A fast meshless sparse imaging method based on two-dimensional weighted atomic norm minimization

Abstract

This invention relates to the field of radar imaging technology, and more particularly to a fast meshless sparse imaging method based on two-dimensional weighted atomic norm minimization. It includes: Step S1, modeling the received echo signal as a linear combination of two-dimensional frequencies; Step S2, establishing a new weighted meshless imaging optimization model based on the two-dimensional framework; Step S3, iteratively executing 2D ANM using an iterative optimization strategy, processing with the ADM algorithm in each iteration, determining the weighting values ​​for 2D frequency selection based on the latest iteration, and obtaining the 2D frequencies contained in the matrix through one-dimensional Vandermonde decomposition. This invention improves the resolution and computational efficiency of the sparse reconstruction method.

Description

Technical Field

[0001] This invention relates to the field of radar imaging technology, and in particular to a fast gridless sparse imaging method based on two-dimensional weighted atomic norm minimization. Background Technology

[0002] Inverse synthetic aperture radar (ISAR) possesses the capability to image non-cooperative moving targets in one, two, and even multiple dimensions, and is widely used in civilian and military fields. In stepped-frequency ISAR (SF-ISAR) systems, the SF signal achieves a large synthetic bandwidth by transmitting a series of narrow-band sub-pulses. Then, multiple pulse trains are continuously transmitted to synthesize a long aperture in the lateral direction. However, due to the non-cooperative motion of the target and the multi-functionality of the radar system, long continuous coherent processing intervals (CPI) are usually not achievable in practical applications, resulting in a lack of echoes in the two-dimensional direction of ISAR echo data, also known as sparse two-dimensional echo data. For such limited bandwidth and short aperture data, it is difficult to obtain high-resolution ISAR images using traditional techniques. Therefore, research on sparse two-dimensional ISAR imaging has received increasing attention.

[0003] Compressed sensing (CS) theory has been introduced into the field of sparse ISAR imaging due to its high-resolution capability. Therefore, many research results have been reported in recent years. However, in general, these CS-based methods can mostly be regarded as sparse reconstruction techniques on a grid, characterized by the assumption that the scatterer of the target can accurately fall on a discrete grid. However, this assumption is often difficult to satisfy in practice, leading to a decrease in imaging performance. Although many grid error correction methods can alleviate the effect of grid mismatch to some extent, they cannot completely eliminate it. Tang et al. proposed the atomic norm minimization (ANM) technique, namely continuous compressed sensing (CCS), which can work directly in a continuous parameter space without discretizing the target space, thus completely avoiding the off-grid effect of traditional CS methods. To improve sparsity and resolution, a reweighting strategy of a continuous dictionary, called reweighted ANM (RAM), was proposed in [i]. When applied to large imaging scenes, the traditional SDPT3 solver becomes very time-consuming. Furthermore, these methods all utilize the atomic l1 norm to represent sparsity, which also suffers from a decrease in resolution. In reality, the atomic norm weighting strategy performs weighting in one-dimensional space and has high computational complexity.

[0004] Chinese Patent Publication No. CN109459752A discloses a resource adaptive scheduling method for two-dimensional sparse imaging using inverse synthetic aperture radar (InSAR), relating to the field of phased array radar InSAR imaging technology. The technical problem it addresses is how to rationally allocate resources within a single radar for multi-target imaging tasks from the perspectives of target azimuth and range, thereby improving overall system performance. This method calculates the pulse resource requirement for each target based on the radar's feature recognition, and then determines the sub-pulse transmission position allocated to each target according to the constraints selected by the radar. By alternately observing the targets and acquiring their echo signals, the InSAR imaging task for multiple targets is completed. This invention can realize resource allocation for a single-step radar facing multiple targets from the perspectives of target azimuth and range, saving radar resources and improving overall system performance. However, it is evident that the resource adaptive scheduling method for two-dimensional sparse imaging using InSAR suffers from insufficient resolution and low computational efficiency due to the limitations of sparse reconstruction methods. Summary of the Invention

[0005] To address this, the present invention provides a fast meshless sparse imaging method based on two-dimensional weighted atomic norm minimization, in order to overcome the problems of reduced resolution and low computational efficiency in the prior art.

[0006] To address this technical issue, this invention provides a fast meshless sparse imaging method based on two-dimensional weighted atomic norm minimization, comprising: step S1, modeling the received echo signal as a linear combination of two-dimensional frequencies; step S2, establishing a new weighted meshless imaging optimization model based on the two-dimensional framework; and step S3, iteratively executing 2D ANM using an iterative optimization strategy, processing with the ADM algorithm in each iteration, determining the weighting value for 2D frequency selection based on the latest iteration, and obtaining the 2D frequencies contained in the matrix through one-dimensional Vandermonde decomposition.

