CN115453536A - Forward-looking azimuth pitching two-dimensional super-resolution imaging method for motion platform - Google Patents

Abstract

The invention discloses a method for two-dimensional super-resolution imaging of a forward-looking direction and a pitching direction of a motion platform. The method solves the problems that the existing real-aperture radar is low in azimuth-elevation two-dimensional resolution and is not suitable for a moving platform, and compared with the existing two-dimensional scanning radar super-resolution method, the method not only can be used for two-dimensional super-resolution imaging of the moving platform, but also has excellent target resolution capability and lower calculation complexity and is suitable for engineering realization.

Description

A forward-looking azimuth and pitch two-dimensional super-resolution imaging method for a moving platform

technical field

The invention belongs to the technical field of radar imaging, and in particular relates to a two-dimensional super-resolution imaging method for forward-looking azimuth and pitch of a moving platform.

Background technique

Real Aperture Radar (RAR) has forward-looking target detection capabilities that Synthetic Aperture Radar (SAR) and Doppler Beam Sharpening (DBS) cannot achieve in terms of imaging mechanism. field. Existing RAR super-resolution methods can improve azimuth imaging resolution, but the result is usually an azimuth-range two-dimensional image with a fixed elevation angle, regardless of the scan modulation of the elevation angle beam antenna pattern.

In order to obtain a high azimuth resolution that matches the distance 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" proposes a sparse iterative covariance-based estimation (SPICE) method based on beam updating, which takes advantage of the sparsity of the target scene and further improves the azimuth resolution compared with the iterative adaptive method rate, but the processing order of the method for the azimuth and elevation directions will seriously deteriorate the imaging results, 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 and Remote Sensing Symposium IGARSS.IEEE, 2021: 6630-6633" proposed a two-dimensional super-resolution imaging based on sparse alternating direction multipliers (Alternating Direction Method of Multipliers, ADMM) Method, using the regularization 1 norm constraint, and using the ADMM method to solve, improve the two-dimensional resolution of RAR, but this method is only effective for fixed platforms, but it is difficult to achieve two-dimensional resolution of radar azimuth and pitch on airborne moving platforms.

Contents of the invention

In order to solve the above technical problems, the present invention proposes a two-dimensional super-resolution imaging method for forward-looking azimuth and pitch of a moving platform.

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

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

为目标相对于雷达平台的俯仰角，θ_P为目标相对于y轴方向的方位角。根据几何关系，t时刻机载平台与目标的距离R(t)可表示为：Assuming that the airborne radar platform is located at point A(0,0,Γ) at t=0, and moves at a constant speed along the y-axis direction at a speed v, Γ represents the height of the airborne platform, and the airborne platform moves to point B at time t, which can be Obtain AB=vt, set a point P(x _P ,y _P ,z _P ) in the scanning area, the 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:

According to the following geometric relations:

Substitute formula (2) into formula (1) to get:

Taylor expansion of R(t) can be obtained:

Among them, O(t ² ) represents the high-order term of t ² , which satisfies vt<<R _P within the coherent integration time (CPI) of an arbitrary point target, so the quadratic term and higher-order term of the distance history can be ignored , formula (4) can be approximated as:

Suppose the scanning radar emits a chirp signal:

Among them, τ is the distance 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 linear frequency modulation slope. According to the process of platform movement, beam scanning, signal transmission and reception, the echo signal of the target point P can be expressed as:

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

Among them, B _r represents the signal bandwidth, satisfying B _r =K _r T _p , sinc(·) represents the range signal envelope after pulse compression, specifically recorded as sinc(γ)=sin(πγ)/πγ, the last item exp{-j4πR(t)/λ} represents the Doppler phase term caused by platform motion, and λ represents the wavelength.

分别表示方位和俯仰波位跃度，Φ_θ和

分别表示方位和俯仰扫描范围，N_θ和

分别表示方位和俯仰采样点数。在机载雷达平台扫描过程中，快时间τ和慢时间t分别满足：The way to scan the target airspace Ω is: for the first pitch dimension, first scan along the azimuth direction, then jump to the second pitch dimension, and then scan along the azimuth, and so on, forming a "Z" intensive scan line by line , where Δθ and

denote the azimuth and elevation wave pitch jumps respectively, Φ _θ and

Respectively represent the azimuth and pitch scanning range, N _θ and

Represents the number of azimuth and elevation sampling points, respectively. During the scanning process of the airborne radar platform, the fast time τ and slow time t satisfy respectively:

表示方位俯仰角度变量，R表示距离变量。where PRF represents the pulse repetition frequency, θ and

Represents the variable of azimuth and pitch angle, and R represents the variable of distance.

