A Two-Dimensional Super-Resolution Imaging Method for the Pitch of the Forward View Azimuth of a Motion Platform

Abstract

The present invention discloses a method for forward-looking azimuth and pitch two-dimensional super-resolution imaging of a moving platform. By establishing an azimuth-pitch two-dimensional scanning echo model of an airborne radar platform, then performing range migration correction on the azimuth-pitch-range echo, approximating the echo model as a two-dimensional convolution model, and finally, using the conjugate gradient method and the Kronecker product property of matrices, the azimuth and pitch two-dimensional target scattering intensity is derived. The method of the present invention solves the problems of low azimuth-pitch two-dimensional resolution of existing real-aperture radars and inapplicability to moving platforms. Compared with existing two-dimensional scanning radar super-resolution methods, it can not only be used for two-dimensional super-resolution imaging of moving platforms, but also has excellent target resolution ability and low computational complexity, and is suitable for engineering implementation.

Description

Technical Field

[0001] The present invention belongs to the technical field of radar imaging, and particularly relates to a two-dimensional super-resolution imaging method for forward-looking azimuth and pitch of a moving platform. Background Art

[0002] A real aperture radar (RAR) has the ability to detect forward-looking targets that cannot be achieved by a synthetic aperture radar (SAR) and a Doppler beam sharpening (DBS) in terms of imaging mechanism. It is widely used in civilian fields such as topographic mapping, autonomous landing, and material airdrop. Existing RAR super-resolution methods can improve the azimuth imaging resolution, but their results are usually azimuth-range two-dimensional images with a fixed pitch angle, without considering the scanning modulation of the pitch angle beam antenna pattern.

[0003] In order to obtain a high azimuth resolution matching the range resolution, the literature "Zhang Y, Li J, Li M, et al. Online Sparse Reconstruction for Scanning Radar Using Beam-Updating q-SPICE. IEEE Geoscience and Remote Sensing Letters, 2021, 19: 1-5" proposed a generalized (sparse iterative covariance-based estimation, SPICE) method based on beam updating. This method utilizes the sparsity of the target scene and further improves the azimuth resolution compared with the iterative adaptive method. However, the processing order of the azimuth and pitch directions of this method will seriously deteriorate the imaging result and cannot be directly used for two-dimensional super-resolution imaging. The literature "Tuo X, Xia Y, Zhang Y, et al. Super-Resolution Imaging for Real Aperture Radarby Two-Dimensional Deconvolution. 2021 IEEE International Geoscience andRemote Sensing Symposium IGARSS. IEEE, 2021: 6630-6633" proposed a two-dimensional super-resolution imaging method based on sparse alternating direction method of multipliers (ADMM), which uses the regularization 1-norm constraint and is solved by the ADMM method, improving the two-dimensional resolution of RAR. However, this method is only effective for a fixed platform and is difficult to achieve two-dimensional resolution of azimuth and pitch for an airborne moving platform radar. Summary of the Invention

[0004] To solve the above technical problems, the present invention proposes a method for two-dimensional super-resolution imaging of the forward-looking azimuth and pitch of a moving platform.

[0005] The technical solution adopted by the present invention is as follows: A method for two-dimensional super-resolution imaging of the forward-looking azimuth and pitch of a moving platform, and the specific steps are as follows:

[0006] Step S1, the azimuth-pitch two-dimensional scanning echo model of the airborne radar platform;

[0007] Assume that the airborne radar platform is located at point A(0, 0, Γ) at t = 0, and moves uniformly along the y-axis direction at a speed v. Γ represents the height of the airborne platform. At time t, the airborne platform moves to point B, and it can be obtained that AB = vt. Assume a point P(x P , y P , z P ) within the scanning area, and its projection on the xOy plane is point Q. R P is the distance between the radar platform and the target at the initial moment, is the pitch angle of the target relative to the radar platform, and θ P is the azimuth angle of the target relative to the y-axis direction. According to the geometric relationship, the distance R(t) between the airborne platform and the target at time t can be expressed as:

[0008]

[0009] According to the following geometric relationship:

[0010]

[0011] Substitute Equation (2) into Equation (1) to obtain:

[0012]

[0013] Perform Taylor expansion on R(t), and it can be obtained that:

[0014]

[0015] where, O(t 2 ) represents the high-order term of t 2 . Within the coherent integration time (CPI) of an arbitrary point target, it satisfies vt << R P . Therefore, the quadratic term and higher-order terms of the distance history can be ignored, and Equation (4) can be approximated as:

[0016]

[0017] Assume that the scanning radar emits a linear frequency modulation signal:

[0018]

[0019] where τ is the range-time variable, T p is the pulse width, rect(·) represents the rectangular window function, f c represents the carrier frequency, and K r represents the chirp rate. According to the platform motion, beam scanning, signal transmission, and reception processes, the echo signal of the target point P can be expressed as:

[0020]

[0021] where t is the platform motion time variable (slow time), τ is the range-time variable (fast time), σ P is the target scattering coefficient of the target P, κ represents a constant containing various gains and losses, c is the electromagnetic wave propagation speed, and h(t) is the antenna pattern modulation function. In terms of range, high range resolution is achieved through pulse compression. After de-carrier frequency and pulse compression processing, the echo can be written as:

[0022]

[0023] where B r represents the signal bandwidth, satisfying B r = K r T p , sinc(·) represents the range-direction signal envelope after pulse compression, specifically denoted as sinc(γ) = sin(πγ) / πγ, and the last term exp{-j4πR(t) / λ} represents the Doppler phase term caused by platform motion, where λ represents the wavelength.

[0024] The scanning method for the target airspace Ω is as follows: for the first elevation dimension, scan along the azimuth direction first, then jump to the second elevation dimension and scan along the azimuth again, and so on, performing a "Z"-shaped dense scan row by row. Here, Δθ and represent the azimuth and elevation wave position jumps respectively, Φ θ and represent the azimuth and elevation scanning ranges respectively, N θ and represent the azimuth and elevation sampling points respectively. During the scanning process of the airborne radar platform, the fast time τ and the slow time t respectively satisfy:

[0025]

[0026]

[0027] where PRF represents the pulse repetition frequency, θ and represent the azimuth-elevation angle variables, and R represents the range variable.

[0028] Step S2, echo preprocessing;

[0029] Substituting Equation (9) and Equation (10) into the echo Equation (8), the three-dimensional echo expression of range-azimuth-elevation can be obtained:

[0030]

[0031] where, represents the two-dimensional antenna pattern. In order to eliminate the range walk caused by the relative motion between the platform and the target, the range walk amount can be expressed as:

[0032]

[0033] Construct the phase compensation factor in the range frequency domain:

[0034]

[0035] where, f R represents the range frequency variable. After performing a fast Fourier transform (FFT) on the range direction of Equation (11), multiplying it by Equation (13), and then performing an inverse fast Fourier transform (IFFT) on the range direction, the echo after walk correction can be obtained:

[0036]

[0037] where, the fourth term in the equation represents a constant term related to the initial slant range, which does not affect the two-dimensional processing of azimuth and elevation and can be approximately ignored. The fifth term represents the Doppler phase modulation generated during the relative motion process. When the target is in the forward-looking range of the airborne platform, in this case, the Doppler histories of each target in the forward-looking area are approximately equal. Therefore, Equation (14) can be approximated as:

[0038]

[0039] Assume that the backscattering coefficient of the target located at in the airspace Ω is R 0 , θ 0 , respectively represent the range, azimuth angle, and elevation angle of the target . Then the total echo of the detection airspace can be written as:

[0040]

[0041] For a fixed range cell, only the azimuth and elevation changes are considered. Scanning the azimuth-elevation two-dimensional of the detection area is equivalent to the two-dimensional convolution of the two-dimensional antenna pattern with the scattering points in the detection area, which can be regarded as the target scattering coefficient and the two-dimensional antenna pattern of azimuth and elevation. Therefore, Equation (16) can be simplified to the echo model of a single range cell as:

[0042]

[0043] Among them, is equivalent to a two-dimensional convolution kernel, indicating the target azimuth elevation scattering distribution corresponding to the distance cell.

[0044] Considering the additive noise, we have:

[0045]

[0046] Among them, * represents two-dimensional convolution, is equivalent to a two-dimensional convolution kernel, representing additive Gaussian noise.