[0007] Furthermore, in the SF ISAR imaging system, Na pulse train signals need to be transmitted, and each pulse train signal contains N sub-pulses. The bandwidth of each pulse train is (N-1)Δf, where Δf represents the sub-pulse bandwidth. Assume the observed target contains K scattering points, and the scattering point coefficient is denoted by σ. k , k=1,2,...K.

[0008] Therefore, the target echo signal after motion compensation is represented as:

[0009]

[0010] Where n = 0, 1, 2, ..., N-1, n a =0,1,2,...,Na-1,T rThis represents the pulse reconstruction time, where f0 is the initial carrier frequency of the sub-pulse, (x k ,y k ) represents the position of the target scattering point in the target coordinate system.

[0011] The above formula (1) is expressed as:

[0012]

[0013] in and Represented as

[0014]

[0015]

[0016] in Considered as range-directed frequency, It is considered as the azimuth frequency.

[0017] The problem of two-dimensional imaging is transformed into the problem of estimating the two-dimensional joint frequency, that is, estimating it from equation (2).

[0018] Furthermore, the steps of the 2D FRAM algorithm are as follows:

[0019] Extending the one-dimensional atomic norm weighting strategy to two-dimensional space, we construct a two-dimensional set of weighted matrix atomic norms. as follows:

[0020]

[0021] in For weighted atoms, For a pair of weighted values ​​in two-dimensional space.

[0022] For sparse signals, the reconstruction process involves finding the minimum number of atoms in the atom set to accurately describe the signal. Therefore, the following two-dimensional atom l0 norm form is obtained:

[0023]

[0024] Where inf{·} denotes the infimum.

[0025] Furthermore, the NP-hard problem of the l0 norm is solved by relaxing it into a two-dimensional atomic l1 norm problem, including:

[0026]

[0027] in, These are the weighting coefficients for the two-dimensional atom A(f).

[0028] Furthermore, the process of solving the two-dimensional atom l1 norm problem also includes: constructing the following matrix Z

[0029]

[0030] Where Z is a positive semi-definite matrix (PSD),

[0031] Based on the positive semidefinite property of matrix Z, the l1 norm problem in equation (7) is transformed into the following positive semidefinite programming problem for solution.

[0032]

[0033] Furthermore, the algorithm for iteratively updating the matrix Z includes: based on the logarithmic function f log (·) is used to approximate the rank of the PSD matrix, and the expression for the logarithmic function is written as:

[0034] f l o g (tr(T(u a )))=ln|T(u a )+ξ a I a | (10)

[0035] Where ξ a ≥0 is used to control f log (·) and tr(T(u) a Equivalent parameter, I a It is a unit diagonal matrix.

[0036] Based on the above equivalence relationship, equation (9) is equivalent to:

[0037]

[0038] Where ξ b and I b With ξ a and I a They have similar functions.

[0039] The matrix Z is minimized iteratively using an optimization maximization algorithm. The specific process of minimization includes:

[0040] Assumption For u a The result of the l-th iteration and the result of the (l+1)-th iteration are represented as T(u) a )+ξ a I a exist The first-order expansion form at:

[0041]

[0042] in As a constant, ignoring the constant term, the two-dimensional weighted l1 norm after the (l+1)th iteration is expressed as:

[0043]

[0044] Among them W a =1 / (T(u) a )+ξ a I a ), W b =1 / (T(u) b )+ξ b I b ) is T(u a ) and T(u b The corresponding weighting function is given by:

[0045]

[0046] In the first iteration, and Given two identity matrices, the first iteration is considered as a traditional 2D ANM method; in the second iteration, the selection of atoms at each step is achieved through a weighted function W. l a and W l b To confirm; during iteration, the weighting function is updated at any time in each loop, and the objective optimization problem is converged to a local minimum through iterative solution. The new iterative solution method is expressed as 2D reweighted ANM (2D RANM).

[0047] Furthermore, the steps to address the two-dimensional sparsity of the generated echo signal in terms of distance and orientation are as follows:

[0048] Setting S Ω Let Ω be a sparse echo signal, where Ω∈{1,2,...,N}×{1,2,...,Na} and |Ω|=M×H, (M≤N,H≤Na). Parameters M and H represent the number of echo data points in a two-dimensional random measurement. In the presence of noise interference, the sparse echo signal is represented by the following expression:

[0049] Y = S Ω +X (15)

[0050] Where X represents random noise.

[0051] Furthermore, under the condition of noise interference, the expression (13) is transformed into the following optimization problem to be solved:

[0052]

[0053] Among them ||·|| F Let F represent the norm, and τ > 0 be the regularization parameter.

