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

Abstract

The invention discloses an 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 an SAR image and two-dimensional echo data, and preprocessing the two-dimensional echo data to obtain a target reflectivity; and based on the linear relationship between the two-dimensional echo data and the target reflectivity, decomposing the SAR image by using a preset method to obtain a sparse component and a low-rank component. A decomposition result is iteratively updated through a preset optimization algorithm, the precision is improved until errors converge, and a sparse and low-rank decomposition algorithm expression is obtained; and solving a preset SAR imaging model by using the sparse and low-rank decomposition algorithm expression, outputting an SAR imaging result, and finally outputting the SAR imaging result. According to the invention, the complexity of SAR imaging can be effectively reduced, and the imaging efficiency is improved. Even if the data sampling rate is reduced, the accuracy of image reconstruction can be ensured.

Description

Technical Field

[0001] The present invention belongs to the field of radar imaging, and more specifically, relates to a SAR imaging method, system and readable storage medium based on sparse and low-rank decomposition. Background Art

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

[0003] SAR imaging faces several challenges. It is difficult to maintain the structural integrity of the original data in sparse synthetic aperture radar imaging for targets with periodic or local correlation, thus failing to guarantee the SAR imaging effect. In addition, the computational complexity is high and it is difficult to adapt to practical application requirements, especially in dynamic or complex scenes.

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

[0005] S3: Based on the image noise, the OMP algorithm is used to calculate the sparse code, and the dictionary is iteratively updated to obtain an updated dictionary and sparse code; S4: The denoised SAR image is sparsely decomposed and reconstructed using the updated dictionary and sparse code to obtain a denoised image. This scheme has the problem of high computational complexity and is difficult to adapt to actual application needs. Summary of the invention

[0006] The present invention aims to overcome the problems faced by SAR imaging in the prior art, especially the difficulty in maintaining the structural integrity of the original data of targets with periodicity or local correlation in sparse synthetic aperture radar imaging, thereby failing to ensure the SAR imaging effect, and the high computational complexity of the prior art, which is difficult to adapt to actual application needs. A SAR imaging method, system and readable storage medium based on sparse and low-rank decomposition are provided.

[0007] The primary purpose of the present invention is to solve the above technical problems. The technical solutions of the present invention are as follows:

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

[0009] Collect SAR images and two-dimensional echo data, pre-process the two-dimensional echo data, and obtain the target reflectivity;

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

[0011] Using the preset optimization algorithm, iteratively update the matrix sparse and low-rank decomposition results to improve the accuracy until the error converges and obtain the sparse and low-rank decomposition algorithm expression;

[0012] The sparse and low-rank decomposition algorithm expression is used to solve the preset SAR imaging model and output the SAR imaging result.

[0013] Furthermore, the optimization algorithm for iteratively updating the matrix sparse and low-rank decomposition results is an alternating multiplier method.

[0014] Furthermore, the preset method for decomposing the SAR image is a factorization of a relaxed nuclear paradigm.

[0015] Furthermore, the decomposition of the SAR image by factorization using the relaxed nuclear paradigm specifically includes:

[0016] The SAR image B is decomposed into surface target X and point target S. The low-rank matrix fitting algorithm is used to initialize X and S. The expression is as follows:

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

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

[0019]

[0020] in It is expressed as:

[0021]

[0022] The SAR data B is decomposed into sparse components and low-rank components under the constraints of the Hankel structured matrix. The sparse and low-rank balance parameters are used to output the decomposition results for model optimization. The expression is as follows:

[0023]

[0024] Among them, the sparse component S represents the point target, the low-rank component L corresponds to the surface target, and τ is the balance parameter between low rank and sparsity. is the Hankel matrix constructed using X.

[0025] Furthermore, the iterative update of the matrix sparse and low-rank decomposition results improves the accuracy until the error converges, and the sparse and low-rank decomposition algorithm expression is obtained, which specifically includes:

[0026] Using the factorization form of the nuclear norm relaxation of the alternating multiplier method, the sparse and low-rank decomposition equations in expression (2) are transformed into the optimization objective equation expression:

[0027]

[0028] in, is the name of the optimization target equation, U and V are The two variables after factorization by nuclear paradigm relaxation, S is the sparse component promoted to 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;

[0029] Based on the optimization objective equation of the SAR imaging problem and its related constraints, the Lagrangian function is constructed using the Lagrangian multiplier method. The ADMM algorithm is used to optimize the sparse and low-rank parts in the SAR imaging model and solve the Lagrangian variables S, X, U, V, Λ and Ω in turn. The Lagrangian function expression is as follows:

[0030]

[0031] Among them, τ, α and β represent equilibrium parameters, Λ, 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.

