A multi-channel radar forward-looking super-resolution imaging method

Abstract

This invention discloses a multi-channel radar forward-looking super-resolution imaging method. First, echo data of the region to be imaged is acquired. Range pulse compression is performed on the acquired data, and the error matrix caused by grid mismatch is considered to obtain an imaging model under grid mismatch conditions. The least squares method is used to reasonably estimate the error matrix and correct the ideal steering matrix. Finally, based on sparse Bayesian learning theory, the approximate posterior distribution function of the target is estimated through the probability distributions of each variable. The target scattering coefficient is then reconstructed using the mean of the distribution function, achieving super-resolution imaging under grid mismatch conditions. This method, by correcting the theoretical steering matrix, reduces the mismatch between the steering matrix and the echo signal, overcoming the problem of poor super-resolution performance caused by grid mismatch, and contributing to robust multi-channel radar forward-looking super-resolution imaging.

Description

Technical Field

[0001] This invention belongs to the field of radar imaging technology, specifically relating to a multi-channel radar forward-looking super-resolution imaging method. Background Technology

[0002] Forward-looking radar imaging has important applications in autonomous landing, autonomous navigation, and forward-looking reconnaissance guidance. However, conventional single-channel SAR or Doppler beam sharpening techniques suffer from forward-looking imaging blind zones due to Doppler symmetry ambiguity and small Doppler variations in the forward-looking region. Bistatic SAR, by separating the transmit and receive terminals, can achieve forward-looking region imaging of the receiving platform. However, the need for external radiation sources limits the autonomy of imaging, and the separation of transmit and receive terminals introduces complex synchronization and motion compensation problems.

[0003] The literature “Hou Haiping, Qu Changwen, Xiang Yingchun, et al. Research on airborne SAR forward-looking imaging based on LFMCW. Journal of Circuits and Systems, 2011, 16(3): 7” proposes that by using one or more transmit channels and multiple receive channels on a single platform, an aperture can be formed in the azimuth direction, which has the potential for forward-looking imaging. However, due to the limitation of platform size, its azimuth resolution is usually very low. The literature “Zhang Jie, Wu Di, Zhu Daiyin. A forward-looking super-resolution imaging algorithm for airborne / missile-borne array radar. Radar Science and Technology, 2018, 16(2): 6” proposes the MUSIC algorithm for forward-looking super-resolution imaging. However, since this algorithm requires knowledge of the number of target sources and usually requires multiple snapshots to obtain good performance. The literature “Cao Kaicheng, Fan Bo, Zhou Xiaoli et al. Research on radar correlation imaging method under model mismatch. Modern Radar, 2017, 39(12): 42-47” proposes to realize radar correlation imaging under model mismatch by using sparse Bayesian algorithm. However, it assumes that the error matrix follows a certain unknown distribution, resulting in poor adaptability to different scenarios and affecting the imaging effect.

[0004] Furthermore, during the imaging process, the grid cannot be drawn densely enough, so grid mismatch is inevitable. The literature “Zhang Tao, Zhong Lunlong, Lai Ran, et al. Algorithm for estimating the spatiotemporal spectrum of clutter based on sparse Bayesian learning. Acta Aeronautica Sinica, 2020, 42(5): 1-12” considers the grid mismatch error and establishes a spatiotemporal dynamic dictionary model. However, the error compensation model used is based on the first-order Taylor series approximation. When the dictionary grid interval is large, the model approximation error increases. Summary of the Invention

[0005] To address the aforementioned technical problems, this invention proposes a multi-channel radar forward-looking super-resolution imaging method. This method incorporates the grid mismatch error matrix into the imaging model, uses the least squares method to reasonably estimate the error matrix, and then uses the estimated error matrix to correct the theoretical steering matrix. This reduces the mismatch between the steering matrix and the echo signal, overcomes the problem of poor super-resolution performance under grid mismatch conditions, and helps to achieve robust multi-channel radar forward-looking super-resolution imaging.