[0054] Based on the optimization of the above formula, u is obtained. a and u b Based on the optimization, T(u) is finally obtained. a ) and T(u b The one-dimensional Hermitian Toeplitz matrix T(u) a ) and T(u b It includes two-dimensional frequencies. The two-dimensional frequency is obtained according to the following one-dimensional Vandermonde decomposition, specifically through the following steps:

[0055] T(u a ) = v a (f a )d a v a (f a ) H ,T(u b ) = v b (f b )d b v b (f b ) H (17)

[0056] in d a ≥0∈C K×K and d b ≥0∈C K×K These are two diagonal matrices.

[0057] After completing the above operations, the two frequencies obtained are paired into K pairs using pairing techniques, with the help of the recovered data matrix, and the maximum correlation method is used to obtain the amplitude information of the scatterer using the least squares method.

[0058] Furthermore, the optimization problem in the ISAR imaging problem applies a fast algorithm based on ADMM, which is as follows:

[0059] The augmented Lagrange function formula of equation (16) can be written in the following form:

[0060]

[0061] Where γ is the penalty parameter, <, > denote the inner product, and Λ l ∈C(N+Na)×(N+Na) Λ is a Lagrangian multiplier. l and Z l The structure is represented as:

[0062]

[0063] Where Λ0∈C N×N Λ1∈C N×Na Λ2∈C Na×Na Z0∈C N×N Z1∈C N×Na Z2∈C Na×Na .

[0064] Without loss of generality, the iteration symbol l is omitted, and ADMM solves by iteratively updating the following formula (19).

[0065]

[0066]

[0067]

[0068]

[0069] Where M a and M b These are two diagonal matrices, with their diagonal elements being respectively and T · (·) denotes the conjugate of T(·), Ω c This represents the complement of Ω. This indicates the corresponding missing data.

[0070] For matrix Z, update it according to the following formula.

[0071]

[0072] It is through the Hermitian matrix It is obtained by performing feature decomposition.

[0073] Furthermore, the computational complexity of the traditional 2DANM method based on SDPT3 is O(N). p (N+Na) 6 The computational complexity of the SDPT3-based 2DRANM method is O(LN). p (N+Na) 6 ), where L is the number of iterations, and the computational complexity of the ADMM-based 2DFRANM method is O(LIN). p (N+Na)3 ), where I is the number of iterations of the ADMM algorithm.

[0074] Compared with the prior art, the beneficial effects of the present invention are that, by modeling the echo signal as a linear combination of two-dimensional frequencies, establishing a new weighted meshless optimization model based on a two-dimensional framework, and performing 2D ANM using an iterative optimization strategy and determining the weighting value of the D frequency selection based on the latest iteration, the present invention achieves improved resolution and computational efficiency compared with other meshless sparse reconstruction methods based on ANM.

[0075] Furthermore, the method described in this invention simplifies the two-dimensional imaging problem by transforming it into a two-dimensional joint frequency estimation problem, thereby further improving resolution and computational efficiency.

[0076] Furthermore, the method described in this invention extends the one-dimensional atomic norm weighting strategy to two-dimensional space and reconstructs sparse signals by accurately describing the signal by finding the fewest atoms in the atomic set, thereby improving the reconstruction capability of sparse signals and further improving resolution and computational efficiency.

[0077] Furthermore, the method described in this invention improves the efficiency and accuracy of solving NP-hard problems of norms by relaxing the norm problem into a two-dimensional atomic norm problem, thereby further improving resolution and computational efficiency.

[0078] Furthermore, the method described in this invention solves the two-dimensional atomic norm problem by constructing a positive semidefinite matrix, which can transform the norm problem into a positive semidefinite programming problem, thereby improving the efficiency and accuracy of solving the two-dimensional atomic norm problem and further improving resolution and computational efficiency.

[0079] Furthermore, the method described in this invention updates the weighting function of the matrix iterative update solution algorithm in each loop, and converges the target optimization problem to a local minimum through iterative rescue, reducing the loss of resolution and reconstruction accuracy, thereby further improving resolution and computational efficiency.

[0080] Furthermore, the method described in this invention optimizes sparse echo signals under noise interference, thereby addressing the two-dimensional sparseness of the echo signal generation distance and orientation, improving the accuracy of sparse echo signal processing, and further enhancing resolution and computational efficiency.

[0081] Furthermore, the method of the present invention obtains a one-dimensional matrix and the two-dimensional frequencies contained in the matrix by optimizing the expression under noise interference conditions, and obtains the amplitude information of the scatterer by means of the recovered data matrix and the least squares method according to the pairing technique, thereby improving the efficiency of obtaining the two-dimensional frequencies and further improving the resolution and computational efficiency.