[0047] For concise representation, the two-dimensional convolution model (18) can be discretized as:

[0048] Y = AXB T + N (19)

[0049] Among them, represents the echo matrix of a certain distance slice, is the backscattering coefficient matrix of the target, K 1 , K 2 respectively represent the number of azimuth and elevation sampling points of the backscattering coefficient matrix, is the additive noise, represents the real number space, T represents the transpose of the matrix, and respectively represent the azimuth and elevation antenna pattern modulation matrices, and are specifically written as:

[0050]

[0051]

[0052] Among them, represents the azimuth dimension antenna pattern sampling points, represents the elevation dimension antenna pattern sampling points, L 1 and L 2 respectively represent the number of azimuth and elevation dimension antenna pattern sampling points, and the number of sampling points M, N, K 1 , K 2 , L 1 , L 2 satisfy the relationship:

[0053]

[0054] Step S3, calculate the autocorrelation matrix R;

[0055] The target scattering result x (q+1) can be written as:

[0056] x (q+1) = P (q) F H (R (q) ) -1 y(23)

[0057] where q represents the number of iterations, and R (q) = F H P (q) F is obtained by calculating P in the q-th iteration, P (q) = diag(x (q) )), diag(·) represents constructing a diagonal matrix according to a certain vector, H represents the conjugate transpose operation, and y = vec(Y) is the one-dimensional vectorized form of the echo Y. For the sake of simplicity of representation, in Equation (24) and subsequent, the iteration number subscript of R is omitted. In addition, due to the special structure of the scanning matrix F, it can be found that R is actually a singular matrix, which will lead to calculation errors of R (q) -1 H . To ensure the accuracy of angle estimation and prevent the problem of noise amplification, diagonal loading is performed on R:

[0058] R = FPF H + μI (24)

[0059] where μ represents a regularization parameter that can balance noise and resolution performance, and I represents the identity matrix. Since the dimension of the matrix R is too large, directly calculating R -1 has too high a computational complexity. First, the computational complexity of R -1 is reduced by the CG algorithm below.

[0060] Step S4: Iteratively update U;

[0061] Define a variable According to the Fletcher-Reeves conjugate gradient algorithm, the variable u can be updated by the following formula:

[0062] u l+1 = u l + α l d l (25)

[0063] where l represents the number of CG iterations, d l represents the conjugate direction with respect to R, α l represents the step size, satisfying ρ l represents an intermediate variable, and the weight vector w l satisfies:

[0064] w l = Rd l+1 (26)

[0065] Write Equation (25) in two - dimensional form:

[0066]

[0067] where, U l , D l represent the two - dimensional forms of u l and d l , satisfying u l = vec(U l ), d l = vec(D l ), so we have:

[0068] U l+1 = U l +α l D l (28)

[0069] where,

[0070]

[0071] where, * represents the conjugate operation, ⊙ represents the Hadamard product of matrices, and respectively represent the i - th elements of the vectors d l+1 and w l , represents a column vector of all 1s with 1 row, represents θ a column vector of all 1s with N l rows, W l satisfies w l = vec(W

[0072] Step S5: Iteratively update W;

[0073] According to the properties of the Kronecker product, write Equation (26) in two - dimensional form:

[0074]

[0075] where, Σ is the target scattering power, satisfying Σ ij = |X ij | 2 , and has a relationship with the matrix P as P = diag{vec(Σ)}, Σ ij represents the element in the i - th row and j - th column of the target scattering power matrix Σ, i = 1,..., K1 , j = 1, ..., K 2 , X ij represents the element in the i-th row and j-th column of the matrix target X. Since P is a diagonal matrix, P is transformed into the Hadamard product relationship between Σ and (A H D l+1 B * ), so we have:

[0076] W l = A(Σ⊙(A H D l+1 B * ))B T + μD l+1 (31)

[0077] Step S6: Iteratively update D;

[0078] d l The one-dimensional update expression can be written as:

[0079] d l+1 = g l + β l d l (32)

[0080] where g l represents the gradient vector, and the intermediate variable β l+1 = ρ l+1 / ρ l .

[0081]

[0082] Write Equation (32) in two-dimensional form:

[0083]

[0084] where G l represents the two-dimensional form of g l , and satisfies g l = vec(G l ), so we have:

[0085] D l+1 = G l + β l D l (35)

[0086] where the update method of β l+1 is the same as Equation (32), and ||·|| F represents the matrix Frobenius norm.