[0006] The technical solution adopted in this invention is: a multi-channel radar forward-looking super-resolution imaging method, the specific steps of which are as follows:

[0007] A. Acquire echo data of the area to be imaged;

[0008] The multi-channel radar forward-looking imaging employs a one-transmit, multiple-receive channel configuration; the transmitted signal is set to a linear frequency modulated pulse, and the echo signals S received by multiple channels... echo (y m ,t r ;r0) can be represented as:

[0009]

[0010] Where A0 represents a pre-defined constant, K r The distance represents the frequency modulation, c represents the speed of light, λ represents the wavelength of the emitted signal, and t represents the distance frequency modulation. r y represents distance over time. m Let r represent the azimuth coordinates of the m-th receiving antenna, and r0 represent the shortest slant distance from the platform to the scene. Then, the distance R from any point P(x0, y0) to the transmitting antenna is... tx and the distance R to different receiving antennas rx It can be expressed as:

[0011]

[0012]

[0013] Where h1 represents the height difference between the transmitting antenna and the receiving antenna, and h represents the flight altitude of the platform.

[0014] B. Perform range-direction pulse compression on the acquired data;

[0015] Set the matching function for pulse compression to S. ref (t r )=exp(-jπK r t r 2 Then the pulse-compressed signal S compress (y m ,t r This can be represented as:

[0016]

[0017] Wherein, IFFT represents the inverse Fourier transform operator, and FFT represents the Fourier transform operator;

[0018] C. Construct an imaging model;

[0019] Based on the knowledge of array signal processing, if the antenna array consists of M channels and N narrowband signals are incident on the spatial array, then the received signal Sr of the m channels... m It can be represented as:

[0020]

[0021] Where, β n Let b represent the scattering coefficient of the nth target. m Let τ represent the noise of the m-th channel. mn This represents the delay relative to the reference channel when the nth signal arrives at the mth channel, i.e., the path delay. Let f0 represent the path phase and f0 represent the carrier frequency of the transmitted signal; then the echo of a range cell from different channels can be written as:

[0022]

[0023] Equation (6) can be written as:

[0024] Sr M×1 =S M×N ·β N×1 +B M×1 (7)

[0025] Among them, Sr M×1 Indicates echo, B M×1 S represents the noise vector. M×N Denotes the guidance matrix, β N×1 This represents the target scattering coefficient vector.

[0026] D. Grid the imaging region, calculate the path delay, and construct an ideal steering matrix;

[0027] When calculating the path delay, the first consideration should be the effect of platform motion on the transmitting antenna T. x and receiving antenna R x The position change of the transmitting antenna T. x and receiving antenna R x The coordinates are as follows:

[0028]

[0029] Among them, t amLet v represent the sampling time of the m-th channel, and v represent the forward flight speed of the platform. s This indicates the switching speed of each array element.

[0030] Then the imaging region is meshed, as shown in the following expression:

[0031]

[0032] Among them, (X) j ,Y i (,0) represents the coordinates of the i-th row and j-th column in the divided grid, L a L represents the azimuth length of the imaging scene. r N represents the distance length of the image scene. a N represents the number of azimuth sampling points. r d represents the distance from the sampling points. r This indicates the projection length from the platform to the center of the scene.

[0033] Therefore, when sampling the m-th channel, the coordinates P1 and P2 of the transceiver antennas are as shown in equation (8), and the coordinates of the i-th row and j-th column in the scene are R(X). j ,Y i If , 0), then the delay time τ of the nth signal is τ. mn It can be represented as:

[0034]

[0035] In calculating τ mn Then, the ideal guiding matrix S is constructed by substituting equation (7). M×N ;

[0036] E. Construct an imaging model for mesh mismatch;

[0037] When considering the mesh mismatch error, equation (10) becomes:

[0038]

[0039] in, This represents the path delay considering mesh mismatch, where Δd represents the equivalent path length caused by mesh mismatch, and c represents the speed of light. This indicates the path delay caused by grid mismatch.