[0082] Furthermore, the method described in this invention introduces a fast algorithm based on ADMM to apply the optimization problem to the ISAR imaging problem during iteration, thereby reducing the amount of computation and further improving resolution and computational efficiency.

[0083] Furthermore, the method described in this invention reduces computational complexity by using the ADMM-based 2DFRANM method, thereby further improving resolution and computational efficiency. Attached Figure Description

[0084] Figure 1 This is a flowchart of a fast meshless sparse imaging method based on two-dimensional weighted atomic norm minimization, as described in an embodiment of the present invention.

[0085] Figure 2 This is a logical schematic diagram of a fast meshless sparse imaging method based on two-dimensional weighted atomic norm minimization according to an embodiment of the present invention;

[0086] Figure 3 This is a comparison of imaging results of different algorithms of the fast meshless sparse imaging method based on two-dimensional weighted atomic norm minimization in this invention, under the condition of a sparsity rate of 0.5.

[0087] Figure 4 The reconstruction performance of different algorithms for the fast meshless sparse imaging method based on two-dimensional weighted atomic norm minimization in this invention is compared under different SPRs conditions, showing the MSE values.

[0088] Figure 5 This is a comparison of imaging results of other different algorithms of the fast meshless sparse imaging method based on two-dimensional weighted atomic norm minimization in this embodiment of the invention, under the condition of sparsity of 0.5. Detailed Implementation

[0089] To make the objectives and advantages of the present invention clearer, the present invention will be further described below with reference to embodiments; it should be understood that the specific embodiments described herein are merely for explaining the present invention and are not intended to limit the present invention.

[0090] Preferred embodiments of the present invention will now be described with reference to the accompanying drawings. Those skilled in the art should understand that these embodiments are merely illustrative of the technical principles of the present invention and are not intended to limit the scope of protection of the present invention.

[0091] Please see Figure 1 As shown, this is a fast meshless sparse imaging method based on minimizing the two-dimensional weighted atomic norm, including:

[0092] Step S1: Model the received echo signal as a linear combination of two-dimensional frequencies;

[0093] Step S2: Establish a new weighted meshless imaging optimization model based on a two-dimensional framework;

[0094] Step S3: The 2D ANM is iteratively executed using an iterative optimization strategy. In each iteration, the ADM algorithm is used for processing. The weighting value of the 2D frequency selection is determined based on the latest iteration, and the 2D frequencies contained in the matrix are obtained through one-dimensional Vandermonde decomposition.

[0095] This invention improves resolution and computational efficiency compared to other ANM-based meshless sparse reconstruction methods by modeling the echo signal as a linear combination of two-dimensional frequencies, establishing a new weighted meshless optimization model based on a two-dimensional framework, and performing 2D ANM using an iterative optimization strategy and determining the weighting value of the D frequency selection based on the latest iteration.

[0096] Specifically, in the SF ISAR imaging system, Na pulse train signals need to be transmitted, and each pulse train signal contains N sub-pulses. Each pulse train can synthesize a bandwidth of (N-1)Δf, where Δf represents the sub-pulse bandwidth. Assume the observed target contains K scattering points, and the scattering point coefficient is denoted by σ. k Let k = 1, 2, ..., K. Then, the target echo signal after motion compensation is expressed as:

[0097]

[0098] Where n = 0, 1, 2, ..., N-1, n a =0,1,2,...,Na-1,T r This represents the pulse reconstruction time, where f0 is the initial carrier frequency of the sub-pulse, (x k ,y k ) represents the position of the target scattering point in the target coordinate system.

[0099] The above formula (1) is expressed as:

[0100]

[0101] in and Represented as

[0102]

[0103]

[0104] in Considered as range-directed frequency, It is considered as the azimuth frequency.

[0105] The problem of two-dimensional imaging is transformed into the problem of estimating the two-dimensional joint frequency, that is, estimating it from equation (2).

[0106] The method described in this invention simplifies the two-dimensional imaging problem by transforming it into a two-dimensional joint frequency estimation problem, thereby improving resolution and computational efficiency.

[0107] Specifically, the steps of the 2D FRAM algorithm are as follows:

[0108] Extending the one-dimensional atomic norm weighting strategy to two-dimensional space, we construct a two-dimensional set of weighted matrix atomic norms. as follows:

[0109]

[0110] in For weighted atoms, For a pair of weighted values ​​in two-dimensional space.

[0111] For sparse signals, the reconstruction process involves finding the minimum number of atoms in the atom set to accurately describe the signal. Therefore, the following two-dimensional atom l0 norm form is obtained:

[0112]

[0113] Where inf{·} denotes the infimum.

