A SAR imaging method, system and readable storage medium based on sparse and low-rank decomposition

Abstract

The application discloses a SAR imaging method and system based on sparse and low-rank decomposition, and a readable storage medium. The imaging method comprises the following steps: collecting a SAR image and two-dimensional echo data, pre-processing the two-dimensional echo data to obtain target reflectivity. Based on the linear relationship between the two-dimensional echo data and the target reflectivity, a preset method is used to decompose the SAR image to obtain sparse components and low-rank components. The decomposition result is iteratively updated by using a preset optimization algorithm to improve the accuracy until the error converges, and a sparse and low-rank decomposition algorithm expression is obtained. The preset SAR imaging model is solved by using the sparse and low-rank decomposition algorithm expression, and the SAR imaging result is output. The application can effectively reduce the complexity of SAR imaging and improve the imaging efficiency. Even in the case of reduced data sampling rate, the accuracy of image reconstruction can be ensured.

Description

TECHNICAL FIELD

[0001] The application belongs to the field of radar imaging, and more particularly relates to a SAR imaging method, system and readable storage medium based on sparse and low-rank decomposition. BACKGROUND

[0002] Synthetic aperture radar high-resolution imaging has always been an important requirement of radar systems, but due to the limitation of the Nyquist sampling theorem, high resolution and system complexity are difficult to balance. Traditional SAR imaging relies on high sampling rate to achieve high resolution, which increases the complexity and cost of the system. Sparse SAR imaging technology is based on the principle of compressive sensing, which utilizes the sparsity of signals to perform imaging from fewer samples, effectively reducing the sampling rate. Imaging methods based on low-rank matrices are an important research direction to obtain high-resolution imaging by utilizing the inherent structural characteristics of SAR images.

[0003] SAR imaging faces several challenges. Targets with periodicity or local correlation are difficult to maintain the structural integrity of the original data in sparse synthetic aperture radar imaging, thus the imaging effect of SAR cannot be guaranteed, and the computational complexity is high, which is difficult to adapt to actual application requirements, especially in dynamic or complex scenes.

[0004] The prior art patent with publication number CN115249216A proposes a SAR image denoising method based on sparse representation and blind estimation combination, comprising: S1: selecting a dictionary and initializing dictionary parameters; S2: obtaining a SAR image to be denoised, and estimating the image noise of the SAR image to be denoised using a blind estimation algorithm;

[0005] S3: according to the image noise, using the OMP algorithm to calculate the sparse coding, updating the dictionary iteratively, obtaining the updated dictionary and sparse coding; S4: using the updated dictionary and sparse coding to perform sparse decomposition and reconstruction on the SAR image to be denoised, obtaining a denoised image. This scheme has the problems of high computational complexity and difficulty in adapting to actual application requirements. SUMMARY

[0006] To overcome the problems of SAR imaging in the prior art, especially the difficulty of maintaining the structural integrity of the original data in sparse synthetic aperture radar imaging for targets with periodicity or local correlation, thus the imaging effect of SAR cannot be guaranteed, and the high computational complexity of the prior art, which is difficult to adapt to actual application requirements, the application provides a SAR imaging method, system and readable storage medium based on sparse and low-rank decomposition.

[0007] The primary object of the present application is to solve the above technical problems. The technical solution of the present application is as follows:

[0008] The first aspect of the present application provides a SAR imaging method based on sparse and low-rank decomposition, comprising the following steps:

[0009] Collecting SAR images and two-dimensional echo data, preprocessing the two-dimensional echo data to obtain target reflectivity;

[0010] Based on the linear relationship between the two-dimensional echo data and the target reflectivity, decomposing the SAR image using a preset method to obtain a sparse component and a low-rank component;

[0011] Using a preset optimization algorithm, iteratively updating the decomposition results of the matrix sparsity and low rank to improve the accuracy until the error converges, obtaining a sparse and low-rank decomposition algorithm expression;

[0012] Using the sparse and low-rank decomposition algorithm expression to solve a preset SAR imaging model, outputting a SAR imaging result.

[0013] Further, the optimization algorithm for iteratively updating the decomposition results of the matrix sparsity and low rank is an alternating multiplier method.

[0014] Further, the preset method for decomposing the SAR image is a kernel norm relaxation factorization.

[0015] Further, the decomposition of the SAR image using the kernel norm relaxation factorization specifically includes:

[0016] Decomposing the SAR image B into a surface target X and a point target S, initializing X and S using a low-rank matrix fitting algorithm, the expression is as follows:

[0017] B=X+S (1);

[0018] Discretizing the surface target X of size M×N at Nyquist sampling rate to obtain X i,j =x[i,j], constructing a corresponding Hankel matrix The expression is as follows:

[0019]

[0020] Where is expressed as:

[0021]

[0022] Decomposing the SAR data B into a sparse component and a low-rank component under the constraint of the Hankel structured matrix, using a sparse and low-rank balance parameter, outputting the decomposition result for model optimization, the expression is as follows:

[0023]

[0024] where S represents the sparse component, L corresponds to the low-rank component, and τ is the balance parameter between the low-rank and sparsity, is a Hankel matrix constructed by X.

[0025] Further, the decomposition results of the iteration update matrix sparsity and low rank are improved in precision until the error converges, and a sparse and low-rank decomposition algorithm expression is obtained, specifically including:

[0026] By using the alternating multiplier method of the core norm relaxation factorization form, the sparse and low-rank decomposition equation in expression (2) is converted into an optimization target equation expression:

[0027]

[0028] where, is the name of the optimization target equation, and U and V are two variables after the core norm relaxation factorization, S is a sparse component raised to a Hankel matrix form, and UV H is a decomposition factor of a low-rank component, and the superscript H is the conjugate transpose of the matrix;

[0029] Based on the optimization target equation of the SAR imaging problem and the related constraint conditions, a Lagrange function is constructed by using the Lagrange multiplier method; the sparse and low-rank parts in the SAR imaging model are optimized by using the ADMM algorithm, and the Lagrange variables S, X, U, V, Λ and Ω are sequentially solved, and the Lagrange function expression is as follows:

[0030]

[0031] where τ, α and β represent balance parameters, Λ, Ω represent Lagrange variables, B is a SAR image to be processed, is a Hankel matrix constructed by X, S and X are point targets and surface targets respectively, U and V are two variables after the core norm relaxation factorization.

[0032] The variables S and X solved are substituted into the optimization target equation to obtain a sparse and low-rank decomposition algorithm expression The expression is:

[0033]

[0034] where B is a SAR image with noise, is a SAR image after denoising by the sparse and low-rank decomposition algorithm.