Step S2, echo preprocessing;

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

表示二维天线方向图，为了消除由平台和目标的相对运动产生的距离走动，距离走动量可表示为：in,

Represents the two-dimensional antenna pattern, in order to eliminate the distance walking caused by the relative motion of the platform and the target, the distance walking can be expressed as:

Construct phase compensation factors in the range-frequency domain:

Among them, f _R represents the range-frequency variable. After the fast Fourier transform (FFT) is performed on the range of formula (11), it is multiplied by formula (13), and then the range-wise inverse fast Fourier transform (IFFT) can be Get the echo corrected by walking:

表示一与初始斜距有关的常数项，不影响方位俯仰二维处理，可近似忽略，第五项表示相对运动过程产生的多普勒相位调制。当目标位于机载平台正前视范围时，

这种情况下前视区域各目标的多普勒历程近似相等，因此式(14)可近似为：Among them, the fourth item in the formula

Represents a constant term related to the initial slope distance, which does not affect the two-dimensional processing of azimuth and pitch, and can be approximately ignored. The fifth term represents the Doppler phase modulation generated by the relative motion process. When the target is in the forward sight range of the airborne platform,

In this case, the Doppler history of each target in the forward-looking area is approximately equal, so formula (14) can be approximated as:

的目标后向散射系数为

R₀,θ₀,

分别表示目标

的距离、方位角、俯仰角，则探测空域的总回波可以写为：Let the airspace Ω be located at

The target backscatter coefficient of is

R ₀ ,θ ₀ ,

Respectively represent the target

distance, azimuth, and elevation angle, the total echo in the detection airspace can be written as:

For a fixed distance cell, only azimuth and elevation changes are considered. The azimuth-elevation two-dimensional scanning of the detection area is equivalent to the two-dimensional antenna pattern. Each scattering point in the detection area can be regarded as the two-dimensional convolution of the target scattering coefficient and the azimuth and elevation two-dimensional antenna pattern. Therefore, formula (16) can be simplified to the echo model of a single range unit as:

等价于一个二维卷积核，

表示对应距离单元的目标方位俯仰散射分布。in,

Equivalent to a two-dimensional convolution kernel,

Indicates the target azimuth pitch scatter distribution for the corresponding range cell.

Considering additive noise, there are:

等价于一个二维卷积核，

表示加性高斯噪声。Among them, * indicates two-dimensional convolution,

Equivalent to a two-dimensional convolution kernel,

Represents additive Gaussian noise.

For concise representation, the two-dimensional convolutional model (18) can be discretized as:

Y=AXB ^T +N (19)

表示某一距离切片的回波矩阵，

为目标的后向散射系数矩阵，K₁、K₂分别表示后向散射系数矩阵方位和俯仰采样点数，

为加性噪声，

表示实数空间，T表示矩阵的转置，

和

分别表示方位和俯仰天线方向图调制矩阵，具体地写为：in,

represents the echo matrix for a range slice,

is the backscatter coefficient matrix of the target, K ₁ and K ₂ represent the azimuth and elevation sampling points of the backscatter coefficient matrix respectively,

is additive noise,

Represents the real number space, T represents the transpose of the matrix,

with

Denote the modulation matrix of the azimuth and elevation antenna pattern respectively, specifically written as:

表示方位维天线方向图采样点，

表示俯仰维天线方向图采样点，L₁和L₂分别表示方位维和俯仰维天线方向图采样点数，并且采样点数M，N，K₁，K₂，L₁，L₂之间满足关系：in,

Indicates the sampling points of the antenna pattern in the azimuth dimension,

Represents the sampling points of the antenna pattern in the elevation dimension, L ₁ and L ₂ represent the sampling points of the antenna pattern in the azimuth dimension and the elevation dimension respectively, and the number of sampling points M, N, K ₁ , K ₂ , L ₁ , and L ₂ satisfy the relationship:

Step S3, calculating the autocorrelation matrix R;