[0114] The method described in this invention extends the one-dimensional atomic norm weighting strategy to two-dimensional space and reconstructs sparse signals by accurately describing the signal by finding the fewest atoms in the atomic set, thereby improving the reconstruction capability of sparse signals and further improving resolution and computational efficiency.

[0115] Specifically, the NP-hard problem of the l0 norm is solved by relaxing it into a two-dimensional atomic l1 norm problem, including:

[0116]

[0117] in, These are the weighting coefficients for the two-dimensional atom A(f).

[0118] The method described in this invention improves the efficiency and accuracy of solving NP-hard problems of norms by relaxing the norm problem into a two-dimensional atomic norm problem, and further improves the resolution and computational efficiency.

[0119] Specifically, the process of solving the two-dimensional atom l1 norm problem also includes: constructing the following matrix Z

[0120]

[0121] Where Z is a positive semi-definite matrix (PSD),

[0122] Based on the positive semidefinite property of matrix Z, the l1 norm problem in equation (7) is transformed into the following positive semidefinite programming problem for solution.

[0123]

[0124] The method described in this invention solves the two-dimensional atomic norm problem by constructing a positive semidefinite matrix, transforming the norm problem into a positive semidefinite programming problem, thereby improving the efficiency and accuracy of solving the two-dimensional atomic norm problem and further enhancing resolution and computational efficiency.

[0125] Specifically, the algorithm for solving the iterative update of matrix Z includes: based on the logarithmic function f log (·) is used to approximate the rank of the PSD matrix, and the expression for the logarithmic function is written as:

[0126] f l o g (tr(T(u a )))=ln|T(u a )+ξ a I a | (10)

[0127] Where ξ a ≥0 is used to control f log (·) and tr(T(u) a Equivalent parameter, I a It is a unit diagonal matrix.

[0128] Based on the above equivalence relationship, equation (9) is equivalent to:

[0129]

[0130] Where ξ b and I b With ξ a and I a They have similar functions.

[0131] The matrix Z is minimized iteratively using an optimization maximization algorithm. The specific process of minimization includes:

[0132] Assumption For u a The result of the l-th iteration and the result of the (l+1)-th iteration are represented as T(u) a )+ξ a I a exist The first-order expansion form at:

[0133]

[0134] in As a constant, ignoring the constant term, the two-dimensional weighted l1 norm after the (l+1)th iteration is expressed as:

[0135]

[0136] Among them W a =1 / (T(u) a )+ξ a I a ), W b =1 / (T(u) b )+ξ b I b ) is T(u a ) and T(u b The corresponding weighting function is given by:

[0137]

[0138] In the first iteration, and Given two identity matrices, the first iteration is considered as a traditional 2D ANM method; in the second iteration, the selection of atoms at each step is achieved through a weighted function W. l a and W l b To confirm; during iteration, the weighting function is updated at any time in each loop, and the objective optimization problem is converged to a local minimum through iterative solution. The new iterative solution method is expressed as 2D reweighted ANM (2D RANM).

[0139] The method described in this invention updates the weighting function of the matrix iterative update solution algorithm in each loop, and converges the target optimization problem to a local minimum through iterative solution, reducing the loss of resolution and reconstruction accuracy, and further improving resolution and computational efficiency.

[0140] Specifically, the steps to address the two-dimensional sparsity of the distance and orientation in the echo signal generation are as follows:

[0141] Setting S Ω Let Ω be a sparse echo signal, where Ω∈{1,2,...,N}×{1,2,...,Na} and |Ω|=M×H, (M≤N,H≤Na). Parameters M and H represent the number of echo data points in a two-dimensional random measurement. In the presence of noise interference, the sparse echo signal is represented by the following expression:

[0142] Y = S Ω +X(15)

[0143] Where X represents random noise.

[0144] The method described in this invention optimizes sparse echo signals under noise interference, thereby addressing the two-dimensional sparsity of the echo signal generation distance and orientation, improving the accuracy of sparse echo signal processing, and further enhancing resolution and computational efficiency.

[0145] Specifically, under the condition of noise interference, the expression (13) is transformed into the following optimization problem to be solved:

[0146]

[0147] Among them ||·|| F Let F represent the norm, and τ > 0 be the regularization parameter.

[0148] Based on the optimization of the above formula, u is obtained. a and u b Based on the optimization, T(u) is finally obtained. a ) and T(u b The one-dimensional Hermitian Toeplitz matrix T(u) a ) and T(u b It includes two-dimensional frequencies. The two-dimensional frequency is obtained according to the following one-dimensional Vandermonde decomposition, specifically through the following steps:

[0149] T(u a ) = v a (f a )d a v a (f a ) H ,T(u b ) = v b (f b )d b v b (f b )H (17)

[0150] in d a ≥0∈C K×K and d b ≥0∈C K×K These are two diagonal matrices.