[0040] Define the error matrix E as the actual steering matrix. The error between the ideal guiding matrix S and the error of the ideal guiding matrix S is then expressed as follows:

[0041]

[0042] Where Sr represents the echo, B represents the noise vector, and β represents the target scattering coefficient vector.

[0043] F. Estimation of grid mismatch error and correction of ideal steering matrix;

[0044] According to the least squares principle, the unconstrained function containing two unknown variables, the target scattering coefficient and the error matrix, is obtained according to equation (12):

[0045]

[0046] Where β and E represent the target scattering coefficient and error matrix, respectively, μ1 and μ2 represent the regularization parameters, and ||·||1, ||·||2 and ||·|| F Let represent the vector 1 norm, vector 2 norm, and vector F norm, respectively.

[0047] The above unconstrained function contains two unknown variables. To achieve the optimal estimation of these variables, the above equation is decomposed into two subproblems each containing only one variable. The estimation of the error matrix E is achieved by solving the following subproblems:

[0048]

[0049] Taking the partial derivative of E in equation (14) and setting the result to zero, we can obtain an estimated value for E:

[0050]

[0051] in,(·) H Let μ2 represent the conjugate transpose operator; μ2 is updated in each iteration, which can be expressed as:

[0052]

[0053] Finally, the ideal steering matrix is ​​corrected according to equation (12).

[0054] G. Target scattering coefficient estimation, to obtain super-resolution imaging results;

[0055] First, a probability distribution model for each variable is established. Then, based on Bayesian theory, an approximate posterior distribution function of the reconstructed signal is derived. Finally, the target scattering coefficient is estimated based on the mean of the posterior distribution function.

[0056] The noise B is modeled as a complex Gaussian white noise, that is:

[0057]

[0058] Where p(·) represents the probability density function of the variable, Representing a complex Gaussian distribution, α0 = σ -2Indicates noise accuracy, σ 2 Let I represent the noise variance and the identity matrix. Since the gamma hyperprior is the conjugate prior of the Gaussian distribution, we model α0 as a gamma distribution:

[0059] p(α0;a,b)=Γ(α0|a,b) (18)

[0060] in, Γ(·) represents the gamma function, where a and b are constants that tend to 0.

[0061] For the linear model of equation (12), the echo signal follows the parameter... and Gaussian distribution:

[0062]

[0063] Furthermore, each element in the target scattering coefficient β is assumed to have zero mean and variance. , where α is a Gaussian distribution. n To represent the accuracy of the Gaussian distribution, similarly, α... n Modeled as a gamma distribution p(α) n ;ρ)=Γ(α) n |1,ρ), let α={α1,α2…,α n}, then we have:

[0064]

[0065] Where ρ represents a constant that approaches 0; Δ = diag(α), and diag(·) represents diagonalizing the vector.

[0066] Therefore, the joint probability density function is:

[0067] p(β|Sr,α0,α)=p(Sr|β,α0)p(β|α)p(α) (21)

[0068] After obtaining the probability distributions of the above variables, based on Bayesian theory, the posterior distribution of the target scattering coefficient β can be derived as follows:

[0069]

[0070] Where μ and Σ represent the mean and variance of β, respectively:

[0071]

[0072]

[0073] Therefore, the reconstruction of the target scattering coefficient β can be obtained according to equations (23) and (24). Since the parameters α0 and α are unknown, an iterative update solution process is usually involved. Based on the expectation maximization idea, the parameter update expression is obtained as follows:

[0074]

[0075]

[0076] Where Ξ=μμ H +Σ, tr(·) represents finding the trace of a matrix.

[0077] In summary, the iterative steps of AG are as follows:

[0078] a. Initialization: Set the gamma distribution parameters a and b, set the maximum number of iterations and the iteration termination threshold, and set the initial value of the reconstructed signal;

[0079] b. Error matrix estimation: Estimate the error matrix E according to equations (15) and (16);

[0080] c. Correction of the guidance matrix: Correct the guidance matrix according to equation (12);

[0081] d. Reconstructed signal estimation: Calculate the mean μ and variance Σ of β according to equations (23) and (24);

[0082] e. Parameter update: Update parameters α and α0 according to equations (25) and (26);

[0083] f. Iteration termination judgment: If If the maximum number of iterations is reached, the output μ is the final reconstructed signal β; otherwise, return to step b for the next iteration.