[0087] Step S7: Iteratively update G;

[0088] In Equation (32), the update formula for the gradient g l is as follows:

[0089] g l+1 = g l - α l w l (36)

[0090] Convert the above formula into a two-dimensional form:

[0091]

[0092] Therefore, we get:

[0093] G l+1 = G l - α l W l (38)

[0094] When is satisfied or the specified number of iterations is reached, terminate the CG iteration. ε represents a small positive number, and output the two-dimensional CG iteration result

[0095] Step S8: The two-dimensional form of the iterative estimation result;

[0096] According to Equation (23), the two-dimensional expression of the target scattering can be written as:

[0097]

[0098] Similar to Equation (30), P (q) = diag{vec(Σ (q) )} is used in the above formula to obtain the two-dimensional target scattering iteration formula:

[0099] X (q+1) = Σ (q) ⊙ (A H U (q) B * ) (40)

[0100] where U (q) satisfies vec(U (q) ) = R -1 y. At the beginning of the target scattering iteration, the variable Σ is initialized as: Σ (0) = A H YB H . At the beginning of the CG iteration, the variables are initialized as: U 0 = 0, D 0 = 0, β 0 = 0, G 0 = Y, Through repeated iterations of two-dimensional target scattering (Equation (40)) and CG (Equations (28), (29), (31), (35), and (38)), X in the model Equation (19) can be directly solved.

[0101] Advantages of the present invention: The method of the present invention establishes an azimuth-pitch two-dimensional scanning echo model for an airborne radar platform, then corrects the range migration of the azimuth-pitch-range echo, approximates the echo model as a two-dimensional convolution model, and finally, uses the conjugate gradient method and the Kronecker product property of matrices to derive the azimuth-pitch two-dimensional target scattering intensity. The method of the present invention solves the problem that the existing real-aperture radar has low azimuth-pitch two-dimensional resolution and is not applicable to moving platforms. Compared with the existing two-dimensional scanning radar super-resolution methods, the method of the present invention can not only be used for two-dimensional super-resolution imaging of moving platforms, but also has excellent target resolution ability and low computational complexity, and is suitable for engineering implementation. Description of the Drawings

[0102] Figure 1 It is a flowchart of a method for forward-looking azimuth-pitch two-dimensional super-resolution imaging of a moving platform according to the present invention.

[0103] Figure 2 It is a moving geometry model of an airborne radar platform in an embodiment of the present invention.

[0104] Figure 3 It is a schematic diagram of a two-dimensional scanning mode of an airspace target in an embodiment of the present invention.

[0105] Figure 4 It is an azimuth-pitch two-dimensional antenna pattern in an embodiment of the present invention.

[0106] Figure 5 It is a layout diagram of three-dimensional point targets in the original scene in an embodiment of the present invention.

[0107] Figure 6 It is a setting diagram of azimuth-pitch two-dimensional point targets in an embodiment of the present invention.

[0108] Figure 7 It is a result diagram of the azimuth-pitch two-dimensional real beam echo after preprocessing in an embodiment of the present invention.

[0109] Figure 8 It is a result diagram of two-dimensional Wiener inverse filtering super-resolution in an embodiment of the present invention.

[0110] Figure 9 It is a result diagram of super-resolution in an embodiment of the present invention. Detailed Embodiments

[0111] The method of the present invention will be further described below in conjunction with the drawings and embodiments.

[0112] In this embodiment, a simulation experiment is conducted to verify the effectiveness of a proposed two-dimensional super-resolution imaging method for the forward-looking azimuth and pitch of a moving platform. The steps and results in this embodiment are verified on the MATLAB 2018b simulation platform.

[0113] As Figure 1 shown, the flowchart of a two-dimensional super-resolution imaging method for the forward-looking azimuth and pitch of a moving platform according to the present invention is as follows:

[0114] Step S1: Establish a signal geometric model;

[0115] As shown in Table 1, the simulation parameters of the radar moving platform are listed. The sampling rate satisfies the Nyquist sampling theorem.

[0116] Table 1

[0117]

[0118] The airborne radar platform is located at point A(0, 0, Γ) at t = 0, where Γ = 1000 m, and moves uniformly along the y-axis direction at a speed of v = 90 m / s. At time t, the airborne platform moves to point B, and AB = vt. Let a point P(x P , y P , z P ) in the scanning area, and its projection on the xOy plane is point Q. R P = 30000 m is the distance between the radar platform and the target at the initial moment, is the pitch angle of the target relative to the radar platform, θ P is the azimuth angle of the target relative to the y-axis direction. As Figure 2 shown, according to the geometric relationship, the distance R(t) between the airborne platform and the target at time t can be written as:

[0119]

[0120] According to Figure 2 , there are the following geometric relationships:

[0121]

[0122] Substitute Equation (42) into Equation (41) to get:

[0123]

[0124] Perform a Taylor expansion on R(t) and neglect the quadratic and higher terms of the distance history to obtain:

[0125]

[0126] According to the platform movement, beam scanning, signal transmission and reception processes, the echo signal of the target point P can be expressed as:

[0127]

[0128] where τ is the range-time variable, rect(·) represents the rectangular window function, K r represents the chirp rate, t is the platform movement time variable (slow time), τ is the range-time variable (fast time), σ P is the target scattering coefficient of the target P, κ represents a constant containing various gains and losses, c is the electromagnetic wave propagation speed, and h(t) is the antenna pattern modulation function.