[0151] After completing the above operations, the two frequencies obtained are paired into K pairs using pairing techniques, with the help of the recovered data matrix, and the maximum correlation method is used to obtain the amplitude information of the scatterer using the least squares method.

[0152] The method described in this invention obtains a one-dimensional matrix and the two-dimensional frequencies contained in the matrix by optimizing the expression under noise interference conditions. It also obtains the amplitude information of the scatterer by using the pairing technique, the recovered data matrix, and the least squares method. This improves the efficiency of obtaining two-dimensional frequencies and further enhances the resolution and computational efficiency.

[0153] Specifically, the optimization problem in the ISAR imaging problem applies a fast algorithm based on ADMM, which is as follows:

[0154] The augmented Lagrange function formula of equation (16) can be written in the following form:

[0155]

[0156] Where γ is the penalty parameter, <, > denote the inner product, and Λ l ∈C (N+Na)×(N+Na) These are Lagrangian multipliers. Where Λl l and Z l The structure is represented as:

[0157]

[0158] Where Λ0∈C N×N Λ1∈C N×Na Λ2∈C Na×Na Z0∈C N×N Z1∈C N×Na Z2∈C Na×Na .

[0159] Without loss of generality, the iteration symbol l is omitted, and ADMM solves by iteratively updating the following formula (19).

[0160]

[0161]

[0162]

[0163]

[0164] Where M a and M b These are two diagonal matrices, with their diagonal elements being respectively and T · (·) denotes the conjugate of T(·), Ω c This represents the complement of Ω. This indicates the corresponding missing data.

[0165] For matrix Z, update it according to the following formula.

[0166]

[0167] It is through the Hermitian matrix It is obtained by performing feature decomposition.

[0168] The method described in this invention introduces a fast algorithm based on ADMM to apply the optimization problem to the ISAR imaging problem during iteration, thereby reducing the amount of computation and further improving resolution and computational efficiency.

[0169] Specifically, the computational complexity of the traditional 2DANM method based on SDPT3 is O(N). p (N+Na) 6 The computational complexity of the SDPT3-based 2DRANM method is O(LN). p (N+Na) 6 ), where L is the number of iterations, and the computational complexity of the ADMM-based 2DFRANM method is O(LIN). p (N+Na) 3 ), where I is the number of iterations of the ADMM algorithm.

[0170] The method described in this invention reduces computational complexity by using the ADMM-based 2DFRANM method, thereby further improving resolution and computational efficiency.

[0171] Specifically, a schematic diagram of the proposed 2D FRAM method is shown below. Figure 2 As shown. The proposed method has two iterations. The inner iteration is used for ADMM, and the outer iteration is mainly used to update the weights. It is important to note that the algorithm is set at the beginning. The stagnation condition for both iterations can be set to the error or the maximum number of iterations. Generally, the ADMM algorithm can obtain acceptable results after one or two hundred iterations, and for the outermost iteration, setting it to around 3-5 is usually sufficient to achieve good convergence.

[0172] Specifically, for the simulation data analysis, it is assumed that the target contains 34 ideal scattering points with the same intensity. The SF radar transmits 96 pulse train signals, each pulse train containing 64 sub-pulses, achieving a synthetic bandwidth of 640MHz. The transmitted SF signal carrier frequency is 10GHz. The sparsity (SPR) of the transmitted signal is assumed to be defined as SPR = MH / NNa. The algorithm parameters are set to γ ​​= 1, ξ... a =ξ b =1. Furthermore, the maximum number of iterations for 2D RANM and 2D FRANM was set to 3, while the number of iterations for ADMM was set to 500. The performance of the proposed method was compared with that of 2D ANM and 2D SL0 methods, and the imaging results are as follows: Figure 3 As shown, the SPR and signal-to-noise ratio (SNR) are set to 0.7 and 5dB, respectively.

[0173] Figure 3 Comparison of imaging results of different algorithms under a sparsity ratio of 0.5. (a) 2D SL0. (b) 2D ANM. (c) 2D RANM. (d) 2D FRANM.

[0174] It can be seen that the imaging results obtained by the 2D SL0 algorithm contain more false reconstruction points. The estimation accuracy of 2D ANM is better than that of the 2D SL0 algorithm. Conversely, due to the adoption of an iterative optimization strategy, the reconstruction results of the 2D RANM and 2D FRAM algorithms are significantly higher than those estimated by the 2D SL0 and 2D ANM algorithms, thus improving the reconstruction accuracy and verifying the effectiveness of the proposed algorithm.