[0084] Finally, the reconstruction estimate of the target scattering coefficient β is the super-resolution imaging result.

[0085] The beneficial effects of this invention are as follows: The method first acquires echo data of the region to be imaged, performs range-direction pulse compression on the acquired data, and considers the error matrix caused by grid mismatch to obtain an imaging model under grid mismatch conditions. The least squares method is used to reasonably estimate the error matrix and correct the ideal steering matrix. Finally, based on sparse Bayesian learning theory, the approximate posterior distribution function of the target is estimated through the probability distribution of each variable. The mean of the distribution function is then used to reconstruct the target scattering coefficient, achieving super-resolution imaging under grid mismatch conditions. This method, by correcting the theoretical steering matrix, reduces the mismatch between the steering matrix and the echo signal, overcoming the problem of poor super-resolution performance caused by grid mismatch, and contributing to robust multi-channel radar forward-looking super-resolution imaging. Furthermore, while this method is designed for single-shot data imaging processing, in practice it can be combined with multi-shot data, i.e., synthetic aperture dimension processing, to obtain even better imaging performance. Attached Figure Description

[0086] Figure 1 This is a flowchart of a multi-channel radar forward-looking super-resolution imaging method according to the present invention.

[0087] Figure 2 This is a geometric model diagram of multi-channel radar forward-looking imaging in an embodiment of the present invention.

[0088] Figure 3 This is a schematic diagram of the observation scene in an embodiment of the present invention.

[0089] Figure 4 This is a schematic diagram of the echo of the observation scene in an embodiment of the present invention.

[0090] Figure 5 This is a result image of pulse compression of the echo of the observed scene in an embodiment of the present invention.

[0091] Figure 6 This is an imaging result diagram of the method of the present invention in an embodiment of the present invention.

[0092] Figure 7 This is an image showing the imaging results of using the existing algorithm (BP) in an embodiment of the present invention.

[0093] Figure 8 This is an image showing the imaging result of the traditional sparse Bayesian algorithm in an embodiment of the present invention. Detailed Implementation

[0094] The method of the present invention will be further described below with reference to the accompanying drawings and embodiments.

[0095] like Figure 1 The flowchart of a multi-channel radar forward-looking super-resolution imaging method of the present invention is shown below, and the specific steps are as follows:

[0096] In this embodiment, the geometric configuration of the forward-looking multi-channel radar is as follows: Figure 2 As shown in Table 1, the parameters of the forward-looking multi-channel radar are as follows:

[0097] Table 1

[0098] Aircraft flight altitude <![CDATA[h e ]]> 1050 m Aircraft flight speed <![CDATA[v r ]]> 20 m / s Pulse duration <![CDATA[T r ]]> 1 μs signal bandwidth B 60 MHz Array antenna length <![CDATA[L e ]]> 4.00 m Pulse repetition frequency modulation PRF 500 Hz Transmitted signal wavelength λ 0.0315 m Distance sampling points <![CDATA[N r ]]> 512 Number of azimuth sampling points <![CDATA[N a ]]> 256

[0099] like Figure 2 As shown, in an xyz spatial coordinate system, O represents the origin of the coordinate system, and the forward-looking multi-channel radar moves at a velocity v r =20m / s, height h e = Flying at a constant speed along the x-axis at a distance of 1050m, with the center of the target scene at X. c =1050m, Y c =0m, Z c =0m, meaning the angle between the aircraft and the center of the scene is 45°. Forward-looking multi-channel radar transmitting antenna T x Located directly below the central receiving antenna at a position h1 = 0.1m, during flight, the transmitting antenna T... x The signal is transmitted at a pulse repetition frequency (PRF) of 500 Hz, and the number of receiving array elements is N. a =256, uniformly arranged with element spacing d=L e / (256-1)≈0.0156m. In this simulation, the wavelength of the radar transmitted signal was set to λ=0.0315m, and the pulse duration T was... r A linear frequency modulated pulse signal with a duration of 1 μs and a bandwidth of B = 60 MHz. The number of sampling points N in the distance direction. r The number of sampling points N in the azimuth direction is 512. a It is 256.