[0129] After de-carrier frequency and pulse compression processing, the echo can be written as:

[0130]

[0131] where sinc(·) represents the range-direction signal envelope after pulse compression, specifically denoted as sinc(γ) = sin(πγ) / πγ, and the last term exp{-j4πR(t) / λ} represents the Doppler phase term caused by platform movement, and λ represents the wavelength.

[0132] As Figure 3 shown, the scanning method for the target airspace Ω is as follows: for the first elevation dimension, scan along the azimuth direction first, then jump to the second elevation dimension, and then scan along the azimuth direction, and so on, scanning densely in a "Z" shape row by row. Among them, N θ and respectively represent the number of azimuth and elevation sampling points. During the scanning process of the airborne radar platform, the fast time τ and the slow time t respectively satisfy:

[0133]

[0134]

[0135] where PRF represents the pulse repetition frequency, θ and represent the azimuth-elevation angle variables, and R represents the range variable.

[0136] Step S2, echo preprocessing;

[0137] From equations (46), (47) and (48), the range-azimuth-elevation three-dimensional echo expression can be written as:

[0138]

[0139] where represents the two-dimensional antenna pattern function, and its energy distribution is as Figure 4As shown in the figure, the phase compensation factor is constructed in the range frequency domain as follows:

[0140]

[0141] where f R represents the range frequency variable. After performing a fast Fourier transform (FFT) on the range direction of Equation (49) and multiplying it by Equation (50), and then performing an inverse fast Fourier transform (IFFT) on the range direction, the corrected echo after motion compensation can be obtained:

[0142]

[0143] When the target is within the forward-looking range of the airborne platform, the Doppler history of each target in the forward-looking area is approximately equal. Therefore, Equation (51) can be approximated as:

[0144]

[0145] Assume that the backscattering coefficient of the target located at in the airspace Ω is R 0 , θ 0 , represent the range, azimuth angle, and elevation angle of the target respectively. The target is set as shown in . The total echo of the detection airspace can be written as: Figure 5 , 6 As shown.

[0146]

[0147] According to the above equation, the two-dimensional scanning convolution model can be discretized as:

[0148] Y = AXB T + N (54)

[0149] where represents the echo matrix of a certain range slice, is the backscattering coefficient matrix of the target. K 1 , K 2 represent the number of azimuth and elevation sampling points of the backscattering coefficient matrix respectively. is the additive noise, represents the real number space, T represents the transpose of the matrix, and represent the azimuth and elevation antenna pattern modulation matrices respectively, and are specifically written as:

[0150]

[0151]

[0152] Among them, represents the sampling points of the azimuth - dimension antenna pattern, represents the sampling points of the elevation - dimension antenna pattern, L 1 and L 2 respectively represent the number of sampling points of the azimuth - dimension and elevation - dimension antenna patterns.

[0153] Step S3: Iteratively update U;

[0154] Define a variable Update the variable u through the following formula:

[0155] u l+1 = u l + α l d l (57)

[0156] where l represents the number of CG iterations, d l represents the conjugate direction with respect to R, α l represents the step size, satisfying ρ l represents an intermediate variable, w l represents the weight vector.

[0157] Write formula (58) in two - dimensional form:

[0158]

[0159] where U l , D l represent the two - dimensional forms of u l and d l , satisfying u l = vec(U l ), d l = vec(D l ), so there is:

[0160] U l+1 = U l + α l D l (59)

[0161] where,

[0162]

[0163] where * represents the conjugate operation, ⊙ represents the Hadamard product, and respectively represent the i - th elements of the vectors d l+1 and w l , represents a column vector of all 1s with 1 row, Denote N θ a vector of all 1s with 1 row and N columns, W l satisfies w l =vec(W l ).