[0175] Furthermore, we further compared the performance of the four methods under different conditions using mean squared error (MSE). Figure 4 (a) shows the MSE values ​​of the reconstruction results under different SPRs with SNR = 5 dB. It can be seen that the proposed 2D FRANM algorithm has the smallest reconstruction error under these SPR conditions. Furthermore, under different SNR conditions, and with SPR set to 0.5, the MSE values ​​of each algorithm are as follows: Figure 4 As shown in (b), it can be seen that 2D RANM and 2D FRANM have lower MSE in all cases, which means better estimation performance, especially 2D FRANM with lower reconstruction error.

[0176] Finally, the running times of the three meshless methods were compared under different SPRs conditions, and the results are as follows: Figure 4 As shown in (c), this experiment was conducted on a laptop equipped with an Intel Core™ i7 CPU and 16GB of RAM. Applying ADMM to 2D RANM significantly reduced the runtime from nearly 700s for 2D RANM to only about 8s for 2D FRANM, demonstrating the significant speed advantage of the proposed algorithm.

[0177] Figure 4 Comparison of reconstruction performance of different algorithms: (a) MSE value under different SPRs. (b) MSE value under different SNRs. (c) Reconstruction time.

[0178] Specifically, for the analysis of measured data, the proposed algorithm was further applied to the imaging processing of actual measurement data obtained from the SF radar system. This system has a total of 128 pulse trains, each containing 64 sub-pulses, with a combined bandwidth of 150MHz. Imaging results obtained through different methods are shown below. Figure 5 As shown, the parameter settings in the experiment are the same as those in the experiment. Figure 2 The parameter settings are the same.

[0179] Figure 5 Comparison of imaging results of different algorithms under a sparsity ratio of 0.5. (a) 2D SL0. (b) 2D ANM. (c) 2D RANM. (d) 2D FRANM.

[0180] As can be seen, many spurious reconstruction points are distributed in the images obtained by 2D SL0. 2D ANM reduces these spurious reconstructions to some extent, while 2D RANM and 2D FRAM further reduce these spurious scattering points, resulting in more focused imaging results. Regarding reconstruction accuracy, since there are no actual labeled images, we use the complete reconstructed data to measure the performance of the methods. The mean squared errors of the four methods are 0.4103, 0.3282, 0.3145, and 0.3070, respectively. Furthermore, the corresponding running times of the three meshless methods are 12.67 s, 726.13 s, and 2184.15 s, respectively, which further demonstrates the feasibility of the proposed 2D FRAM method in practical applications.

[0181] Specifically, a meshless ISAR imaging method employing an iterative optimization strategy is presented. This method iteratively executes 2DANM, selecting superior atoms in each iteration using a weighted strategy, thereby improving atom selection accuracy and reconstruction performance. Furthermore, the ADMM algorithm is used in each iteration to further reduce computational complexity. This method is suitable for practical ISAR applications while maintaining high estimation accuracy.

[0182] The technical solution of the present invention has been described above with reference to the preferred embodiments shown in the accompanying drawings. However, it will be readily understood by those skilled in the art that the scope of protection of the present invention is obviously not limited to these specific embodiments. Without departing from the principles 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 all fall within the scope of protection of the present invention.

[0183] The above description is merely a preferred embodiment of the present invention and is not intended to limit the invention. Various modifications and variations can be made to the present invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the present invention should be included within the scope of protection of the present invention.