[0032] Substitute the solved variables S and X into the optimization objective equation to obtain the sparse and low-rank decomposition algorithm expression Its expression is:

[0033]

[0034] Where, B is a noisy SAR image, This is a SAR image denoised by the sparse low-rank decomposition algorithm.

[0035] Furthermore, the specific steps for solving the Lagrangian variables include:

[0036] Defining the S subproblem as minimizing expression (6) yields a quadratic subproblem with respect to the S variable:

[0037]

[0038] Among them, X (k) is the surface target output after the kth iteration, k is the number of iterations, 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;

[0039] Simplifying expression (9) with respect to S, we can get τ / α Representing pixel-by-pixel soft thresholding, we obtain the following solution:

[0040] S (k+1) =Γ τ / α (BX (k) -Λ (k) ) (10)

[0041] Among them, S (k+1) is the point target output after the k+1th iteration, k is the number of iterations, α is the trade-off parameter, Λ is the Lagrangian variable, B is the SAR image to be processed, and 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.

[0042] Defining the X subproblem as minimizing expression (6) yields a quadratic subproblem with respect to the X variable:

[0043]

[0044] Among them, X (k+1) is the surface target output after the k+1th iteration, k is the number of iterations, α and β represent the trade-off parameters, Λ and Ω represent the Lagrangian variables, B is the SAR image to be processed, S is the point target, It represents the Hankel matrix constructed for X, U and V are Two variables after factorization with nuclear paradigm relaxation;

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

[0046]

[0047] Among them, X (k+1)is the surface target output after the k+1th iteration, k is the number of iterations, α and β represent the trade-off parameters, Λ and Ω represent the Lagrangian variables, B is the SAR image to be processed, S is the point target, U and V are After the two variables are factorized by nuclear paradigm relaxation, represents the pseudo-inverse mapping from the Hankel matrix back to the initial state, and its expression is as follows:

[0048]

[0049] represents the pseudo-inverse mapping from the Hankel structure back to the initial state, the operator Express Sum the multiple elements and put them back to their original coordinates. represents the number of divisions by multiple corresponding terms;

[0050] Defining the U subproblem as minimizing expression (6) yields a quadratic subproblem with respect to the U variable:

[0051]

[0052] 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;

[0053] Taking the derivative of matrix U with respect to expression (11) and equating it to zero yields the following closed form solution:

[0054]

[0055] 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;

[0056] Defining the V subproblem as minimizing expression (6) yields a quadratic subproblem with respect to the V variable:

[0057]

[0058] 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;

[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] Furthermore, the preset SAR imaging model expression is:

[0070]

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

[0072] Furthermore, the outputting of SAR imaging results by using sparse and low-rank decomposition algorithm expressions specifically includes:

[0073] For equation (17), based on the results of the ADMM algorithm, the gradual alternation between estimating |F| and v under the constraint |F| = v is expressed as:

[0074]

[0075] in, is the shape parameter of the reflectivity F, η is the penalty parameter that affects the convergence speed, u is an auxiliary variable, |F| is the size of the reflectivity F, 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, Φ is a diagonal matrix, λ is the regularization parameter that represents the balance between the control data consistency term and the regularization term, v and u are auxiliary variables, R ANK (·) is the rank of the matrix, η is the penalty parameter, The Hankel matrix constructed for the variable v;

[0076] Minimize expression (18) to solve the quadratic subproblem and obtain the reflectivity F after the k+1th iteration, which is expressed as:

[0077]

[0078] 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, 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:

[0079]

[0080] 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;

[0081] In order to separate the phase and amplitude, the reflectivity F is represented 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. The expression is as follows:

[0082] 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 about the Ψ variable:

[0083]

[0084] 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 of the two-dimensional echo data after vectorization, Α (k) is the diagonal matrix containing the reflectivity output after the kth iteration, 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, and is solved iteratively as follows:

[0085]

[0086] 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 of the two-dimensional echo data after vectorization, Α is the diagonal matrix containing the reflectivity, 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 unit matrix. Since the other parameters in formula (22) are all known quantities, 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] Where k is the number of iterations, |F| is the reflectivity F, and 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] A 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, the steps of a SAR imaging method based on sparse and low rank decomposition are implemented.

