Tang, S.C. Yan, and Z.C. Lin, “Nonconvex nonsmooth low-rank minimization via iteratively reweighted nuclear norm,” IEEE Trans. Image Process., vol. 25, no. 2, pp. 829–839, Feb. 2016.] are also used for comparison. First, the imaging results under three SPR conditions are shown as Figure 5 shown, with the SNR set to 10 dB.

[0197] As can be seen from the imaging results, the images obtained by the SVT method contain more false reconstructed scattering points. Especially at low sampling rates (i.e., SPR = 0.3), the images are contaminated by a large number of false reconstructed scattering points. This is because it only uses singular value decomposition to process singular values. In contrast, the other two low-rank based methods can obtain cleaner images with fewer pseudo-scattering points in most cases. However, the reconstruction performance is also affected by missing data. Especially when SPR is 0.3, the noise floor of the reconstructed image is significantly higher than other cases. Generally speaking, compared with other methods, the images obtained by the proposed RMaC are cleaner, better focused, and have a lower noise floor. More notably, when SPR is as low as 0.3, only a small number of false scattering points are generated in the obtained images, further verifying its robustness under low sampling conditions.

[0198] Please refer to Figure 6 shown, when the SNR is 10 dB, the corresponding reconstruction errors and running times when SPR varies from 0.1 to 0.9. The error situations of different algorithms are as Figure 6 (a) shown, and the corresponding running times are as Figure 6 (b) shown (AMD Ryzen 9 4900H @ 3.30 GHz). It can be clearly seen that the SVT method has the highest MSE value in all given cases, indicating its worst imaging performance. Additionally, except for the cases with relatively high SPR, IRNN and RMaC have the best and similar imaging performances. The reason is that both algorithms adopt the same weighting strategy when selecting singular values. However, the calculation time shows that IRNN requires the longest running time, which is more time-consuming than other methods. Due to the fast convergence ability of ADMM, RMaC only needs less than 1.5 seconds, which verifies the higher computational efficiency and imaging performance of RMaC.

[0199] Please refer to Figure 7 shown, which compares the performance of four methods under different SNR conditions, where SPR is fixed at 0.5. At the same time, Figure 8 (a) and Figure 8 (b) respectively give the relationships between the MSE curve and the running time and SNR. The simulation results show that the images obtained by SVT and IALM have poor focusing performance under low signal-to-noise ratio conditions. Especially when the signal-to-noise ratio is as low as 3 dB, the reconstructed images contain a large amount of noise floor. Similarly, IRNN and RMaC achieve good focusing under the given SNR conditions and have the fewest false reconstructed points. Additionally, under low signal-to-noise ratio conditions, the performance of RMaC is slightly better than that of IRNN, indicating its robustness to noise. Furthermore, the running time curve shows that RMaC is close to SVT, further demonstrating the high computational efficiency of the proposed algorithm.

[0200] So far, the technical solution of the present invention has been described in conjunction with the preferred embodiments shown in the accompanying drawings. However, those skilled in the art can easily understand that the protection scope of the present invention is obviously not limited to these specific embodiments. Without departing from the principle of the present invention, those skilled in the art can make equivalent changes or substitutions to the relevant technical features, and the technical solutions after these changes or substitutions will fall within the protection scope of the present invention.

[0201] The above are only the preferred embodiments of the present invention and are not intended to limit the present invention. For those skilled in the art, the present invention can have various changes and modifications. Any modification, equivalent substitution, improvement, etc. made within the spirit and principle of the present invention shall be included in the protection scope of the present invention.

Claims

1. A two-dimensional sparse ISAR imaging method based on weighted matrix completion, characterized in that Including: Step S1: Construct the ISAR echo imaging model of random sampling into a matrix containing singular values; Step S2: Adopt a weighting scheme to protect small singular values and constrain large singular values, and use an iterative weighted method based on the alternating direction multiplier method to solve the matrix; Step S3: When all data is restored, use the two-dimensional inverse fast Fourier transform to obtain the final ISAR image, so as to accurately restore the missing data in the random sampling mode; The nuclear norm of the full data echo matrix S is regarded as the L0 norm of the singular values of S. Set different eigenvalues to use different thresholds to obtain the weighted nuclear norm optimization model formula (1): where, w i is the weight corresponding to the singular value σ i of S. Let i = 1, 2, 3,..., I and 0 ≤ w1 ≤... ≤ w i ≤... ≤ w L . When setting the weights, if 0.3σ1 ≥ σ i , then the weight w i satisfies w i ≥ 0.7w L . If 0.7σ1 ≤ σ i , then the weight w i satisfies w i ≤ 0.3w L . w i has no linear relationship with σ i . The formula (1) is relaxed to formula (2): where ||·|| F is the F-norm, E is the noise matrix, and λ is the regularization parameter; Introduce an auxiliary variable X, then the formula (2) is equivalent to formula (3): For the formula (3), construct the augmented Lagrangian function as formula (4): where is the Lagrange multiplier and μ is the penalty parameter.