[0164] Step S4: Iteratively update W;

[0165] For the variable in Equation (57):

[0166] w l =Rd l+1 (61)

[0167] Write Equation (61) in two-dimensional form:

[0168]

[0169] where Σ is the target scattering power, satisfying Σ ij =|X ij | 2 , and has a relationship with matrix P as P = diag{vec(Σ)}, Σ ij represents the element in the i-th row and j-th column of the target scattering power matrix Σ, i = 1,..., K 1 , j = 1,..., K 2 , X ij represents the element in the i-th row and j-th column of matrix target X. Since P is a diagonal matrix, P is transformed into the Hadamard product relationship between Σ and (A H D l+1 B * ), so there is:

[0170] W l =A(Σ⊙(A H D l+1 B * ))B T +μD l+1 (63)

[0171] Step S5: Iteratively update D;

[0172] The relationship between d l+1 and d l is updated by the following formula:

[0173] d l+1 =g l +β l d l (64)

[0174] where g l represents the gradient vector, and the intermediate variable β l+1 =ρ l+1 / ρl 。

[0175]

[0176] Write Equation (64) in two - dimensional form:

[0177]

[0178] where G l represents the two - dimensional form of g l , and satisfies g l = vec(G l ). Thus, we have:

[0179] D l+1 = G l +β l D l (67)

[0180] where β l+1 is updated in the same way as Equation (64), and ||·|| F represents the matrix Frobenius norm.

[0181] Step S6: Iteratively update G;

[0182] In Equation (64), the gradient g l can be updated by the formula:

[0183] g l+1 = g l -α l w l (68)

[0184] Convert the above formula into two - dimensional form:

[0185]

[0186] Thus, we get:

[0187] G l+1 = G l -α l W l (70)

[0188] When is satisfied or after 10 iterations, terminate the CG iteration. ε = 10 -3 represents a small positive number, and output the two - dimensional CG iteration result

[0189] Step S7: Iteratively estimate the two - dimensional form of the result;

[0190] After terminating the CG iteration, output the CG iteration result u(q) = u l and substituting it into Equation (23) to obtain the q-th target scattering iteration result:

[0191]

[0192] where u (q) represents the R in the q-th target scattering iteration formula (23) obtained by CG iteration -1 y.

[0193] According to Equation (23), the two-dimensional iteration expression of the target scattering can be written as:

[0194]

[0195] Therefore, there is:

[0196] X (q+1) = Σ (q) ⊙ (A H U (q) B * ) (73)

[0197] where U (q) satisfies vec(U (q) ) = R -1 y. At the beginning of the target scattering iteration, the variable Σ is initialized as: Σ (0) = A H YB H . At the beginning of the CG iteration, the variables are initialized as: U 0 = 0, D 0 = 0, β 0 = 0, G 0 = Y, Through the repeated iteration of two-dimensional target scattering (Equation (73)) and CG (Equations (59), (60), (63), (67) and (70)), X in the model Equation (54) can be directly solved. This method, on the one hand, does not require matrix inversion, and on the other hand, it also does not require constructing and calculating the multiplication of high-dimensional matrices and vectors, which can greatly reduce the computational complexity of the target scattering iteration method.

[0198] Specifically analyze the complexity of this method: In the CG iteration, four matrices need to be stored, namely U l , W l , D l and G l . Each CG iteration requires 4 matrix-matrix multiplications (A(Σ⊙(A H D l+1 B * ))B T), the second matrix-matrix Hadamard multiplication (once as Σ⊙(A in Equation (63) H D l+1 B * ), and the other time as in Equation (60) ), four matrix-scalar multiplications (U l , D l , G l update in and μD in Equation (63) l+1 ) and one scalar multiplication Suppose L iterations are required, then the computational complexity of the two-dimensional CG algorithm is Each target scattering iteration requires two matrix multiplications (Σ (q) ⊙(A H U (q) B * )) and one scalar multiplication (Σ (q) = X (q) ⊙X (q) ). Assume Q target scattering iterations are performed, then the overall computational complexity of the algorithm is:[[]]

[0199]

[0200] Figure 7 , Figure 8 , Figure 9 The point target super-resolution results of different methods are given. Limited by the platform memory, the azimuth and elevation sampling points are reduced. Among them, Figure 7 is the real beam result. It can be seen that the four targets cannot be resolved at all. Figure 8 is the processing result of the two-dimensional Wiener inverse filtering method. This method has a fast processing speed. It can be seen that although the four targets are basically separated, the resolution performance of this method is poor, and there are strong sidelobes near the 0° azimuth and 0° elevation dimensions. Both of these methods can be quickly implemented in the frequency domain through FFT, but the resolution is limited. Figure 9 is the processing result of the method proposed by the present invention. It can be seen that its resolution performance is significantly better than the real beam result and the result of the two-dimensional Wiener filtering method.