[0035] Further, the specific steps of solving the Lagrange variables include:

[0036] The S sub-problem is defined as the quadratic sub-problem of minimizing expression (6) with respect to S variables:

[0037]

[0038] where X (k) is the output of the surface target after the kth iteration, k is the iteration number, a represents the trade-off parameter, A represents the Lagrange variable, B is the SAR image to be processed, S is the point target, and t represents the balance parameter between low rank and sparsity;

[0039] Simplify expression (9) with respect to S, where G τ / α represents the soft threshold of each pixel, to obtain the following solution:

[0040] S (k+1) = G τ / α (B-X (k) -A (k) ) (10)

[0041] where S (k+1) is the output of the point target after the k+1th iteration, k is the iteration number, a represents the trade-off parameter, A represents the Lagrange variable, B is the SAR image to be processed, X (k) is the output of the surface target after the kth iteration, t represents the balance parameter between low rank and sparsity, and G τ / α represents the soft threshold of each pixel in S.

[0042] The X sub-problem is defined as the quadratic sub-problem of minimizing expression (6) with respect to X variables:

[0043]

[0044] where X (k+1) is the output of the surface target after the k+1th iteration, k is the iteration number, a and b represent the trade-off parameters, A and W represent the Lagrange variables, B is the SAR image to be processed, S is the point target, represents the Hankel matrix constructed for X, and U and V are two variables after factorization of the kernel norm relaxation;

[0045] Simplify expression (7) with respect to X to obtain the following solution:

[0046]

[0047] where X (k+1)Xk+1= Xk- αΛk- βΩk, k = 1, 2,..., (6) where Xk is the output surface target after the kth iteration, k is the iteration number, α and β are trade-off parameters, Λ and Ω are Lagrange multipliers, B is the SAR image to be processed, S is the point target, U and V are two variables after factorization of the kernel norm relaxation, the superscript H denotes the conjugate transpose of a matrix, and I is the identity matrix. denotes the pseudo-inverse mapping from the Hankel structure back to the initial state, the operator

[0048]

[0049] denotes the pseudo-inverse mapping from the Hankel structure back to the initial state, the operator denotes the summation of multiple elements and puts them back to the initial coordinates, denotes the division by the number of corresponding items;

[0050] The U subproblem is defined as the quadratic subproblem with respect to the U variable by minimizing the expression (6):

[0051]

[0052] where Xk is the output surface target after the kth iteration, k is the iteration number, β is the trade-off parameter, and Ω is the Lagrange multiplier, (k+1) denotes the summation of multiple elements and puts them back to the initial coordinates, (k+1) denotes the summation of multiple elements and puts them back to the initial coordinates, denotes the summation of multiple elements and puts them back to the initial coordinates, two variables after factorization of the kernel norm relaxation, the superscript H denotes the conjugate transpose of a matrix, and I is the identity matrix.

[0053] The derivative of the expression (11) with respect to the matrix U is taken and set to zero to obtain the following closed-form solution:

[0054]

[0055] where Xk is the output surface target after the kth iteration, k is the iteration number, β is the trade-off parameter, and Ω is the Lagrange multiplier, (k+1) denotes the summation of multiple elements and puts them back to the initial coordinates, (k+1) denotes the summation of multiple elements and puts them back to the initial coordinates, denotes the summation of multiple elements and puts them back to the initial coordinates, two variables after factorization of the kernel norm relaxation, the superscript H denotes the conjugate transpose of a matrix, and I is the identity matrix.

[0056] The V subproblem is defined as the quadratic subproblem with respect to the V variable by minimizing the expression (6):

[0057]

[0058] where Xk is the output surface target after the kth iteration, k is the iteration number, β is the trade-off parameter, and Ω is the Lagrange multiplier,​(k+1) is the surface target output after the k+1th iteration, k is the number of iterations, β represents the trade-off parameter, Ω represents the Lagrangian variable, Indicates X (k+1) The constructed Hankel matrices, U and V, are The two variables after factorization by nuclear norm relaxation, the superscript H is the conjugate transpose of the matrix;

[0059] Taking the derivative of the matrix V with respect to expression (13) and equating it to zero yields the following closed-form solution:

[0060]

[0061] Among them, X (k+1) is the surface target output after the k+1th iteration, k is the number of iterations, β represents the trade-off parameter, Ω represents the Lagrangian variable, Indicates X (k+1) The constructed Hankel matrices, U and V, are The two variables after factorization by nuclear norm relaxation, the superscript H is the conjugate transpose of the matrix, and I is the identity matrix;

[0062] The method for solving the Lagrangian function Λ variable specifically includes:

[0063] Λ (k+1) =X (k+1) +S (k+1) -B+Λ (k) (15)

[0064] Among them, X (k+1) is the surface target output after the k+1th iteration, k is the number of iterations, S (k+1) represents the point target output after the k+1th iteration, Λ (k) represents the Lagrangian variable after the kth iteration, and B is the SAR image to be processed.

[0065] The method to parse the Ω variable is as follows:

[0066] The method for solving the Lagrangian function Ω variable specifically includes:

[0067]

[0068] Among them, X (k+1) is the surface target output after the k+1th iteration, k is the number of iterations, Ω represents the Lagrangian variable, Indicates X (k+1) The corresponding Hankel matrices U and V are constructed as The two variables after factorization by nuclear norm relaxation, the superscript H is the conjugate transpose of the matrix.

[0069] Further, the preset SAR imaging model expression is:

[0070]

[0071] wherein, is a parameter of reflectivity F, Θ is a measurement matrix determined by the SAR system, Φ = diag(e jφ(F) ) represents a diagonal matrix, is a parameter of auxiliary variable v, is a parameter of auxiliary variable Φ, Y is a one-dimensional vector form of two-dimensional echo data after vectorization, F is a one-dimensional vector form of two-dimensional reflectivity matrix of the target area after vectorization, is a Hankel matrix constructed for variable v, λ is a regularization parameter, R ANK (rank(·)) is the rank of a matrix.