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

[0096] The present invention can effectively reduce the complexity of SAR imaging and improve the imaging speed by combining sparse and low-rank decomposition techniques. When processing echo data, the sparse signal processing technology is used to ensure the reconstruction accuracy of the image when the data sampling rate is reduced. Compared with the traditional SAR imaging method, the present invention can still maintain good image quality when processing noisy or missing data, and has strong adaptability to different imaging scenes. BRIEF DESCRIPTION OF THE DRAWINGS

[0097] In order to make the purpose and technical solution of the present invention clearer, the present invention provides the following drawings and descriptions:

[0098] Figure 1 A flow chart of a method provided by an embodiment of the present invention. DETAILED DESCRIPTION

[0099] In order to more clearly understand the above-mentioned purpose, features and advantages of the present invention, the present invention is further described in detail below in conjunction with the accompanying drawings and specific embodiments. It should be noted that the embodiments of the present application and the features in the embodiments can be combined with each other without conflict.

[0100] In the following description, many specific details are set forth to facilitate a full understanding of the present invention. However, the present invention may also be implemented in other ways different from those described herein. Therefore, the protection scope of the present invention is not limited to the specific embodiments disclosed below.

[0101] Embodiment 1:

[0102] The present invention provides a SAR imaging method based on sparse and low-rank decomposition, such as Figure 1 The figure shows a flow chart of a SAR imaging method based on sparse and low-rank decomposition. The specific steps are as follows:

[0103] S1: Using the SAR system, according to the geometric relationship between the observation platform and the imaging scene, collect SAR images and two-dimensional echo data, pre-process the two-dimensional echo data, and obtain the 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, where the SAR image contains sparse point targets and extended surface targets of the target scene. Assume that this size is 512×512 pixels, and 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 are extended area targets.

[0106] S2: Based on the linear relationship between the two-dimensional echo data and the target reflectivity, the SAR image is decomposed using the factorization method of nuclear paradigm relaxation to obtain sparse components and low-rank components. By combining sparse and low-rank decomposition techniques, the complexity of SAR imaging can be effectively reduced and the imaging speed can be improved. When processing echo data, sparse signal processing technology is used to ensure the image reconstruction accuracy when the data sampling rate is reduced.

[0107] More specifically, the factorization method of the nuclear paradigm relaxation decomposes the SAR image, specifically including:

[0108] S2.1: Decompose the SAR image B into surface targets X and point targets S, and initialize X and S using the low-rank matrix fitting algorithm. The expressions are as follows:

[0109] B=X+S (1)

[0110] S2.2: Discretize the surface target X of size M×N at the Nyquist sampling rate to obtain X i,j =x[i,j], construct the corresponding Hankel matrix The expression is as follows:

[0111]

[0112] in It is expressed as:

[0113]

[0114] The lifting scheme for the Hankel structured matrix is ​​linear, therefore, the sparse components representing point targets in SAR imaging are also enhanced in the enhanced Hankel structured matrix. In the lifted Hankel matrix, our algorithm focuses on decomposing the data into two sparse and low-rank components;

[0115] S2.3: Decompose the SAR data B into sparse components and low-rank components under the constraints of the Hankel structured matrix. Use the sparse and low-rank balance parameters to output the decomposition results for model optimization. The expression is as follows:

[0116]

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

[0118] According to the sparse and low-rank decomposition method proposed in the present invention, the Hankel matrix B constructed based on the SAR image is decomposed into a low-rank matrix X and a sparse matrix S, where: X represents the low-rank part of the extended target, which can capture the large-area regional features in the target scene. S represents the part of the sparse point target in the scene, which retains the high-resolution characteristics of the point target.

[0119] In actual operation, the nuclear norm minimization method without singular value decomposition (SVD) is used to decompose the matrix B, avoiding the singular value decomposition process and greatly speeding up the calculation. The low-rank matrix fitting algorithm (LMaFit) is used to initialize X and S, and the alternating multiplier method (ADMM) algorithm is used for iterative optimization to obtain accurate decomposition results.