[0100] A. Acquire echo data of the area to be imaged;

[0101] In this embodiment, the azimuth length of the scene is 512m, and the distance length is 512m. The number of sampling points N is based on the azimuth and distance. a and N r The scene is set to be uniformly divided into a 256×512 grid. If the target does not fall on a grid in the azimuth direction, the target's specific location is as follows: Figure 3 As shown, using equation (9), the coordinates of each target in the scene can be obtained.

[0102] In addition, considering the aircraft's flight speed v r Using equation (8), the transmitting antenna T can be obtained. x and receiving antenna R x The coordinates are then determined. Finally, the launch path R can be calculated using equations (2) and (3). tx and receiving path R rxThen, the echo signal is calculated using equation (1), such as Figure 4 As shown.

[0103] B. Perform range-direction pulse compression on the acquired data;

[0104] Assume the center of the target scene is X. c =0, Y c =0m, Z c =0, use equations (2) and (3) to calculate the reference transmission path and receiving path, and then obtain the distance-time t. r Matching function S of pulse compression ref (t r )=exp(-jπK r t r 2 Considering the echo data generated in step A of this embodiment, the result of echo pulse compression can be obtained using equation (4), such as... Figure 5 As shown.

[0105] C. Calculate the path delay and construct the ideal steering matrix;

[0106] Based on the target's location, the reference transmission and reception paths are calculated using equations (2) and (3). The delay time τ of the nth signal when sampling the mth channel is calculated using equation (10). mn Thus, the ideal guiding matrix S can be constructed according to equation (6).

[0107] D. Estimating grid mismatch error and correcting the ideal steering matrix;

[0108] In actual imaging, the target will not fall exactly at the center of the gridded imaging area, resulting in grid mismatch error. Since the steering matrix is ​​constructed from the path delay, the impact of grid mismatch error on radar imaging is mainly reflected in the steering matrix, causing a mismatch with reality and thus affecting imaging quality.

[0109] Typically, the steering matrix is ​​designed based on the distance from each channel to the observation scene. However, the observed target usually does not fall exactly on the grid divided in step A of this embodiment. If this error is not considered, it will lead to a mismatch between the steering matrix and the echo signal, thus affecting the reconstruction performance. Therefore, the steering matrix reconstructed in this embodiment takes into account the grid mismatch error. The error matrix E is estimated according to equations (15) and (16), and then the steering matrix can be corrected according to equation (12).

[0110] E. Target scattering coefficient estimation, to obtain super-resolution imaging results;

[0111] In obtaining the corrected steering matrix in step D of this embodiment Then, target scattering sparse estimation is performed according to the iterative steps in step G of the invention. When the iteration termination condition is met, a better super-resolution imaging result can be obtained. During initialization, the parameters a = 0.0001, b = 0.0001, ρ = 0.01 are set, the maximum iteration parameter is set to 300, and the iteration termination threshold is set to 10. -4 The initial value of the reconstructed signal is set to 0. For example... Figure 6 The image shown is the final imaging result using this method, as follows: Figure 7 The image shown is the imaging result using the existing imaging algorithm BP, such as... Figure 8 The imaging results are obtained using the traditional sparse Bayesian learning algorithm. Obviously, the multi-channel radar forward-looking super-resolution imaging method proposed in this embodiment achieves azimuth super-resolution under grid mismatch conditions.

[0112] In summary, the method of this invention, by modifying the theoretical steering matrix, reduces the mismatch between the steering matrix and the echo signal, overcoming the problem of poor super-resolution performance caused by grid mismatch. This helps to achieve super-resolution forward-looking imaging of multi-channel radar considering grid mismatch. Furthermore, the method of this invention is designed for imaging processing of single-shot data; in practice, it can be combined with multi-shot data, i.e., processing of the synthetic aperture dimension, to obtain even better imaging performance.