[0072] Further, the SAR imaging result output by the sparse and low-rank decomposition algorithm expression is specifically:

[0073] The equation (17) is estimated between |F| and v under the constraint |F| = v based on the ADMM algorithm result, and the expression is:

[0074]

[0075] wherein, is a parameter of reflectivity F, η is a penalty parameter affecting the convergence speed, u is an auxiliary variable, |F| is the size of reflectivity F, Y is a one-dimensional vector form of two-dimensional echo data after vectorization, Θ is a measurement matrix determined by the SAR system, Φ is a diagonal matrix, λ is a regularization parameter representing the balance between the data consistency term and the regularization term, v and u are auxiliary variables, R ANK (rank(·)) is the rank of a matrix, and η is a penalty parameter, is a Hankel matrix constructed for variable v;

[0076] The quadratic sub-problem is solved by minimizing the expression (18), and the size of reflectivity F after the k+1th iteration is obtained, and the expression is:

[0077]

[0078] wherein, |F| (k+1) is the size of reflectivity F output after the k+1th iteration, k is the iteration number, Y is a one-dimensional vector form of two-dimensional echo data after vectorization, Θ is a measurement matrix determined by the SAR system, Φ (k) is a diagonal matrix output after the kth iteration, η is a penalty parameter, v(k) and u (k) is the auxiliary variable output after the kth iteration, which is solved by the conjugate gradient method as follows:

[0079]

[0080] where |F| (k+1) is the size of reflectivity F output after the k+1th iteration, k is the iteration number, Y is a one-dimensional vector form of two-dimensional echo data after vectorization, Θ is the measurement matrix determined by the SAR system, λ is the regularization parameter representing the balance between the consistent term and the regularization term of the control data, v (k) and u (k) is the auxiliary variable output after the kth iteration, the superscript H is the conjugate transpose of the matrix, and I is the unit matrix;

[0081] In order to separate the phase and amplitude, the reflectivity F is expressed as ΨΑ, where Ψ is a diagonal matrix containing phase information, and Α is a vector containing amplitude information. The ΘΦ|F| in formula (18) is converted to ΘΑΨ, so that the optimization problem (18) is divided into two independent steps for phase and amplitude, and the expression is as follows:

[0082] Two variables Ψ and Α are diagonal elements containing phase and reflectivity size, respectively, and then the expression (18) is minimized to obtain a quadratic sub-problem about the variable Ψ:

[0083]

[0084] where λ Ψ is the regularization parameter, Ψ (k+1) is the diagonal matrix containing phase output after the k+1th iteration, k is the iteration number, Y is a one-dimensional vector form of two-dimensional echo data after vectorization, Α (k) is the diagonal matrix containing reflectivity size output after the kth iteration, Θ is the measurement matrix determined by the SAR system, λ Ψ is the regularization parameter, and N is the sum of all observation data. |Ψ i | is the size of the i-th element of the diagonal matrix containing phase, which is solved by iteration as follows:

[0085]

[0086] where A is the diagonal matrix containing reflectivity size, j is the imaginary unit in complex number, Ψ (k+1) is the diagonal matrix containing phase output after the k+1th iteration, k is the iteration number, Y is a one-dimensional vector form of two-dimensional echo data after vectorization, Α is the diagonal matrix containing reflectivity size, Θ is the measurement matrix determined by the SAR system, λ Ψis the regularization parameter, N is the sum of all observed data, |Ψ i | is the size of the i-th element of the diagonal matrix containing the phase, φ(·) represents the phase of F, the superscript H is the conjugate transpose of the matrix, and I is the identity matrix. Since the other parameters in formula (22) are all known, formula (22) can be used to obtain Ψ in formula (21): (k+1) ;

[0087] Using the phase and amplitude matrices obtained above, the auxiliary variable v is calculated and the solution for algorithm convergence and stability is output. The expression is as follows:

[0088] u (k+1) =u (k) +|F| (k+1) -v (k+1) (twenty three)

[0089] Among them, k is the number of iterations, |F| is the size of the reflectivity F, v (k+1) and u (k+1) It is the auxiliary variable output after the k+1th iteration;

[0090] Consider subproblem (23) as an image processing problem and let |F| (k+1) +u (k) As the image to be processed, the proposed sparse and low-rank decomposition algorithm is used Replacing the denoising algorithm as a despeckle and image detail preservation term, the following expression is obtained:

[0091]

[0092] Repeat the above steps until the convergence condition is reached, and substitute the result into formula (20) to obtain the SAR imaging result |F| (k+1) .

[0093] A second aspect of the present invention provides a SAR imaging system based on sparse and low-rank decomposition, comprising a memory and a processor, wherein the memory comprises a SAR imaging method program based on sparse and low-rank decomposition, and when the SAR imaging method program based on sparse and low-rank decomposition is executed by the processor, the steps of a SAR imaging method based on sparse and low-rank decomposition are implemented.

[0094] The third aspect of the present invention provides a computer-readable storage medium, which includes a SAR imaging method program based on sparse and low-rank decomposition. When the SAR imaging method program based on sparse and low-rank decomposition is executed by a processor, it implements the steps of a SAR imaging method based on sparse and low-rank decomposition.

[0095] Compared with the prior art, the technical solution of the present invention has the following beneficial effects:

[0096] The present application can effectively reduce the complexity of SAR imaging and improve the imaging speed by combining sparse and low-rank decomposition technology. When processing echo data, the sparse signal processing technology is used to ensure the reconstruction accuracy of the image under the condition of reducing the data sampling rate. Compared with the traditional SAR imaging method, the present application can maintain good image quality in the presence of noise or data loss, and has strong adaptability to different imaging scenes. BRIEF DESCRIPTION OF DRAWINGS

[0097] In order to make the purpose, technical scheme of the present application more clear, the present application provides the following drawings and makes the following description:

[0098] Figure 1 The method flowchart provided for the embodiments of the present application. DETAILED DESCRIPTION

[0099] In order to make the purpose, technical scheme of the present application more clear, the present application provides the following drawings and makes the following description:

[0100] In the following description, many specific details are set forth in order to provide a thorough understanding of the present application, however, the present application can also be implemented in other ways different from those described herein, therefore, the protection scope of the present application is not limited by the specific embodiments disclosed below.

[0101] Embodiment 1:

[0102] The present application provides a SAR imaging method based on sparse and low-rank decomposition, as shown in Figure 1 The present application provides a SAR imaging method based on sparse and low-rank decomposition, as shown in

[0103] S1: using a SAR system, according to the geometric relationship between the observation platform and the imaging scene, collecting SAR images and two-dimensional echo data, preprocessing the two-dimensional echo data to obtain target reflectivity;

[0104] In the SAR imaging process, according to the geometric relationship between the observation platform and the imaging scene, the SAR system collects two-dimensional echo data related to the target area, and the SAR image contains sparse point targets and extended surface targets of the target scene. Assuming that the size is 512x512 pixels, the scene contains several targets with different distributions and different backscattering coefficients.

[0105] For example, in a specific scene, the scene contains four targets, two of which are strong scattering point targets and two of which are extended surface targets.