[0120] S3: Using the preset optimization algorithm, iteratively update the matrix sparse and low-rank decomposition results to improve the accuracy until the error converges and obtain the sparse and low-rank decomposition algorithm expression;

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

[0122] More specifically, the alternating multiplier method algorithm is used to iteratively update the sparse and low-rank decomposition results of the matrix to improve the accuracy until the error converges, and the sparse and low-rank decomposition algorithm expression is obtained, which specifically includes:

[0123] S3.1: Using the factorization form of the nuclear paradigm relaxation of ADMM, the sparse and low-rank decomposition equations in expression (2) are converted into the optimization objective equation expression:

[0124]

[0125] in, is the name of the optimization target equation, U and V are The two variables after factorization by nuclear paradigm relaxation, S is the sparse component promoted to 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;

[0126] S3.2: Based on the optimization objective equation of the SAR imaging problem and its related constraints, the Lagrangian function is constructed using the Lagrangian multiplier method. The ADMM algorithm is used to optimize the sparse and low-rank parts of the SAR imaging model and solve the Lagrangian variables S, X, U, V, Λ and Ω in turn. The Lagrangian function expression is as follows:

[0127]

[0128] Among them, τ, α and β represent balance parameters, τ takes the value of 0.3, α and β take the values ​​of 1 and 1, Λ, 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: solve S first, then use the solved S to solve X, then use X to solve U, solve U and X to solve V, then solve Λ from S and X, 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] Simplifying expression (9) with respect to S, we can get τ / α Representing pixel-by-pixel soft thresholding, we obtain 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 trade-off parameter, Λ is the Lagrangian variable, B is the SAR image to be processed, and 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)is the surface target output after the k+1th iteration, k is the number of iterations, α and β represent the trade-off parameters, Λ and Ω represent the Lagrangian variables, B is the SAR image to be processed, S is the point target, It represents the Hankel matrix constructed for X, U and V are Two variables after factorization with nuclear paradigm relaxation;

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

[0140]

[0141] Among them, X (k+1) is the surface target output after the k+1th iteration, k is the number of iterations, α and β represent the trade-off parameters, Λ and Ω represent the Lagrangian variables, B is the SAR image to be processed, S is the point target, U and V are After the two variables are factorized by nuclear paradigm relaxation, represents the pseudo-inverse mapping from the Hankel matrix back to the initial state, and its expression is as follows:

[0142]

[0143] represents the pseudo-inverse mapping from the Hankel structure back to the initial state, the operator Express Sum the multiple elements and put them back to their original coordinates. represents the number of divisions by multiple corresponding terms;

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

[0145]

[0146] 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;

[0147] Taking the derivative of matrix U with respect to expression (11) and equating it to zero yields the following closed form solution:

[0148]

[0149] 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;

[0150] S3.2.4: Define the V subproblem as minimizing expression (6) to obtain a quadratic subproblem with respect to the V variable:

[0151]

[0152] 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;

[0153] Taking the derivative of the matrix V with respect to expression (13) and equating it to zero yields the following closed form solution:

[0154]

[0155] 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;

[0156] S3.2.5: Methods for solving the Lagrangian function Λ variable, specifically including:

[0157] Λ (k+1) =X (k+1) +S (k+1) -B+Λ (k) (15)

[0158] 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.

[0159] The method to parse the Ω variable is as follows:

[0160] S3.2.6: Methods for solving the Lagrangian function Ω variable, including:

[0161]

[0162] 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.

[0163] S3.2.7: Substitute the solved variables S and X into the optimization objective equation to obtain the sparse and low-rank decomposition algorithm expression Its expression is:

[0164]

[0165] Where, B is a noisy SAR image, This is a SAR image denoised by the sparse low-rank decomposition algorithm.

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

[0167] For example, in the above scenario, the decomposition results show that the point target has high brightness and clear structure in the image, while the extended target area can still maintain complete structural characteristics at low resolution, achieving efficient SAR image reconstruction.

[0168] S4: Solve the preset SAR imaging model using the sparse and low-rank decomposition algorithm expression and output the SAR imaging result.