The target scattering result x ^(q+1) can be written as:

x ^(q+1) = P ^(q) F ^H (R ^(q) ) ^-1 y (23)

Among them, q represents the number of iterations, R ^(q) = F ^H P ^(q) F is calculated by P ^(q) of the qth iteration, P ^(q) = diag(x ^(q) ), diag(·) Indicates that a diagonal matrix is constructed according to a certain vector, H indicates a conjugate transpose operation, and y=vec(Y) is a one-dimensional vectorized form of echo Y. For the sake of brevity, in Equation (24) and the following, the subscript of the number of iterations 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 an error in the calculation of R ^-1 . To ensure the accuracy of angle estimation and prevent noise amplification problems, R is loaded diagonally:

R = FPF ^H +μI (24)

where μ denotes a regularization parameter that balances noise and resolution performance, and I denotes the identity matrix. Since the dimension of the matrix R is too large, the computational complexity of directly calculating R ^-1 is too high. Firstly, the computational complexity of R ^-1 is reduced by the CG algorithm.

Step S4, iteratively updating U;

根据Fletcher-Reeves共轭梯度算法，变量u可以通过以下公式更新：define variables

According to the Fletcher-Reeves conjugate gradient algorithm, the variable u can be updated by the following formula:

u _l+1 =u _l +α _l d _l (25)

ρ_l表示一中间变量，权重矢量w_l满足：Among them, l represents the number of CG iterations, d _l represents the conjugate direction about 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 formula (25) in two-dimensional form:

Among them, U _l and D _l represent the two-dimensional form of u _l and d _l , satisfying u _l =vec(U _l ), d _l =vec(D _l ), so:

U _l+1 ＝ U _l +α _l D _l (28)

in,

和

分别表示矢量d_l+1和w_l的第i个元素，

表示

行1列的全1向量，

表示N_θ行1列的全1向量，W_l满足w_l＝vec(W_l)，并可通过下面步骤导出。Among them, * represents the conjugate operation, ⊙ represents the Hadamard product of the matrix,

with

represent the i-th element of the vector d _l+1 and w _l respectively,

express

A vector of all 1s by row and column,

A vector of all 1s representing row and column of N _θ , W _l satisfies w _l =vec(W _l ), and can be derived through the following steps.

Step S5, iteratively updating W;

According to the Kronecker product property, formula (26) is written in two-dimensional form:

Among them, Σ is the target scattering power, which satisfies Σ _ij =|X _ij | ² , and satisfies the relationship P=diag{vec(Σ)} with the matrix P, and Σ _ij represents the i-th row and j-th column element of the target scattering power matrix Σ, i=1,...,K ₁ , j=1,...,K ₂ , X _ij represents row i and column j of matrix object X. Since P is a pair of diagonal matrices, P is transformed into the Hadamard product relationship of Σ and (A ^H D _l+1 B ^* ), so there are:

W _l ＝A(Σ⊙(A ^H D _l+1 B ^* ))B ^T +μD _l+1 (31)

Step S6, iteratively updating D;

The d _l one-dimensional update expression can be written as:

d _l+1 ＝g _l +β _l d _l (32)

Wherein, g _l represents the gradient vector, and the intermediate variable β _l+1 =ρ _l+1 /ρ _l .

Write formula (32) in two-dimensional form:

Among them, G _l represents the two-dimensional form of g _l , which satisfies g _l =vec(G _l ), so:

_{Dl +1} ＝Gl+ _{βlDl (} 35)

||·||_F表示矩阵Frobenius范数。Among them, the update method of β _l+1 is the same as formula (32), and

||·|| _F represents the Frobenius norm of the matrix.

Step S7, iteratively updating G;

In formula (32), the update formula of gradient g _l is as follows:

g _l+1 ＝g _l -α _l w _l (36)

Transform the above formula into two-dimensional form:

and thus get:

G _l+1 = G _l -α _l W _l (38)

或达到指定迭代次数后终止CG迭代，ε表示一个小正数，输出二维CG迭代结果

when satisfied

Or terminate the CG iteration 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 formula (23), the two-dimensional expression of target scattering can be written as:

Similar to formula (30), P ^(q) = diag{vec(Σ ^(q) )} is used in the above formula, and the iterative formula of two-dimensional target scattering is obtained:

X ^(q+1) = Σ ^(q) ⊙(A ^H U ^(q) B ^* ) (40)

通过二维目标散射(式(40))和CG(式(28)、(29)、(31)、(35)和(38))的重复迭代，模型式(19)中的X可被直接求解。Wherein, 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 a 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 (Eq. (40)) and CG (Eq. (28), (29), (31), (35) and (38)), the X in the model Eq. (19) can be directly solve.