[0106] S2: based on the linear relationship between the two-dimensional echo data and the target reflectivity, a kernel norm relaxed factorization method is used to decompose the SAR image to obtain sparse components and low rank components, by combining sparse and low rank decomposition technology, the complexity of SAR imaging can be effectively reduced, and the imaging speed is improved, when processing echo data, using sparse signal processing technology, the reconstruction accuracy of the image can be ensured under the condition of reducing the data sampling rate.

[0107] More specifically, the kernel norm relaxed factorization method decomposes the SAR image, specifically including:

[0108] S2.1: the SAR image B is decomposed into a surface target X and a point target S, the low rank matrix fitting algorithm is used to initialize X and S, and the expression is as follows:

[0109] B=X+S (1)

[0110] S2.2: the surface target X with a size of MxN is discretized at the Nyquist sampling rate to obtain X i,j =x[i,j], the corresponding Hankel matrix is constructed The expression is as follows:

[0111]

[0112] Wherein is expressed as:

[0113]

[0114] The lifting scheme of the Hankel structured matrix is linear, therefore, the sparse component representing the point target in the SAR imaging is also enhanced in the enhanced Hankel structured matrix, in the lifted Hankel matrix, our algorithm focuses on decomposing the data into sparse and low rank parts;

[0115] S2.3: the SAR data B is decomposed into sparse components and low rank components under the constraint of the Hankel structured matrix, the sparse and low rank balance parameters are used to output the decomposition results for model optimization, and the expression is as follows:

[0116]

[0117] Wherein, the sparse component S represents the point target, the low rank component L corresponds to the surface target, τ is the balance parameter between the low rank and the sparsity, is the Hankel matrix constructed by X.

[0118] According to the sparse and low-rank decomposition method provided in the application, the Hankel matrix B constructed according to the SAR image is decomposed into a low-rank matrix X and a sparse matrix S, wherein X represents a low-rank part of an extended target and can capture the characteristics of a large area in a target scene, and S represents a part of a sparse point target in the scene and retains the high-resolution characteristics of the point target.

[0119] In actual operation, the matrix B is decomposed by using the nuclear norm minimization method without singular value decomposition (SVD), thereby avoiding the singular value decomposition process and greatly accelerating the calculation speed. The low-rank matrix fitting algorithm (LMaFit) is used to initialize X and S, and the alternating direction method of multipliers (ADMM) algorithm is used for iterative optimization to obtain an accurate decomposition result.

[0120] S3: The decomposition result of the matrix sparse and low-rank is iteratively updated by using a preset optimization algorithm, the accuracy is improved, and the decomposition algorithm expression is obtained until the error converges.

[0121] It should be noted that the optimization algorithm for iteratively updating the decomposition result of the matrix sparse and low-rank is the alternating direction method of multipliers (ADMM).

[0122] More specifically, the decomposition result of the matrix sparse and low-rank is iteratively updated by using the alternating direction method of multipliers algorithm, the accuracy is improved, and the decomposition algorithm expression is obtained until the error converges, and specifically includes:

[0123] S3.1: The sparse and low-rank decomposition equation in expression (2) is converted into an optimization objective equation expression by using the kernel norm relaxation factorization form of ADMM, and the expression is as follows:

[0124]

[0125] wherein, is the name of the optimization objective equation, U and V are two variables after the kernel norm relaxation factorization, S is a sparse component raised to the Hankel matrix form, UV H is a decomposition factor of a low-rank component, and the superscript H is the conjugate transpose of the matrix;

[0126] S3.2: Based on the optimization objective equation of the SAR imaging problem and the related constraint conditions, a Lagrange function is constructed by using the Lagrange multiplier method; the sparse and low-rank parts in the SAR imaging model are optimized by using the ADMM algorithm, and the Lagrange variables S, X, U, V, Lambda and Omega are sequentially solved, and the Lagrange function expression is as follows:

[0127]

[0128] Among them, τ, α and β represent balance parameters, τ is 0.3, α and β are 1 and 1, Λ and Ω represent Lagrangian variables, B is the SAR image to be processed, is the Hankel matrix constructed using X, where S and X are point targets and surface targets respectively, and U and V are Two variables after factorization using nuclear paradigm relaxation.

[0129] The order of solving the Lagrangian variables is as follows: first solve S, then use the solved S to solve X, then use X to solve U, and then use the solved U and X to solve V, then use S and X to solve Λ, and then use the solved X, U, and V to solve Ω. The specific steps include:

[0130] S3.2.1: Define the S subproblem as minimizing expression (6) to obtain a quadratic subproblem with respect to the S variables:

[0131]

[0132] Among them, X (k) is the surface target output after the kth iteration, k is the number of iterations, which is set to 60, α represents the trade-off parameter, Λ represents the Lagrangian variable, B is the SAR image to be processed, S is the point target, and τ represents the balance parameter between low rank and sparsity;

[0133] Simplify expression (9) with respect to S, and let Γ τ / α Denotes pixel-by-pixel soft thresholding, resulting in the following solution:

[0134] S (k+1) =Γ τ / α (BX (k) -Λ (k) ) (10)

[0135] Among them, S (k+1) is the point target output after the k+1th iteration, k is the number of iterations, α is the weight parameter, Λ is the Lagrangian variable, B is the SAR image to be processed, X (k) is the face target output after the kth iteration, τ represents the balance parameter between low rank and sparsity, Γ τ / α represents the pixel-by-pixel soft threshold in S.

[0136] S3.2.2: Define the X subproblem as minimizing expression (6) to obtain a quadratic subproblem with respect to the X variable:

[0137]

[0138] Among them, X (k+1)S3.2.1 : Define the U subproblem as minimizing expression (6) to obtain a quadratic subproblem in the U variables: is the Hankel matrix constructed from X, and U and V are the two variables after factorization with kernel norm relaxation,

[0139] Simplify expression (7) with respect to X to obtain the following solution:

[0140]

[0141] where X (k+1) is the output surface target after the k+1 iteration, k is the iteration number, and a and b are the trade-off parameters, and A and W are the Lagrange multipliers, B is the SAR image to be processed, and S is the point target, are the two variables after factorization with kernel norm relaxation, is the pseudo-inverse mapping from the Hankel matrix back to the initial state, whose expression is as follows:

[0142]

[0143] is the pseudo-inverse mapping from the Hankel structure back to the initial state, and the operator represents the summation of multiple elements in and puts them back to the initial coordinates, represents dividing by the number of corresponding items;

[0144] S3.2.3: Define the U subproblem as minimizing expression (6) to obtain a quadratic subproblem in the U variables:

[0145]

[0146] where X (k+1) is the output surface target after the k+1 iteration, k is the iteration number, and b is the trade-off parameter, and W is the Lagrange multiplier, is the Hankel matrix constructed from X (k+1) , and U and V are the two variables after factorization with kernel norm relaxation, with superscript H being the conjugate transpose of the matrix;

[0147] Take the derivative of expression (11) with respect to matrix U and set it equal to zero to obtain the following closed-form solution:

[0148]

[0149] where X (k+1) ​is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, and Ω represents the Lagrange variable, is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, and Ω represents the Lagrange variable, (k+1) is the constructed Hankel matrix, U and V are respectively are two variables after factorization by relaxing the kernel norm, the superscript H is the conjugate transpose of the matrix, and I is the unit matrix.