[0169] It should be noted that the SAR imaging model of the present invention is based on the sparse and low-rank decomposition method, combined with the characteristics of the low-rank structure of the Hankel matrix, to achieve high-precision reconstruction of SAR images. First, the model introduces the linear relationship between the reflectivity matrix and the observed data, and constructs a reconstruction error term based on the measurement matrix to quantify the accuracy of data reconstruction during SAR imaging. Then, in order to utilize the sparsity and low-rank nature of SAR data, the reflectivity matrix is ​​converted into a Hankel matrix and a low-rank constraint is imposed, and a balance parameter is used to control the weight of the low-rank component, thereby enhancing the imaging detail retention capability. 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] in, is the shape parameter of the reflectivity F, Θ is the measurement matrix determined by the SAR system, represents a diagonal matrix, is the parameter of the auxiliary variable v, is the parameter of the auxiliary variable Φ, Y is the one-dimensional vector form of the two-dimensional echo data after vectorization, F is the one-dimensional vector form of the two-dimensional reflectivity matrix of the target area after vectorization, is the Hankel matrix constructed for the variable v, λ is the regularization parameter, R ANK (·) is the rank of the matrix.

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

[0174] S5.1: For equation (17), based on the ADMM algorithm results, the asymptotic alternation between estimating |F| and v under the constraint |F| = v is expressed as:

[0175]

[0176] in, is the shape parameter of the reflectivity F, η is the penalty parameter that affects the convergence speed, and its value is 0.1, u is an auxiliary variable, |F| is the size of the reflectivity F, 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, Φ is a diagonal matrix, λ is the regularization parameter that represents the balance between the control data consistency term and the regularization term, and its value is 0.5, v and u are auxiliary variables, R ANK (·) is the rank of the matrix, set to 1, η is the penalty parameter, The Hankel matrix constructed for the variable v;

[0177] S5.2: Minimize expression (18) and obtain the quadratic subproblem about the |F| variable, 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 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, 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: 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, thereby dividing the optimization problem (18) into two independent steps for phase and amplitude. The expression is as follows:

[0183] The two variables Ψ and Α are diagonal elements containing the magnitude of phase and reflectivity, respectively. Then, by minimizing expression (18), we obtain the quadratic subproblem about the Ψ variable:

[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 of the two-dimensional echo data after vectorization, Α (k) is the diagonal matrix containing the reflectivity output after the kth iteration, 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, 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 of the two-dimensional echo data after vectorization, Α is the diagonal matrix containing the reflectivity, 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, φ(·) represents the phase of F, the superscript H is the conjugate transpose of the matrix, and I is the unit matrix. Since the other parameters in formula (22) are all known quantities, 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] Where k is the number of iterations, |F| is the reflectivity F, and 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 replace |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 the convergence condition is reached, and substitute the result into formula (20) to obtain the SAR imaging result |F| (k+1) .

[0194] Finally, the generated SAR images are evaluated and the effectiveness of the method is verified using the equivalent number of views (ENL) and edge preservation index (EPI) indicators. Compared with traditional SAR imaging methods, the present invention can still maintain good image quality when processing noisy or missing data, and has strong adaptability to different imaging scenarios.

[0195] This embodiment evaluates the effectiveness of the imaging method by the equivalent number of views (ENL) and the edge preservation index (EPI). When the sampling rate is 0.4, the ENL and EPI values ​​of this 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, 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 this 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] Compared with other methods, RDA has larger errors and obvious noise. Both DnCNN and this imaging method show a trend of improved imaging effect when the sampling rate increases, but this method is superior to the existing technology in all indicators, showing the best ENL and EPI values, generating clearer images, and having excellent denoising effect. Therefore, compared with the existing technology, this imaging method can effectively maintain the integrity of the target structure and significantly improve the adaptability to targets with low structural clarity, thereby ensuring the SAR imaging effect and the reconstruction accuracy of the target area.

[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 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 as described in Example 1 are implemented.

[0199] Embodiment 3:

[0200] The present embodiment 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, the steps of a SAR imaging method based on sparse and low-rank decomposition as described in Example 1 are implemented.

[0201] Obviously, the above embodiments of the present invention are merely examples for clearly illustrating the present invention, and are not intended to limit the embodiments of the present invention. For those skilled in the art, other different forms of changes or modifications can be made based on the above description. It is not necessary and impossible to list all the embodiments here. Any modifications, equivalent substitutions and improvements made within the spirit and principles of the present invention should be included in the protection scope of the claims of the present invention.