[0113] Those skilled in the art will recognize that the embodiments described herein are intended to help the reader understand the principles of the invention, and should be understood that the scope of protection of the invention is not limited to such specific statements and embodiments. Various modifications and variations can be made to the invention by those skilled in the art. Any modifications, equivalent substitutions, improvements, etc., made within the spirit and principles of the invention should be included within the scope of the claims of the invention.

Claims

1. A multi-channel radar forward-looking super-resolution imaging method, the specific steps of which are as follows: A. Acquire echo data of the area to be imaged; The multi-channel radar forward-looking imaging employs a one-transmit, multiple-receive channel configuration; the transmitted signal is set to a linear frequency modulated pulse, and the echo signals S received by multiple channels... echo (y m ,t r ;r0) can be represented as: in, A0 represents a pre-defined constant, K r The distance represents the frequency modulation, c represents the speed of light, λ represents the wavelength of the emitted signal, and t represents the distance frequency modulation. r y represents distance over time. m Let r represent the azimuth coordinates of the m-th receiving antenna, and r0 represent the shortest slant distance from the platform to the scene. Then, the distance R from any point P(x0, y0) to the transmitting antenna is... tx and the distance R to different receiving antennas rx It can be expressed as: Where h1 represents the height difference between the transmitting antenna and the receiving antenna, and h represents the flight altitude of the platform; B. Perform range-direction pulse compression on the acquired data; Set the matching function for pulse compression to S. ref (t r )=exp(-jπK r t r 2 Then the pulse-compressed signal S compress (y m ,t r This can be represented as: Wherein, IFFT represents the inverse Fourier transform operator, and FFT represents the Fourier transform operator; C. Construct an imaging model; Based on the knowledge of array signal processing, if the antenna array consists of M channels and N narrowband signals are incident on the spatial array, then the received signal Sr of the m channels... m It can be represented as: Where, β n Let b represent the scattering coefficient of the nth target. m Let τ represent the noise of the m-th channel. mn This represents the delay relative to the reference channel when the nth signal arrives at the mth channel, i.e., the path delay. Let f0 represent the path phase and f0 represent the carrier frequency of the transmitted signal; then the echo of a range cell from different channels can be written as: Equation (6) can be written as: Sr M×1 =S M×N ·β N×1 +B M×1 (7) Among them, Sr M×1 Indicates echo, B M×1 S represents the noise vector. M×N Denotes the guidance matrix, β N×1 Represents the target scattering coefficient vector; D. Grid the imaging region, calculate the path delay, and construct an ideal steering matrix; When calculating the path delay, the first consideration should be the effect of platform motion on the transmitting antenna T. x and receiving antenna R x Position change, transmitting antenna T x and receiving antenna R x The coordinates are as follows: Among them, t am Let v represent the sampling time of the m-th channel, and v represent the forward flight speed of the platform. s This indicates the switching speed of each array element; Then the imaging region is meshed, as shown in the following expression: Among them, (X) j ,Y i (,0) represents the coordinates of the i-th row and j-th column in the divided grid, L a L represents the azimuth length of the imaging scene. r N represents the distance length of the image scene. a N represents the number of azimuth sampling points. r d represents the distance from the sampling points. r This represents the projection length from the platform to the center of the scene; Therefore, when sampling the m-th channel, the coordinates P1 and P2 of the transceiver antennas are as shown in equation (8), and the coordinates of the i-th row and j-th column in the scene are R(X). j ,Y i If , 0), then the delay time τ of the nth signal is τ. mn It can be represented as: In calculating τ mn Then, the ideal guiding matrix S is constructed by substituting equation (7). M×N ; E. Construct an imaging model for mesh mismatch; When considering the mesh mismatch error, equation (10) becomes: in, This represents the path delay considering mesh mismatch, where Δd represents the equivalent path length caused by