Beneficial effects of the present invention: the method of the present invention establishes the azimuth-pitch two-dimensional scanning echo model of the airborne radar platform, and then performs distance walk correction on the azimuth-pitch-distance echo, and approximates the echo model as a two-dimensional volume Finally, using the conjugate gradient method and the Kronecker product property of the matrix, the scattering intensity of the two-dimensional target in azimuth and elevation is derived. 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 suitable for moving platforms. Compared with the existing two-dimensional scanning radar super-resolution method, it can not only be used for two-dimensional super-resolution on moving platforms Imaging, and has excellent target resolution and low computational complexity, suitable for engineering implementation.

Description of drawings

FIG. 1 is a flow chart of a two-dimensional super-resolution imaging method for forward-looking azimuth and pitch of a moving platform according to the present invention.

Fig. 2 is a motion geometric model of the airborne radar platform in the embodiment of the present invention.

Fig. 3 is a schematic diagram of a two-dimensional scanning mode of an airspace target in an embodiment of the present invention.

Fig. 4 is an azimuth and elevation two-dimensional antenna pattern in an embodiment of the present invention.

Fig. 5 is a layout diagram of three-dimensional point objects in an original scene in an embodiment of the present invention.

Fig. 6 is a diagram of azimuth and pitch two-dimensional point target setting in the embodiment of the present invention.

Fig. 7 is a graph of the two-dimensional real beam echo results in azimuth and elevation after preprocessing in an embodiment of the present invention.

FIG. 8 is a diagram of super-resolution results of two-dimensional Wiener inverse filtering in an embodiment of the present invention.

FIG. 9 is a diagram of super-resolution results in an embodiment of the present invention.

detailed description

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

In this embodiment, the effectiveness of a proposed two-dimensional super-resolution imaging method for forward-looking azimuth and pitch of a moving platform is verified through simulation experiments. The steps and results in this embodiment are all verified on the MATLAB 2018b simulation platform.

As shown in Figure 1, a flow chart of a two-dimensional super-resolution imaging method for forward-looking azimuth and pitch of a moving platform according to the present invention, the specific steps are as follows:

Step S1, establishing a signal geometric model;

As shown in Table 1, the simulation parameters of the radar motion platform are listed. The sampling rate satisfies the Nyquist sampling law.

Table 1

为目标相对于雷达平台的俯仰角，θ_P为目标相对于y轴方向的方位角，如图2所示，根据几何关系，t时刻机载平台与目标的距离R(t)可以写为：The airborne radar platform is located at point A(0,0,Γ) at t=0, Γ=1000m, and moves at a constant speed along the y-axis at a speed of v=90m/s. When the airborne platform moves to point B at time t, AB=vt can be obtained. Suppose a point P(x _P , y _P , z _P ) in the scanning area is projected on the xOy plane as point Q. R _P =30000m 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 shown in Figure 2, according to the geometric relationship, the distance R(t) between the airborne platform and the target at time t can be written as:

According to Figure 2, there are the following geometric relations:

Substitute formula (42) into formula (41):

Taking Taylor expansion of R(t) and ignoring the quadratic and higher terms of the distance history, we can get:

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

Among them, τ is the distance time variable, rect( ) represents the rectangular window function, K _r represents the chirp slope, t is the platform motion time variable (slow time), τ is the distance time variable (fast time), σ _P is the target P The target scattering coefficient of , κ represents a constant including various gains and losses, c is the propagation speed of electromagnetic waves, and h(t) is the antenna pattern modulation function.

After decarrier frequency and pulse compression processing, the echo can be written as:

Among them, sinc(·) represents the range signal envelope after pulse compression, specifically recorded as sinc(γ)=sin(πγ)/πγ, and the last term exp{-j4πR(t)/λ} represents the Doppler phase term, λ represents the wavelength.