[0150] S3.2.4: The V sub-problem is defined as minimizing expression (6) to obtain a quadratic sub-problem about the V variable:

[0151]

[0152] is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, and Ω represents the Lagrange variable, (k+1) is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, and Ω represents the Lagrange variable, is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, and Ω represents the Lagrange variable, (k+1) is the constructed Hankel matrix, U and V are respectively are two variables after factorization by relaxing the kernel norm, the superscript H is the conjugate transpose of the matrix.

[0153] The derivative of expression (13) with respect to matrix V is taken and set to zero to obtain the following closed-form solution:

[0154]

[0155] is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, and Ω represents the Lagrange variable, (k+1) is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, and Ω represents the Lagrange variable, is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, and Ω represents the Lagrange variable, (k+1) is the constructed Hankel matrix, U and V are respectively are two variables after factorization by relaxing the kernel norm, the superscript H is the conjugate transpose of the matrix, and I is the unit matrix.

[0156] S3.2.5: The method for solving the Lagrange function Λ variable, specifically includes:

[0157] Λ (k+1) is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, and Ω represents the Lagrange variable, (k+1) is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, and Ω represents the Lagrange variable, (k+1) is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, and Ω represents the Lagrange variable, (k) (15)

[0158] is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, and Ω represents the Lagrange variable, (k+1) is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, and Ω represents the Lagrange variable, (k+1) is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, and Ω represents the Lagrange variable, (k) is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, and Ω represents the Lagrange variable,

[0159] The method for solving the variable Ω is as follows:

[0160] S3.2.6: the method for solving the variable of the Lagrange function Ω, specifically comprising:

[0161]

[0162] wherein X (k+1) is the output surface target after the k+1th iteration, k is the iteration number, Ω represents the Lagrange variable, represents the update of X (k+1) is the corresponding Hankel matrix constructed, U and V are respectively two variables after the factorization of the kernel norm relaxation, the superscript H is the conjugate transpose of the matrix.

[0163] S3.2.7: substituting the solved variable S and X into the optimization target equation to obtain the expression of the sparse and low-rank decomposition algorithm The expression is:

[0164]

[0165] wherein B is a SAR image with noise, is the SAR image after denoising by the sparse and low-rank decomposition algorithm.

[0166] The SAR image is reconstructed by using the decomposed X and S matrices. Specifically, the sparse matrix S is used to reconstruct the high-resolution point targets in the SAR image, ensuring that the fine structure of the point targets is preserved. The low-rank matrix X provides the extended target area information in the SAR image, ensuring the imaging quality of the large-area region.

[0167] For example, in the above scenario, the decomposition result shows that the point targets have high brightness and clear structure in the image, and the extended target area still maintains complete structural characteristics under low resolution, realizing efficient SAR image reconstruction.

[0168] S4: solving the preset SAR imaging model by using the expression of the sparse and low-rank decomposition algorithm, and outputting the SAR imaging result.

[0169] It should be noted that the SAR imaging model of the present application is based on sparse and low-rank decomposition method, combined with the characteristics of low-rank structure of Hankel matrix, to realize high-precision reconstruction of SAR image. First, the model introduces the linear relationship between reflectivity matrix and observation data, and constructs a reconstruction error term based on the measurement matrix, to quantify the accuracy of data reconstruction in the SAR imaging process. Then, in order to utilize the sparsity and low-rank of SAR data, the reflectivity matrix is converted into Hankel matrix and low-rank constraint is applied, and balanced parameters are used to control the weight of low-rank component, so as to enhance the detail preservation ability of imaging. Finally, by solving the optimization problem model, high-precision SAR imaging results can be obtained under the balance of sparse and low-rank components.

[0170] More specifically, the model expression for establishing SAR imaging is:

[0171]

[0172] wherein, is a parameter of reflectivity F, Θ is a measurement matrix determined by the SAR system, represents a diagonal matrix, is a parameter of auxiliary variable v, is a parameter of auxiliary variable Φ, Y is a one-dimensional vector form after vectorization of two-dimensional echo data, F is a one-dimensional vector form after vectorization of two-dimensional reflectivity matrix of target area, is a Hankel matrix constructed for variable v, λ is a regularization parameter, R ANK (·) is the rank of the matrix.

[0173] More specifically, the SAR imaging result output by the sparse and low-rank decomposition algorithm expression specifically includes:

[0174] S5.1: based on the ADMM algorithm result, the gradual alternation between |F| and v is estimated under the constraint |F|=v for equation (17), and the expression is:

[0175]

[0176] wherein, is a parameter of reflectivity F, η is a penalty parameter affecting the convergence speed, taking a value of 0.1, u is an auxiliary variable, |F| is the size of reflectivity F, Y is a one-dimensional vector form after vectorization of two-dimensional echo data, Θ is a measurement matrix determined by the SAR system, Φ is a diagonal matrix, λ is a regularization parameter representing the balance between data consistency term and regularization term, taking a value of 0.5, v and u are auxiliary variables, R ANK (·) is the rank of the matrix, set to 1, and η is a penalty parameter, is a Hankel matrix constructed for variable v;

[0177] S5.2: Minimize expression (18) and obtain the quadratic subproblem with respect to the variable |F|, which is expressed as:

[0178]

[0179] Among them, |F| (k+1) is the reflectivity F output after the k+1th iteration, k is the number of iterations, Y is the one-dimensional vector form after the two-dimensional echo data is vectorized, Θ is the measurement matrix determined by the SAR system, Φ (k) is the diagonal matrix output after the kth iteration, η is the penalty parameter, v (k) and u (k) is the auxiliary variable output after the kth iteration, and is solved by the conjugate gradient method as follows:

[0180] |F| (k+1) =(Θ H Θ+λI) -1 (Θ H Y+λ(v (k) -u (k) )) (20)