[0201] Those of ordinary skill in the art will realize that the embodiments described herein are for helping the reader understand the principles of the present invention and should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. For those skilled in the art, various changes and modifications can be made to the present invention. Any modification, equivalent replacement, improvement, etc. made within the spirit and principle of the present invention shall be included within the scope of the claims of the present invention.

Claims

1. A method for two-dimensional super-resolution imaging of the forward-looking azimuth and pitch of a moving platform, the specific steps are as follows: Step S1, the azimuth-pitch two-dimensional scanning echo model of the airborne radar platform; Assume that the airborne radar platform is located at point A(0, 0, Γ) at t = 0 and moves uniformly along the y-axis direction with a speed of v. Γ represents the altitude of the airborne platform. At time t, the airborne platform moves to point B, and it can be obtained that AB = vt. Assume that a point P(x P , y P , z P ) in the scanning area, and its projection on the xOy plane is point Q. R P is the distance between the radar platform and the target at the initial moment, is the elevation angle of the target relative to the radar platform, θ P is the azimuth angle of the target relative to the y-axis direction. According to the geometric relationship, the distance R(t) between the airborne platform and the target at time t is expressed as: According to the following geometric relationship: Substitute Equation (2) into Equation (1) to get: Perform Taylor expansion on R(t), and we can get: Where, O(t 2 ) represents the high-order term of t 2 . During the coherent integration time of an arbitrary point target, vt << R is satisfied P . Equation (4) is approximated as: Suppose the scanning radar emits a linear frequency modulation signal: where τ is the range-time variable, T p is the pulse width, rect(·) represents the rectangular window function, f c represents the carrier frequency, K r represents the chirp rate, and the echo signal of the target point P is expressed as: where \(t\) is the platform motion time variable corresponding to the slow time, \(\tau\) is the range time variable corresponding to the fast time, \(\sigma\) P is the target scattering coefficient of target \(P\), \(\kappa\) represents a constant containing various gains and losses, \(c\) is the electromagnetic wave propagation speed, \(h(t)\) is the antenna pattern modulation function. In the range dimension, high range resolution is achieved through pulse compression. After de - carrier frequency and pulse compression processing, the echo is written as: Among them, B r represents the signal bandwidth, satisfying B r = K r T p , sinc(·) represents the range-direction signal envelope after pulse compression, specifically denoted as sinc(γ) = sin(πγ) / πγ, and the last term exp{-j4πR(t) / λ} represents the Doppler phase term caused by platform motion, where λ represents the wavelength; Scan the target airspace Ω, where Δθ and represent the azimuth and elevation wave position jumps respectively, Φ θ and represent the azimuth and elevation scan ranges respectively, N θ and represent the azimuth and elevation sampling points respectively. During the scanning process of the airborne radar platform, the fast time τ and the slow time t respectively satisfy: wherein, PRF represents the pulse repetition frequency, θ and represent the azimuth and elevation angle variables, and R represents the range variable; Step S2, echo preprocessing; Substitute Equation (9) and Equation (10) into the echo Equation (8), and we can get the range-azimuth-pitch three-dimensional echo expression: Among them, represents the two-dimensional antenna pattern, and the distance movement amount is expressed as: Construct a phase compensation factor in the range frequency domain: where f R represents the range frequency variable. After performing a fast Fourier transform (FFT) on the range direction of Equation (11), multiplying it by Equation (13), and then performing an inverse fast Fourier transform (IFFT) on the range direction, the corrected echo after motion compensation can be obtained: Among them, the fourth term in the formula represents a constant term related to the initial slant range, and the fifth term represents the Doppler phase modulation generated during the relative motion process. When the target is within the forward-looking range of the airborne platform, Equation (14) is approximated as: Let the backscattering coefficient of the target located in the airspace Ω be and be respectively representing the distance, azimuth angle, and elevation angle of the target , and the total echo of the detection airspace is written as: Equation (16) is simplified to the echo model of a single range cell: Among them, is equivalent to a two-dimensional convolution kernel, indicating the target