分别表示方位和俯仰采样点数。在机载雷达平台扫描过程中，快时间τ和慢时间t分别满足：As shown in Figure 3, the way to scan the target airspace Ω is: for the first pitch dimension, first scan along the azimuth direction, then jump to the second pitch dimension, and then scan along the azimuth, and so on. Z" font intensive scanning. where N _θ and

Represents the number of azimuth and elevation sampling points, respectively. During the scanning process of the airborne radar platform, the fast time τ and slow time t satisfy respectively:

表示方位俯仰角度变量，R表示距离变量。where PRF represents the pulse repetition frequency, θ and

Represents the variable of azimuth and pitch angle, and R represents the variable of distance.

Step S2, echo preprocessing;

From equations (46), (47) and (48), it can be seen that the range-azimuth-elevation three-dimensional echo expression can be written as:

表示二维天线方向图函数，其能量分布如图4所示。在距离频域构造相位补偿因子有：in,

Represents the two-dimensional antenna pattern function, and its energy distribution is shown in Figure 4. The phase compensation factors constructed in the distance frequency domain are:

Among them, f _R represents the range-frequency variable. After the fast Fourier transform (FFT) is performed on the range of formula (49), it is multiplied by formula (50), and then the range inverse fast Fourier transform (IFFT) can be Get the echo corrected by walking:

前视区域各目标的多普勒历程近似相等，因此式(51)可近似为：When the target is in the forward sight range of the airborne platform,

The Doppler history of each target in the forward-looking area is approximately equal, so formula (51) can be approximated as:

的目标后向散射系数为

R₀,θ₀,

分别表示目标

的距离、方位角、俯仰角，目标设定如图5、6所示。探测空域的总回波可以写为：Let the airspace Ω be located at

The target backscatter coefficient of is

R ₀ ,θ ₀ ,

Respectively represent the target

The distance, azimuth, and elevation angle of the target are set as shown in Figures 5 and 6. The total echo in the detection space can be written as:

According to the above formula, the two-dimensional scanning convolution model can be discretized as:

Y=AXB ^T +N (54)

表示某一距离切片的回波矩阵，

为目标的后向散射系数矩阵，K₁、K₂分别表示后向散射系数矩阵方位和俯仰采样点数，

为加性噪声，

表示实数空间，T表示矩阵的转置，

和

分别表示方位和俯仰天线方向图调制矩阵，具体地写为：in,

represents the echo matrix for a range slice,

is the backscatter coefficient matrix of the target, K ₁ and K ₂ represent the azimuth and elevation sampling points of the backscatter coefficient matrix respectively,

is additive noise,

Represents the real number space, T represents the transpose of the matrix,

with

Denote the modulation matrix of the azimuth and elevation antenna pattern respectively, specifically written as:

表示方位维天线方向图采样点，

表示俯仰维天线方向图采样点，L₁和L₂分别表示方位维和俯仰维天线方向图采样点数。in,

Indicates the sampling points of the antenna pattern in the azimuth dimension,

Indicates the sampling points of the antenna pattern in the elevation dimension, and L ₁ and L ₂ represent the sampling points in the antenna pattern in the azimuth and elevation dimensions, respectively.

Step S3, iteratively updating U;

通过以下公式更新变量u：define variable

The variable u is updated by the following formula:

u _l+1 =u _l +α _l d _l (57)

ρ_l表示一中间变量，w_l表示权重矢量。Among them, l represents the number of CG iterations, d _l represents the conjugate direction about R, and α _l represents the step size, satisfying

ρ _l represents an intermediate variable, w _l represents a weight vector.

Write formula (58) in two-dimensional form:

Among them, U _l and D _l represent the two-dimensional form of u _l and d _l , satisfying u _l =vec(U _l ), d _l =vec(D _l ), so:

Ul ₊₁ ＝Ul+ _{αlDl (} 59 ₎

in,

和

分别表示矢量d_l+1和w_l的第i个元素，

表示

行1列的全1向量，

表示N_θ行1列的全1向量，W_l满足w_l＝vec(W_l)。Among them, * represents the conjugate operation, ⊙ represents the Hadamard product,

with

represent the i-th element of the vector d _l+1 and w _l respectively,

express

A vector of all 1s by row and column,

Represents a vector of all 1s in row and column of N _θ , and W _l satisfies w _l =vec(W _l ).