mesh mismatch, and c represents the speed of light. This indicates the path delay caused by mesh mismatch; Define the error matrix E as the actual steering matrix. The error between the ideal guiding matrix S and the error of the ideal guiding matrix S is then expressed as follows: Where Sr represents the echo, B represents the noise vector, and β represents the target scattering coefficient vector; F. Estimation of grid mismatch error and correction of ideal steering matrix; According to the least squares principle, the unconstrained function containing two unknown variables, the target scattering coefficient and the error matrix, is obtained according to equation (12): Where β and E represent the target scattering coefficient and error matrix, respectively, μ1 and μ2 represent the regularization parameters, and ||·||1, ||·||2 and ||·|| F Let represent the vector 1-norm, vector 2-norm, and vector F-norm, respectively; The above unconstrained function contains two unknown variables. To achieve the optimal estimation of these variables, the above equation is decomposed into two subproblems each containing only one variable. The estimation of the error matrix E is achieved by solving the following subproblems: Taking the partial derivative of E in equation (14) and setting the result to zero, we can obtain an estimated value for E: in,(·) H Let μ2 represent the conjugate transpose operator; μ2 is updated in each iteration, which can be expressed as: Finally, the ideal guidance matrix is ​​corrected according to equation (12); G. Target scattering coefficient estimation, to obtain super-resolution imaging results; First, a probability distribution model for each variable is established. Then, based on Bayesian theory, the approximate posterior distribution function of the reconstructed signal is derived. Finally, the target scattering coefficient estimation result is obtained based on the mean of the posterior distribution function. The noise B is modeled as a complex Gaussian white noise, that is: Where p(·) represents the probability density function of the variable, Representing a complex Gaussian distribution, α0 = σ -2 Indicates noise accuracy, σ 2 Let I represent the noise variance and the identity matrix. Since the gamma hyperprior is the conjugate prior of the Gaussian distribution, we model α0 as a gamma distribution: p(α0;a,b)=Γ(α0|a,b) (18) in, Γ(·) represents the gamma function, where a and b are constants that tend to 0; For the linear model of equation (12), the echo signal follows the parameter... and Gaussian distribution: Furthermore, each element in the target scattering coefficient β is assumed to have zero mean and variance. , where α is a Gaussian distribution. n To represent the accuracy of the Gaussian distribution, similarly, α... n Modeled as a gamma distribution p(α) n ;ρ)=Γ(α) n |1,ρ), let α={α1,α2…,α n }, then we have: Where ρ represents a constant approaching 0; Δ = diag(α), and diag(·) represents diagonalizing the vector; Therefore, the joint probability density function is: p(β|Sr,α0,α)=p(Sr|β,α0)p(β|α)p(α) (21) After obtaining the probability distributions of the above variables, based on Bayesian theory, the posterior distribution of the target scattering coefficient β can be derived as follows: Where μ and Σ represent the mean and variance of β, respectively: Therefore, the reconstruction of the target scattering coefficient β can be obtained according to equations (23) and (24); since the parameters α0 and α are unknown, iterative update solutions are usually involved; based on the expectation maximization idea, the parameter update expression is obtained as follows: Where Ξ=μμ H +Σ, tr(·) represents finding the trace of a matrix; In summary, the iterative steps of AG are as follows: a. Initialization: Set the gamma distribution parameters a and b, set the maximum number of iterations and the iteration termination threshold, and set the initial value of the reconstructed signal; b. Error matrix estimation: Estimate the error matrix E according to equations (15) and (16); c. Correction of the guidance matrix: Correct the guidance matrix according to equation (12); d. Reconstructed signal estimation: Calculate the mean μ and variance Σ of β according to equations (23) and (24); e. Parameter update: Update parameters α and α0 according to equations (25) and (26); f. Iteration termination judgment: If If the maximum number of iterations is reached, the output μ is the final reconstructed signal β; otherwise, return to step b for the next iteration. Finally, the reconstruction estimate of the target scattering coefficient β is the super-resolution imaging result.

Images