Claims

1. A SAR imaging method based on sparse and low-rank decomposition, characterized in that: The steps include: Collect SAR images and two-dimensional echo data, pre-process the two-dimensional echo data, and obtain the target reflectivity; Based on the linear relationship between the two-dimensional echo data and the target reflectivity, the SAR image is decomposed using a preset method to obtain a sparse component and a low-rank component; Using the preset optimization algorithm, iteratively update the matrix sparse and low-rank decomposition results to improve the accuracy until the error converges and obtain the sparse and low-rank decomposition algorithm expression; The sparse and low-rank decomposition algorithm expression is used to solve the preset SAR imaging model and output the SAR imaging result.

2. The SAR imaging method based on sparse and low-rank decomposition according to claim 1, characterized in that: The optimization algorithm for iteratively updating the matrix sparse and low-rank decomposition results is the alternating multiplier method.

3. The SAR imaging method based on sparse and low-rank decomposition according to claim 1, characterized in that: The preset method for decomposing the SAR image is a factorization of a relaxed nuclear paradigm.

4. The SAR imaging method based on sparse and low-rank decomposition according to claim 3, characterized in that: The SAR image is decomposed by factorization using nuclear paradigm relaxation, specifically including: The SAR image B is decomposed into surface target X and point target S. The low-rank matrix fitting algorithm is used to initialize X and S. The expression is as follows: B=X+S (1) Discretize the surface target X of size M×N at the Nyquist sampling rate to obtain X i,j =x[i,j], construct the corresponding Hankel matrix The expression is as follows: in It is expressed as: The SAR data B is decomposed into sparse components and low-rank components under the constraints of the Hankel structured matrix. The sparse and low-rank balance parameters are used to output the decomposition results for model optimization. The expression is as follows: Among them, the sparse component S represents the point target, the low-rank component L corresponds to the surface target, and τ is the balance parameter between low rank and sparsity. is the Hankel matrix constructed using X.

5. The SAR imaging method based on sparse and low-rank decomposition according to claim 4, characterized in that: The iterative update of the matrix sparse and low-rank decomposition results improves the accuracy until the error converges, and the sparse and low-rank decomposition algorithm expression is obtained, which specifically includes: Using the factorization form of the nuclear norm relaxation of the alternating multiplier method, the sparse and low-rank decomposition equations in expression (2) are transformed into the optimization objective equation expression: in, is the name of the optimization target equation, U and V are The two variables after factorization by nuclear paradigm relaxation, S is the sparse component promoted to 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 of the SAR imaging problem and its related constraints, the Lagrangian function is constructed using the Lagrangian multiplier method. The ADMM algorithm is used to optimize the sparse and low-rank parts in the SAR imaging model and solve the Lagrangian variables S, X, U, V, Λ and Ω in turn. The Lagrangian function expression is as follows: Among them, τ, α and β represent equilibrium parameters, Λ, 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 with nuclear paradigm relaxation; Substitute the solved variables S and X into the optimization objective equation to obtain the sparse and low-rank decomposition algorithm expression Its expression is: Where, B is a noisy SAR image, This is a SAR image denoised by the sparse low-rank decomposition algorithm.