Claims

1. A fast meshless sparse imaging method based on two-dimensional weighted atomic norm minimization, characterized in that, include: Step S1: Model the received echo signal as a linear combination of two-dimensional frequencies; Step S2: Establish a new weighted meshless imaging optimization model based on a two-dimensional framework; Step S3: Iterative optimization strategy is adopted to perform 2D ANM iteratively. In each iteration, ADMM algorithm is used for processing. The weighting value of 2D frequency selection is determined according to the latest iteration. The 2D frequencies contained in the matrix are obtained through one-dimensional Vandermonde decomposition. In the SF ISAR imaging system, Na pulse train signals need to be transmitted, and each pulse train signal contains N sub-pulses. Each pulse train is synthesized. The bandwidth, of which Let represent the sub-pulse bandwidth, assuming the observed target contains K scattering points, and the scattering point coefficients are expressed as . Therefore, the target echo signal after motion compensation is represented as: （1）； in , , Indicates the pulse reconstruction time. The initial carrier frequency of the sub-pulse. This represents the position of the target scattering point in the target coordinate system. The above formula (1) is expressed as: （2）； in and Represented as: （3）； （4）； in , Considered as range-directed frequency, , Considered as azimuth frequency; The problem of two-dimensional imaging is transformed into the problem of estimating the two-dimensional joint frequency, that is, estimating it from equation (2). ; The steps of the 2D FRAM algorithm are as follows: Extending the one-dimensional atomic norm weighting strategy to two-dimensional space, we construct a two-dimensional set of weighted matrix atomic norms. as follows: （5）； in For weighted atoms, For a pair of weighted values ​​in two-dimensional space; For sparse signals, the reconstruction process involves finding the minimum number of atoms in the atom set to accurately describe the signal. Therefore, the following two-dimensional atom structure is obtained. Norm form: （6）； in Indicates the infimum; The The NP-hard problem of norms can be solved by relaxing it into two-dimensional atoms. Solving norm problems includes: （7）； in, It is for two-dimensional atoms The weighting coefficients; Solving for two-dimensional atoms The process of solving the norm problem also includes: constructing the following matrix ; （8）； in It is a positive semi-definite matrix (PSD). , , , ; Based on matrix The positive semi-definite property, in equation (7) The norm problem is transformed into the following positive semidefinite programming problem for solution; （9）。 2. The fast meshless sparse imaging method based on two-dimensional weighted atomic norm minimization according to claim 1, characterized in that, The algorithm for solving the iterative update of matrix Z includes: based on the logarithmic function To approximate the rank of the PSD matrix, the expression for the logarithmic function is written as: （10）； in For use in control and Equivalent parameters, It is a unit diagonal matrix; Based on the above equivalent parameters, equation (9) is equivalent to: （11）； in and and and They have the same function; The matrix Z is minimized iteratively using an optimization maximization algorithm. The specific process of minimization includes: Assumption for No. l The result of the iteration, the... l The result of +1 iterations is represented as exist The first-order expansion form at: （12）； in As a constant, ignoring the constant term, in the th... l Two-dimensional weighted sum after +1 iterations Norm is represented as (13); in , for and The corresponding weighting function, and the weights are expressed as: （14）； In the first iteration, and Given two identity matrices, the first iteration is considered as a traditional 2D ANM method; in the second iteration, the selection of atoms is performed each time using a weighted function. and To confirm; during iteration, the weighting function is updated at any time in each loop, and the objective optimization problem is converged to a local minimum through iterative solution. The new iterative solution method is represented as 2D reweighted ANM, i.e. 2D RANM.

3. The fast meshless sparse imaging method based on two-dimensional weighted atomic norm minimization according to claim 2, characterized in that, The steps to address the two-dimensional sparsity of the distance and azimuth in the echo signal generation are as follows: set up It is a sparse echo signal, in which ,and ,parameter M and H This represents the number of echo data points in a two-dimensional random measurement; sparse echo signals in the presence of noise interference are represented by the following expression: (15); in This represents random noise.

4. The fast meshless sparse imaging method based on two-dimensional weighted atomic norm minimization according to claim 3, characterized in that, Under the condition of noise interference, expression (13) is transformed into the following optimization problem to be solved: (16); in Describing the F-norm, For regularization parameters; Based on the optimization of the above formula, we obtain... as well as Based on the optimization, the final result is obtained and One-dimensional Hermitian Toeplitz matrix and It includes two-dimensional frequency The two-dimensional frequency is obtained according to the following one-dimensional Vandermonde decomposition, the specific steps of which are as follows: （17）； in , , and These are two diagonal matrices; After completing the above operations, the two frequencies obtained are paired using pairing techniques, with the aid of the recovered data matrix, and through the maximum correlation method. K The amplitude information of the scatterer is obtained by using the least squares method.

5. The fast meshless sparse imaging method based on two-dimensional weighted atomic norm minimization according to claim 4, characterized in that, The optimization problem described uses a fast algorithm based on ADMM in the ISAR imaging problem. The fast algorithm based on ADMM is as follows: The augmented Lagrange function formula of equation (16) can be written in the following form: （18）； in For penalty parameters, Indicates the inner product. For Lagrangian multipliers, where and The structure is represented as: （19）； in , Without loss of generality, the iteration symbol Omitted, ADMM solves by iteratively updating the following formula (19); (20)； in and These are two diagonal matrices, with their diagonal elements being respectively and , express conjugate, express The supplement, This indicates the corresponding lost data; For matrix Update using the following formula: (21)； It is through the Hermitian matrix It is obtained by performing feature decomposition.

6. The fast meshless sparse imaging method based on two-dimensional weighted atomic norm minimization according to claim 1, characterized in that, The computational complexity of solving 2DANM based on SDPT3 is O(n). The computational complexity of the 2DRANM method based on SDPT3 is... Where L is the number of iterations, the computational complexity of the ADMM-based 2DFRANM method is O(L). ,in This represents the number of iterations in the ADMM algorithm.

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