[0181] Among them, |F| (k+1) is the reflectivity F output after the k+1th iteration, k is the number of iterations, Y is the one-dimensional vector form of the two-dimensional echo data after vectorization, Θ is the measurement matrix determined by the SAR system, λ is the regularization parameter that represents the balance between the control data consistency term and the regularization term, v (k) and u (k) is the auxiliary variable output after the kth iteration, the superscript H is the conjugate transpose of the matrix, and I is the identity matrix;

[0182] S5.3: To separate the phase and amplitude, the reflectivity F is expressed as ΨΑ, where Ψ is a diagonal matrix containing phase information and Α is a vector containing amplitude information. The ΘΦ|F| in formula (18) is converted to ΘΑΨ, thereby dividing the optimization problem (18) into two independent steps for phase and amplitude, as shown below:

[0183] The two variables Ψ and Α are diagonal elements containing the magnitude of phase and reflectivity, respectively. Then, minimizing expression (18) yields a quadratic subproblem with respect to the variable Ψ:

[0184]

[0185] Among them, λ Ψ is the regularization parameter, with a value of 0.5, (k+1)is the diagonal matrix containing the phase output after the k+1th iteration, k is the number of iterations, Y is the one-dimensional vector form after the two-dimensional echo data is vectorized, and A (k) is the diagonal matrix containing the reflectivity output after the kth iteration, Θ is the measurement matrix determined by the SAR system, λ Ψ is the regularization parameter, N is the sum of all observed data, |Ψ i | is the size of the i-th element of the diagonal matrix containing the phase, and is solved iteratively as follows:

[0186]

[0187] Where A is a diagonal matrix containing the reflectivity, j is the imaginary unit in the complex number, and Ψ (k+1) is the diagonal matrix containing the phase output after the k+1th iteration, k is the number of iterations, Y is the one-dimensional vector form after the two-dimensional echo data is vectorized, A is the diagonal matrix containing the reflectivity, Θ is the measurement matrix determined by the SAR system, and λ Ψ is the regularization parameter, N is the sum of all observed data, |Ψ i | is the size of the i-th element of the diagonal matrix containing the phase, φ(·) represents the phase of F, the superscript H is the conjugate transpose of the matrix, and I is the identity matrix. Since the other parameters in formula (22) are all known, formula (22) can be used to obtain Ψ in formula (21): (k+1) ;

[0188] S5.4: Using the phase and amplitude matrices obtained in S5.3, calculate the auxiliary variable v and output the solution for algorithm convergence and stability. The expression is as follows:

[0189] u (k+1) =u (k) +|F| (k+1) -v (k+1) (twenty three)

[0190] Among them, k is the number of iterations, |F| is the size of the reflectivity F, v (k+1) and u (k+1) It is the auxiliary variable output after the k+1th iteration;

[0191] S5.5: Treat subproblem (23) as an image processing problem and let |F| (k+1) +u (k) As the image to be processed, the proposed sparse and low-rank decomposition algorithm is used Replacing the denoising algorithm as a despeckle and image detail preservation term, the following expression is obtained:

[0192]

[0193] S5.6: Repeat the above steps S5.2 to S5.5 until a convergence condition is reached, and substitute the results into formula (20) to obtain the SAR imaging result |F| (k+1) .

[0194] Finally, the effectiveness of the method is verified by evaluating the generated SAR image using the equivalent number of looks (ENL) and edge preservation index (EPI) indicators. Compared with the traditional SAR imaging method, the present application can still maintain good image quality in the presence of noise or data loss, and has strong adaptability to different imaging scenarios.

[0195] The effectiveness of the imaging method is evaluated by the equivalent number of looks (ENL) and edge preservation index (EPI). When the sampling rate is 0.4, the ENL and EPI values of the present method are 34.1703 and 0.5537 respectively, while the ENL value of the prior art RDA is 0.7034, the ENL and EPI values of BM3D are 7.7295 and 0.4021 respectively, and the ENL and EPI values of DnCNN are 30.3605 and 0.5389 respectively. When the sampling rate is 0.8, the ENL and EPI values of the present method are 36.1703 and 0.6033 respectively; the ENL value of RDA is 0.8071, the ENL and EPI values of BM3D are 8.2215 and 0.4611 respectively, and the ENL and EPI values of DnCNN are 33.3247 and 0.5879 respectively.

[0196] RDA has larger error and obvious noise compared with other methods. DnCNN and the present imaging method both show a trend of improved imaging effect as the sampling rate increases, but the present method is superior to the prior art in all indicators, exhibiting the best ENL and EPI values, generating clearer images, and having excellent denoising effect. Therefore, compared with the prior art, the present imaging method can effectively maintain the integrity of the target structure, significantly improve the adaptability to low-structure-definition targets, and thus ensure the SAR imaging effect and the reconstruction accuracy of the target region.

[0197] Embodiment 2:

[0198] The present embodiment provides a SAR imaging system based on sparse and low-rank decomposition, comprising a memory and a processor, wherein the memory comprises a SAR imaging method program based on sparse and low-rank decomposition, and the SAR imaging method program based on sparse and low-rank decomposition is executed by the processor to realize the steps of the SAR imaging method based on sparse and low-rank decomposition as described in embodiment 1.

[0199] Embodiment 3:

[0200] The embodiment provides a computer readable storage medium, wherein the computer readable storage medium comprises a sparse and low-rank decomposition based SAR imaging method program, and the sparse and low-rank decomposition based SAR imaging method program is executed by a processor to realize the steps of the sparse and low-rank decomposition based SAR imaging method as described in the embodiment 1.

[0201] Obviously, the above embodiments of the present application are only examples for clearly illustrating the present application, and are not intended to limit the implementation modes of the present application. Based on the above description, other different forms of changes or variations can be made by those skilled in the art. All the implementation modes do not need to be enumerated here. Any modification, equivalent replacement and improvement within the spirit and principle of the present application should be included in the protection scope of the claims of the present application.