2. The two-dimensional sparse ISAR imaging method based on weighted matrix completion according to claim 1, wherein The echo imaging model decomposes the motion of the target into two components of translation and rotation, and compensates the translation component to avoid defocusing of the ISAR imaging result; The distance between the radar and the target scattering point (x k , y k ) at time t is approximated by Equation (5) as follows: where (x k , y k ) is the position of the scattering point in the target coordinate system, R o is the initial distance between the target center and the radar, ω is the equivalent rotational angular velocity of the target, T r is the pulse repetition time; For a stepped-frequency ISAR system, the transmitted waveform includes Na groups of pulse trains, and each group of pulse trains contains N sub-pulses. Assuming the bandwidth of each sub-pulse is Δf and the synthesized bandwidth is denoted as NΔf, at this time, the echo of the nth sub-pulse in the nth a group of pulse trains can be approximated by Equation (6) as follows: δ k is the intensity of the k-th scattering point, where δ k , k = 1, 2, 3, ..., K, f0 is the center frequency, c is the speed of light. Substituting the said formula (5) into the said formula (6), formula (7) is obtained: According to the ISAR imaging resolution, the range resolution is Δy and the azimuth resolution is Δx. Set Δy = c / 2NΔf and Δx ≈ c / 2f0ωn a NT r ; At this time, the echo signal shown in the formula (7) is as follows through the formula (8): The formula (8) is written in matrix form to obtain formula (9): S = AGB T (9) wherein is the full data echo matrix, is the range dictionary matrix, is the azimuth dictionary matrix, is the two-dimensional imaging result.

3. The two-dimensional sparse ISAR imaging method based on weighted matrix filling according to claim 2, wherein In the random sampling mode, set ⊙ as the Hadamard product, initially construct a matrix containing only 0 and 1 elements, corresponding to the random sampling rule of the data, and after vectorization processing, the extraction process of the data is formula (10): Θ⊙S = Θ⊙(AXB T )(10) After vectorization processing, the formula (10) is formula (11): Θ⊙S = Θ⊙(AXB T ) → s = Φx(11) where vec(·) is the vectorization operation, diag(·) is the operation of obtaining a diagonal matrix, is the Kronecker product, P ≥ N, Q ≥ Na; Based on the sparse prior information of the target, the target scene x is transformed into formula (12): where ||x||0 is the L0 norm of the target scenario x, ||·|| F is the F norm, and ξ is a constant related to the noise level.

4. The two-dimensional sparse ISAR imaging method based on weighted matrix filling according to claim 3, wherein When the full data echo matrix has K scattering points in the target, the formula (7) is formula (13): where a n,k = exp(-j4πnΔfy k / c), b na,k = exp(j2πf0x k ωn a NT r / c), According to the characteristics of the low-rank matrix, the rank of the echo matrix satisfies formula (14): For sparse echo data, it is fitted to the random sparse sampling of the full data echo matrix S. Let Γ be the set of positions of the sampled data, then the sparse echo data Z is formula (15): If the matrix has the low-rank property, the recovery of the full data S is simplified to formula (16): where P Γ (·) is the element operation at the corresponding position of the matrix; The formula (16) is relaxed to formula (17): wherein is the nuclear norm of the full data echo matrix S.

5. The two-dimensional sparse ISAR imaging method based on weighted matrix completion according to claim 4, wherein The formula (4), for the full data echo matrix S, is transformed into formula (18): Where H = X + Y2 / μ, and the optimal solution of S is formula (19): Among them, the singular value decomposition of H is performed to obtain U∑V H , diag(·) is an operation to obtain a diagonal matrix, and (·) + is to take the elements greater than zero.

6. The two-dimensional sparse ISAR imaging method based on weighted matrix filling according to claim 5, wherein The noise matrix E is transformed into formula (20): The optimal solution of E in the formula (20) is formula (21):

7. The two-dimensional sparse ISAR imaging method based on weighted matrix completion according to claim 6, characterized in that The auxiliary variable X is equivalent to formula (22): In the formula (22), there are sampled values and sparse values, which are updated respectively, and the update formula is written as formula (23): Where Γ′ is the complement of the set Γ.

8. The two-dimensional sparse ISAR imaging method based on weighted matrix filling according to claim 7, wherein The weight vector W = [w1, w2,..., w I is written as Equation (24): where α is a positive constant, is an extremely small positive number.

9. The two-dimensional sparse ISAR imaging method based on weighted matrix completion according to claim 8, characterized in that, The iterative optimization includes the following process: Input: Γ, Initialization quantities: i = 1, 2, 3, ..., I, λ > 0, μ > 0, l = 1, ρ > 0, and the loop iteration process is as follows Step T1, Step T2, Step T3, Step T4, Step T5, Step T6, Step T7, Y1 l+1 = Y1 l + μ(P Γ (X l+1 ) + E l+1 - Z)(31); Step T8, Step T9, μ l+1 = ρυ l (33); Step T10, l = l + 1 (34); The iteration ends, Output results: S, X.

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