6. The SAR imaging method based on sparse and low-rank decomposition according to claim 5, characterized in that: The specific steps to solve the Lagrangian variables include: Defining the S subproblem as minimizing expression (6) yields a quadratic subproblem with respect to the S variable: Among them, X (k) is the surface target output after the kth iteration, k is the number of iterations, 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; Simplifying expression (9) with respect to S, we can get τ / α Representing pixel-by-pixel soft thresholding, we obtain the following solution: S (k+1) =C τ / α (BX (k) -L (k) ) (10) Among them, S (k+1) is the point target output after the k+1th iteration, k is the number of iterations, α is the trade-off parameter, Λ is the Lagrangian variable, B is the SAR image to be processed, and 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; Defining the X subproblem as minimizing expression (6) yields a quadratic subproblem with respect to the X variable: Among them, X (k+1) is the surface target output after the k+1th iteration, k is the number of iterations, α and β represent the trade-off parameters, Λ and Ω represent the Lagrangian variables, B is the SAR image to be processed, S is the point target, It represents the Hankel matrix constructed for X, U and V are Two variables after factorization with nuclear paradigm relaxation; Simplifying expression (7) with respect to X, we obtain the following solution: Among them, X (k+1) is the surface target output after the k+1th iteration, k is the number of iterations, α and β represent the trade-off parameters, Λ and Ω represent the Lagrangian variables, B is the SAR image to be processed, S is the point target, U and V are After the two variables are factorized by nuclear paradigm relaxation, represents the pseudo-inverse mapping from the Hankel matrix back to the initial state, and its expression is as follows: represents the pseudo-inverse mapping from the Hankel structure back to the initial state, the operator Express Sum the multiple elements and put them back to their original coordinates. represents the number of divisions by multiple corresponding terms; Defining the U subproblem as minimizing expression (6) yields a quadratic subproblem with respect to the U variable: 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; Taking the derivative of matrix U with respect to expression (11) and equating it to zero yields the following closed form solution: 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; Defining the V subproblem as minimizing expression (6) yields a quadratic subproblem with respect to the V variable: 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; Taking the derivative of the matrix V with respect to expression (13) and equating it to zero yields the following closed form solution: 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; The method for solving the Lagrangian function Λ variable specifically includes: L (k+1) =X (k+1) +S (k+1) -B+L (k) (15) 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; The method to parse the Ω variable is as follows: The method for solving the Lagrangian function Ω variable specifically includes: 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.

7. The SAR imaging method based on sparse and low-rank decomposition according to claim 1, characterized in that: The preset SAR imaging model expression is: in, is the shape parameter of the reflectivity F, Θ is the measurement matrix determined by the SAR system, Φ = diag(e jφ(F) ) represents a diagonal matrix, is the parameter of the auxiliary variable v, is the parameter of the auxiliary variable Φ, Y is the one-dimensional vector form of the two-dimensional echo data after vectorization, F is the one-dimensional vector form of the two-dimensional reflectivity matrix of the target area after vectorization, is the Hankel matrix constructed for the variable v, λ is the regularization parameter, R ANK (·) is the rank of the matrix.

8. The SAR imaging method based on sparse and low-rank decomposition according to claim 7, characterized in that: The method of outputting SAR imaging results by using sparse and low-rank decomposition algorithm expressions specifically includes: For equation (17), based on the results of the ADMM algorithm, the gradual alternation between estimating |F| and v under the constraint |F| = v is expressed as: in, is the shape parameter of the reflectivity F, η is the penalty parameter that affects the convergence speed, u is an auxiliary variable, F| is the size of the reflectivity F, 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, Φ is a diagonal matrix, λ is the regularization parameter that represents the balance between the control data consistency term and the regularization term, v and u are auxiliary variables, R ANK (·) is the rank of the matrix, η is the penalty parameter, The Hankel matrix constructed for the variable v; Minimize expression (18) to solve the quadratic subproblem and obtain the reflectivity F after the k+1th iteration, which is expressed as: 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, 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: F (k+1) =(Θ H (Θ+λI) -1 (I H Y+λ(v (k) -u (k) )) (20) 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; In order to separate the phase and amplitude, the reflectivity F is represented 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. The expression is as follows: 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 about the Ψ variable: 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 of the two-dimensional echo data after vectorization, Α (k) is the diagonal matrix containing the reflectivity output after the kth iteration, 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, and is solved iteratively as follows: 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 of the two-dimensional echo data after vectorization, Α is the diagonal matrix containing the reflectivity, 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, φ(·) represents the phase of F, the superscript H is the conjugate transpose of the matrix, and I is the unit matrix. Since the other parameters in formula (22) are all known quantities, formula (22) can be used to obtain Ψ in formula (21): (k+1) ; Using the phase and amplitude matrices obtained from the above calculations, the auxiliary variable v is calculated and the solution for algorithm convergence and stability is output. The expression is as follows: u (k+1) =u (k) +|F (k+1) -v (k+1) (23) 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; Consider subproblem (23) as an image processing problem and replace |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: 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) .

9. A SAR imaging system based on sparse and low-rank decomposition, characterized in that: The system includes: a memory and a processor, wherein the memory includes 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 as described in any one of claims 1 to 8 are implemented.

10. A computer-readable storage medium, characterized in that: The computer-readable storage medium 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, the steps of a SAR imaging method based on sparse and low-rank decomposition as described in any one of claims 1 to 8 are implemented.

Images