Claims

1. A SAR imaging method based on sparse and low-rank decomposition, characterized in that, The method comprises the following steps: Collecting SAR images and two-dimensional echo data, preprocessing the two-dimensional echo data to obtain target reflectivity; Based on the linear relationship between the two-dimensional echo data and the target reflectivity, the SAR image is decomposed by using a preset method to obtain sparse components and low-rank components; An optimization algorithm is used to iteratively update the decomposition results of the matrix sparsity and low rank, improve the accuracy, and obtain a sparse and low-rank decomposition algorithm expression until the error converges; The sparse and low-rank decomposition algorithm expression is used to solve a preset SAR imaging model to output a SAR imaging result; The optimization algorithm for iteratively updating the decomposition results of the matrix sparsity and low rank is an alternating multiplier method; The expression of the preset SAR imaging model is: wherein, is a parameter of reflectivity F, Θ is a measurement matrix determined by the SAR system, denotes a diagonal matrix, is a parameter of auxiliary variable v, is a parameter of auxiliary variable Φ, Y is a one-dimensional vector form of the two-dimensional echo data after vectorization, F is a one-dimensional vector form of the two-dimensional reflectivity matrix of the target area after vectorization, H(v) is a Hankel matrix constructed for variable v, λ is a regularization parameter, R ANK (·) is the rank of a matrix; The sparse and low-rank decomposition algorithm expression is used to output a SAR imaging result, which specifically comprises: Based on the ADMM algorithm result, the equation (17) is used to estimate the gradual alternation between |F| and v under the constraint |F| = v, and the expression is: wherein, is a parameter of reflectivity F, η is a penalty parameter affecting the convergence speed, u is an auxiliary variable, |F| is the size of reflectivity F, Y is a one-dimensional vector form after vectorization of two-dimensional echo data, Θ is a measurement matrix determined by the SAR system, Φ is a diagonal matrix, v and u are auxiliary variables, R ANK (·) is the rank of the matrix, H(v) is a Hankel matrix constructed for variable v; The quadratic sub-problem is minimized to solve the size of the reflectivity F after the k+1 iteration, and the expression is: where |F| (k+1) is the size of the reflectivity F output after the k+1th iteration, k is the iteration number, Y is a one-dimensional vector form of the two-dimensional echo data after vectorization, Θ is the measurement matrix determined by the SAR system, Φ (k) is the diagonal matrix output after the kth iteration, v (k) and u (k) are auxiliary variables output after the kth iteration, which are solved by the conjugate gradient method as follows: |F| (k+1) = (Θ H Θ + λI) -1 (Θ H Y + λ(v (k) -u (k) )) (20) where |F| is the size of the reflectivity F, k is the iteration number, Y is the one-dimensional vector form of the two-dimensional echo data after vectorization, Θ is the measurement matrix determined by the SAR system, v (k+1) is the size of the reflectivity F output after the k+1th iteration, k is the iteration number, Y is the one-dimensional vector form of the two-dimensional echo data after vectorization, Θ is the measurement matrix determined by the SAR system, v (k) and u (k) is the auxiliary variable output after the kth iteration, the superscript H is the conjugate transpose of the matrix, and I is the unit matrix; In order to separate the phase and amplitude, the reflectivity F is expressed as ΨΑ, wherein Ψ is a diagonal matrix containing phase information, and Α is a diagonal matrix containing reflectivity size. ΘΦ|F| in formula (18) is converted into ΘΑΨ, so that the optimization problem (18) is divided into two independent steps for phase and amplitude, and the expression is as follows: Two variables Ψ and Α are diagonal elements containing phase and reflectivity size, and then the quadratic sub-problem about the variable Ψ is obtained by minimizing the expression (18): Among them, λ Ψ is the regularization parameter, Ψ (k+1) is the diagonal matrix containing the phase output after the k+1th iteration, k is the number of iterations, Y is the one-dimensional vector form after the two-dimensional echo data is vectorized, and A (k) is the diagonal matrix containing the reflectivity output after the kth iteration, Θ is the measurement matrix determined by the SAR system, N is the sum of all observation data, |Ψ i | is the size of the i-th element of the diagonal matrix containing the phase, and is solved iteratively as follows: where A is a diagonal matrix containing the magnitude of reflectivity, j is the imaginary unit in complex number, Ψ (k+1) is the diagonal matrix containing the phase after the k+1th iteration, k is the iteration number, Y is the one-dimensional vector form of the two-dimensional echo data after vectorization, Θ is the measurement matrix determined by the SAR system, N is the sum of all observation data, |Ψ i | is the magnitude of the i-th element of the diagonal matrix containing the phase, φ(·) represents the phase of F, the superscript H is the conjugate transpose of the matrix, I is the unit matrix, because other parameters in formula (22) are known quantities, Ψ (k+1) in formula (21) can be obtained through formula (22). The phase and amplitude matrix obtained by the above calculation is used to calculate the auxiliary variable v, and the solving result for algorithm convergence and stability is output, and the expression is as follows: u (k+1) = u (k) + |F| (k+1) - v (k+1) (23) where k is the iteration number, |F| is the size of reflectance F, v (k+1) and u (k+1) is the auxiliary variable output after the k+1 iteration. Considering sub-problem (23) as an image processing problem, |F| (k+1) +u (k) as the image to be processed, and replacing the denoising algorithm with the proposed sparse and low-rank decomposition algorithm D(·) as the despeckling and image detail preservation term, we have the following expression: v (k+1) = D ( | F | (k+1) + u (k) ) (24) Repeat the above steps until the convergence condition is reached, and substitute the result into formula (20) to obtain the SAR imaging result |F| (k+1) .

2. The SAR imaging method based on sparse and low-rank decomposition according to claim 1, characterized in that, The SAR image is decomposed by using the kernel norm relaxation factorization, specifically comprising: The SAR image B is decomposed into a surface target X and a point target S, and the low-rank matrix fitting algorithm is used to initialize X and S, and the expression is as follows: B=X+S (1) The face target X of size M x N is discretized at the Nyquist sampling rate to obtain X i,j = x[i, j], a corresponding Hankel matrix H(X) is constructed, and the expression is as follows: wherein is represented by: The SAR image B to be processed is decomposed into a sparse component and a low-rank component under the constraint of a Hankel structured matrix, and a sparse and low-rank balance parameter is used to output the decomposition result for model optimization, and the expression is as follows: Wherein, the sparse component S represents a point target, the low-rank component L corresponds to a surface target, τ is a balance parameter between low-rank and sparsity, and H(X) is a Hankel matrix constructed by using X.