azimuth and elevation scattering distribution of the corresponding range cell; Considering additive noise, we have: where * represents a two-dimensional convolution, is equivalent to a two-dimensional convolution kernel, represents additive Gaussian noise; The discretization of the two-dimensional convolution model (18) is: Y = AXB T + N (19) Among them, represents the echo matrix of a certain range slice, is the backscattering coefficient matrix of the target, K 1 and K 2 respectively represent the azimuth and elevation sampling points of the backscattering coefficient matrix, is the additive noise, represents the real number space, T represents the transpose of the matrix, and respectively represent the azimuth and elevation antenna pattern modulation matrices, specifically written as: Among them, represents the sampling points of the azimuthal dimension antenna pattern, represents the sampling points of the elevation dimension antenna pattern, L 1 and L 2 respectively represent the number of sampling points of the azimuthal dimension and elevation dimension antenna patterns, the number of sampling points M, N, K 1 , K 2 , L 1 , L 2 satisfy the relationship: Step S3, calculate the autocorrelation matrix R; Target scattering result x (q+1) Written as: x (q+1) = P (q) F H (R (q) ) -1 y (23) where q represents the number of iterations, R (q) = F H P (q) F is obtained by calculating P in the q-th iteration (q) P (q) = diag(x (q) ), diag(·) represents constructing a diagonal matrix according to a certain vector, H represents the conjugate transpose operation, y = vec(Y) is the one-dimensional vectorized form of the echo Y, and diagonal loading is performed on R: R = FPF H + μI (24) where μ represents a regularization parameter, I represents the identity matrix, and the computational complexity of R is reduced by the CG algorithm -1 ; Step S4, iteratively update U; Define variables The variable u is updated by the following formula: u l+1 = u l + α l d l (25) where l represents the number of CG iterations, d l represents the conjugate direction with respect to R, and α l represents the step size, satisfying ρ l represents an intermediate variable, and the weight vector w l satisfies: w l = Rd l+1 (26) Write Equation (25) in two-dimensional form: Among them, U l , D l represent the two-dimensional forms of u l and d l , satisfying u l = vec(U l ), d l = vec(D l ), so there is: U l+1 = U l + α l D l (28) Where, where, * represents the conjugate operation, ⊙ represents the Hadamard product of matrices, and represent the i-th elements of the vectors d l+1 and w l respectively, denotes a vector of all 1s with 1 row and 1 column, denotes an all-1 vector with N θ rows and 1 column, and W l satisfies w l = vec(W l ); Step S5, iteratively update W; Write Equation (26) in two-dimensional form: where, Σ is the target scattering power, satisfying Σ ij = |X ij 2 , and the matrix P satisfies the relation P = diag{vec(Σ)}, where Σ ij represents the element in the i-th row and j-th column of the target scattering power matrix Σ, i = 1,..., K 1 , j = 1,..., K 2 , X ij represents the i-th row and j-th column of the matrix target X, P is a diagonal matrix, and P is transformed into the Hadamard product relationship with (A H D l+1 B * ), so there is: W l = A(Σ⊙(A H D l+1 B * ))B T + μD l+1 (31) Step S6, iteratively update D; d l The one-dimensional update expression is written as: d l+1 = g l + β l d l (32) where, g l represents the gradient vector, and the intermediate variable β l+1 = ρ l+1 / ρ l ; Write Equation (32) in two-dimensional form: Among them, G l represents the two-dimensional form of g l , satisfying g l = vec(G l ), so we have: D l+1 = G l + β l D l (35) Among them, β l+1 is updated in the same way as Equation (32), and ||·|| F represents the matrix Frobenius norm; Step S7, iteratively update G; In Equation (32), the update formula for the gradient g l is as follows: g l+1 = g l - α l w l (36) Convert the above equation into two-dimensional form: Get: G l+1 = G l - α l W l (38) Terminate the CG iteration when is satisfied or after reaching the specified number of iterations. ε represents a small positive number, and output the two-dimensional CG iteration result Step S8, the two-dimensional form of the iterative estimation result; According to Equation (23), the two-dimensional expression of the target scattering is written as: Get the two-dimensional target scattering iterative formula: X (q+1) = Σ (q) ⊙(A H U (q) B * ) (40) Among them, U (q) satisfies vec(U (q) ) = R -1 y. At the beginning of the target scattering iteration, the variable Σ is initialized as: Σ (0) = A H YB H . At the beginning of the CG iteration, the variables are initialized as: U 0 = 0, D 0 = 0, β 0 = 0, G 0 = Y, Through repeated iterations of two-dimensional target scattering (Equation (40)) and CG (Equations (28), (29), (31), (35), and (38)), X in the model Equation (19) can be directly solved.

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