3. The SAR imaging method based on sparse and low-rank decomposition according to claim 2, characterized in that, The decomposition results of the matrix sparsity and low rank are iteratively updated to improve the accuracy until the error converges, and a sparse and low-rank decomposition algorithm expression is obtained, specifically comprising: The sparse and low-rank decomposition equation in the expression (2) is converted into an optimization target equation by using the kernel norm relaxation factorization form of the alternating multiplier method, and the expression is as follows: where D τ (X) is the name of the optimization target equation, U and V are two variables after factorization of H(X) by kernel norm relaxation, S is the sparse component raised to the Hankel matrix form, UV H is the decomposition factor of the low-rank component, and the superscript H is the conjugate transpose of the matrix; Based on the optimization objective equation and its related constraint conditions of the SAR imaging problem, a Lagrange function is constructed by using the Lagrange multiplier method; the sparse and low-rank parts in the SAR imaging model are optimized by using the ADMM algorithm, and the Lagrange variables S, X, U, V, Λ and Ω are sequentially solved, and the Lagrange function expression is as follows: Wherein, τ, α and β represent the balance parameters, Λ, Ω represent the Lagrange variables, B is the SAR image to be processed, H(X) is the Hankel matrix constructed by using X, S and X are point targets and surface targets respectively, U and V are two variables after factorization of H(X) by using the kernel norm relaxation; The solved variables S and X are substituted into the optimization objective equation to obtain the sparse and low-rank decomposition algorithm expression D(·), and the expression is as follows: wherein B is a noisy SAR image, is a SAR image denoised by the sparse low-rank decomposition algorithm.

4. The SAR imaging method based on sparse and low-rank decomposition according to claim 3, characterized in that: The specific steps for solving the Lagrange variables include: The S sub-problem is defined as minimizing the expression (6) to obtain the quadratic sub-problem about the S variable: where X (k) is the output surface target after the kth iteration, k is the iteration number, a represents the trade-off parameter, A represents the Lagrange variable, B is the SAR image to be processed, S is the point target, and τ represents the balance parameter between low rank and sparsity. Simplify expression (9) with respect to S, by Γ τ / α denotes the pixel-wise soft threshold, resulting in the following solution: S (k+1) = Γ τ / α (B - X (k) - Λ (k) ) (10) where S (k+1) is the point target output after the k+1th iteration, k is the iteration number, a represents the trade-off parameter, Λ represents the Lagrange variable, B is the SAR image to be processed, X (k) is the surface target output after the kth iteration, τ represents the balance parameter between low rank and sparsity, Γ τ / α represents the soft threshold of each pixel in S; The X sub-problem is defined as minimizing the expression (6) to obtain the quadratic sub-problem about the X variable: where X (k+1) is the output surface target after the k+1 iteration, k is the iteration number, a and b represent the trade-off parameters, A and W represent the Lagrange variables, B is the SAR image to be processed, S is the point target, H(X) represents the Hankel matrix constructed for X, and U and V are two variables after the factorization of H(X) through the kernel norm relaxation. The expression (7) about X is simplified to obtain the following solution: where X (k+1) is the output of the (k + 1)th iteration, k is the iteration number, a and b are trade-off parameters, A and W are Lagrange multipliers, B is the SAR image to be processed, S is the point target, U and V are two variables after factorization of H(X) by kernel norm relaxation, denotes the pseudo-inverse mapping from the Hankel matrix back to the initial state, and its expression is as follows: denotes the pseudo-inverse mapping from the Hankel structure back to the initial state, the operator H * denotes the summing of multiple elements in H and putting them back into the initial coordinates, (H * H) -1 denotes dividing by the number of corresponding items; The U sub-problem is defined as minimizing the expression (6) to obtain the quadratic sub-problem about the U variable: where X (k+1) is the output surface target after the k+1 iteration, k is the iteration number, β represents the trade-off parameter, Ω represents the Lagrange variable, H(X (k+1) ) represents the Hankel matrix constructed for X (k+1) , U and V are two variables after the factorization of H(X) through the kernel norm relaxation, and the superscript H is the conjugate transpose of the matrix. The derivative of the matrix U of the expression (11) is taken and is equal to zero to obtain the following closed-form solution: U (k+1) = β(H(X (k+1) )+ Ω (k) )V (k) (βV (k)H V (k) +I) -1 (12) wherein X (k+1) is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, Ω represents the Lagrange variable, H(X (k+1) ) represents the Hankel matrix constructed for X (k+1) , U and V are two variables after factorization of H(X) by kernel norm relaxation, the superscript H is the conjugate transpose of the matrix, and I is the identity matrix; The V sub-problem is defined as minimizing the expression (6) to obtain the quadratic sub-problem about the V variable: where X (k+1) is the output surface target after the k+1 iteration, k is the iteration number, β represents the trade-off parameter, Ω represents the Lagrange variable, H(X (k+1) ) represents the Hankel matrix constructed for X (k+1) , U and V are two variables after the factorization of H(X) through the kernel norm relaxation, and the superscript H is the conjugate transpose of the matrix. The derivative of the matrix V of the expression (13) is taken and is equal to zero to obtain the following closed-form solution: V (k+1) = β(H(X (k+1) )+ Ω (k) ) H U (k+1) (βU (k+1)H U (k+1) +I) -1 (14) where X (k+1) is the output surface target after the k+1th iteration, k is the iteration number, β represents the trade-off parameter, Ω represents the Lagrange variable, H(X (k+1) ) represents the Hankel matrix constructed for X (k+1) , U and V are two variables after factorization of H(X) by kernel norm relaxation, the superscript H is the conjugate transpose of the matrix, and I is the identity matrix; The method for solving the Lagrange function Λ variable includes: Λ (k+1) = X (k+1) + S (k+1) - B + Λ (k) (15) wherein X (k+1) is the surface target output after the k+1th iteration, k is the iteration number, S (k+1) represents the point target output after the k+1th iteration, Λ (k) represents the Lagrange variable after the kth iteration, and B is the SAR image to be processed. The method for analyzing the Ω variable is as follows: The method for solving the Lagrange function Ω variable includes: Ω (k+1) = H(X (k+1) )- U (k+1) V (k+1)H + Ω (k) (16) where X (k+1) is the output surface target after the k+1th iteration, k is the iteration number, Ω represents the Lagrange variable, H(X (k +1) ) represents the corresponding Hankel matrix constructed for X (k+1) , U and V are two variables after the factorization of H(X) by the kernel norm relaxation, and the superscript H is the conjugate transpose of the matrix.

5. A SAR imaging system based on sparse and low-rank decomposition, characterized in that, The system includes a memory and a processor, the memory includes a sparse and low-rank decomposition-based SAR imaging method program, and the sparse and low-rank decomposition-based SAR imaging method program is executed by the processor to realize the steps of the sparse and low-rank decomposition-based SAR imaging method in any one of claims 1 to 4.

6. A computer-readable storage medium, characterized in that, The computer readable storage medium includes a sparse and low-rank decomposition-based SAR imaging method program, and the sparse and low-rank decomposition-based SAR imaging method program is executed by the processor to realize the steps of the sparse and low-rank decomposition-based SAR imaging method in any one of claims 1 to 4.

Images