Step S4, iteratively updating W;

Variables in formula (57):

w _l =Rd _l+1 (61)

Write formula (61) in two-dimensional form:

Among them, Σ is the target scattering power, which satisfies Σ _ij =|X _ij | ² , and satisfies the relationship P=diag{vec(Σ)} with the matrix P, and Σ _ij represents the i-th row and j-th column element of the target scattering power matrix Σ, i=1,...,K ₁ , j=1,...,K ₂ , X _ij represents row i and column j of matrix object X. Since P is a pair of diagonal matrices, P is transformed into the Hadamard product relationship of Σ and (A ^H D _l+1 B ^* ), so there are:

W _l ＝A(Σ⊙(A ^H D _l+1 B ^* ))B ^T +μD _l+1 (63)

Step S5, iteratively updating D;

The relationship between d _l+1 and d _l is updated by the following formula:

d _l+1 ＝g _l +β _l d _l (64)

Wherein, g _l represents the gradient vector, and the intermediate variable β _l+1 =ρ _l+1 /ρ _l .

Write formula (64) in two-dimensional form:

Among them, G _l represents the two-dimensional form of g _l , which satisfies g _l =vec(G _l ), so:

_{Dl +1} ＝Gl+ _{βlDl (} 67)

||·||_F表示矩阵Frobenius范数。Among them, the update method of β _l+1 is the same as formula (64), and

||·|| _F represents the Frobenius norm of the matrix.

Step S6, iteratively updating G;

In formula (64), the gradient g _l can be updated by the formula:

g _l+1 ＝g _l -α _l w _l (68)

Transform the above formula into two-dimensional form:

and thus get:

G _l+1 ＝G _l -α _l W _l (70)

或达到10次迭代后终止CG迭代，ε＝10^-3表示一个小正数，输出二维CG迭代结果

when satisfied

Or terminate the CG iteration after reaching 10 iterations, ε=10 ^-3 represents a small positive number, and output the two-dimensional CG iteration result

Step S7, the two-dimensional form of the iterative estimation result;

After terminating the CG iteration, output the CG iteration result u ^(q) = u _l and bring it into formula (23) to obtain the qth target scattering iteration result:

Among them, u ^(q) represents R ⁻¹ y in the qth target scattering iteration formula (23) obtained through CG iteration.

According to formula (23), the two-dimensional iterative expression of target scattering can be written as:

Thus there are:

X ^(q+1) = Σ ^(q) ⊙(A ^H U ^(q) B ^* ) (73)

通过二维目标散射(式(73))和CG(式(59)、(60)、(63)、(67)和(70))的重复迭代，模型式(54)中的X可被直接求解。这种方法一方面无需矩阵求逆，另一方面也无需构造并计算高维矩阵和矢量的乘法，能够极大的降低目标散射迭代方法的计算复杂度。Wherein, 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 a 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 (Eq. (73)) and CG (Eq. (59), (60), (63), (67) and (70)), the X in the model Eq. (54) can be directly solve. On the one hand, this method does not require matrix inversion, and on the other hand, it does not need to construct and calculate the multiplication of high-dimensional matrices and vectors, which can greatly reduce the computational complexity of the target scattering iterative method.

)，4次矩阵-标量乘法(U_l,D_l,G_l的更新和式(63)中μD_l+1)和1次标量乘法

设需要迭代L次，则二维CG算法的计算复杂度为

每次目标散射迭代需要两次矩阵乘法(Σ^(q)⊙(A^HU^(q)B^*))和一次标量乘法(Σ^(q)＝X^(q)⊙X^(q))。假设进行Q次目标散射迭代，则算法整体的计算复杂度为：Concrete analysis of the complexity of this method: In the CG iteration, four matrices need to be stored, which are 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 ), and 2 matrix-matrix Hadamard multiplications (one is Σ⊙(A ^H D _l+1 B ^* ), another time is in formula (60)

), 4 matrix-scalar multiplications (updates of U _l , D _l , G _l and μD _l+1 in Equation (63)) and 1 scalar multiplication

Assuming that it needs to iterate L times, 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) ). Assuming that Q times of target scattering iterations are performed, the overall computational complexity of the algorithm is:

图7为实波束结果，可以看到，4个目标完全无法分辨。图8为二维维纳逆滤波方法的处理结果。该方法处理速度较快。可以看到虽然四个目标基本被分开，但是该方法分辨率性能较差，并且在0°方位和0°俯仰维附近还有强副瓣影响。这两种方法均可通过FFT在频域快速实现，但是分辨率有限。图9为本发明提出方法的处理结果，可以看到，其分辨性能明显优于实波束结果和二维维纳滤波方法的结果。Figure 7, Figure 8, and Figure 9 show the super-resolution results of point targets by different methods. Limited by platform memory, the number of azimuth and pitch sampling points is reduced, among which,

Figure 7 shows the real beam results. It can be seen that the four targets are completely indistinguishable. Fig. 8 is the processing result of the two-dimensional Wiener inverse filtering method. This method has a faster 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 methods can be quickly implemented in the frequency domain by FFT, but the resolution is limited. Fig. 9 is the processing result of the method proposed by the present invention. It can be seen that its resolution performance is obviously better than the real beam result and the result of the two-dimensional Wiener filtering method.

Those skilled in the art will appreciate that the embodiments described here are to help readers understand the principles of the present invention, and it should be understood that the protection scope of the present invention is not limited to such specific statements and embodiments. Various modifications and variations of the present invention will occur to those skilled in the art. Any modifications, equivalent replacements, improvements, etc. made within the spirit and principles of the present invention shall be included within the scope of the claims of the present invention.

Claims (1)

1. A method for two-dimensional super-resolution imaging of pitching of a forward-looking azimuth of a motion platform comprises the following specific steps:

s1, an orientation-pitching two-dimensional scanning echo model of an airborne radar platform;

assuming that the airborne radar platform is located at a point A (0, gamma) when t =0 and moves at a constant speed along the y-axis direction at a speed v, gamma represents the height of the airborne platform, and the airborne platform moves to a point B at the time t, AB = vt can be obtained, and a point P (x) in a scanning area is assumed _P ,y _P ,z _P ) The projection on the xOy plane is the Q point, R _P The distance between the radar platform and the target at the initial moment,

is the pitch angle, θ, of the target relative to the radar platform _P According to the geometrical relationship, the distance R (t) between the airborne platform and the target at the time t is expressed as follows:

according to the following geometrical relationship:

substituting formula (2) into formula (1) to obtain:

taylor unfolding R (t) gives:

wherein, O (t) ² ) Denotes t ² Satisfies vt & lt R within the coherent accumulation time (CPI) of an arbitrary point target _P Equation (4) is approximated as:

setting a scanning radar to transmit a chirp signal:

where τ is the distance time variable, T _p For pulse width, rect (·)Representing a rectangular window function, f _c Denotes the carrier frequency, K _r Representing the chirp slope, the echo signal at target P point is represented as:

where t is the platform motion time variable (slow time), τ is the distance time variable (fast time), σ _P For the target scattering coefficient of the target P, κ represents a constant containing various gains and losses, c is the electromagnetic wave propagation speed, h (t) is the antenna pattern modulation function, high range resolution is achieved by pulse compression over distance, and after carrier frequency removal and pulse compression processing, the echo is written as:

wherein, B _r Represents the signal bandwidth and satisfies B _r ＝K _r T _p Sine (·) represents the distance-wise signal envelope after pulse compression, and is specifically denoted as sine (γ) = sin (π γ)/π γ, and the last term exp { -j4 π R (t)/λ } represents the Doppler phase term due to stage motion, and λ represents wavelength;

scanning a target space domain omega, wherein delta theta and

respectively representing the azimuth and elevation wave position jump, phi _θ And

respectively representing azimuth and elevation sweep ranges, N _θ And

respectively representing the number of azimuth sampling points and the number of pitching sampling points, wherein in the scanning process of the airborne radar platform, the fast time tau and the slow time t respectively meet the following requirements:

wherein PRF denotes the pulse repetition frequency, θ and

representing an azimuth pitch angle variable, and R representing a distance variable;

s2, echo pretreatment;

substituting the formula (9) and the formula (10) into the echo formula (8) can obtain a distance-azimuth-pitch three-dimensional echo expression:

wherein,

representing a two-dimensional antenna pattern, the distance walk is represented as:

constructing a phase compensation factor in the distance frequency domain:

wherein f is _R And (3) representing a distance frequency variable, performing Fast Fourier Transform (FFT) on the distance direction of the formula (11), multiplying the distance direction by the formula (13), and performing Inverse Fast Fourier Transform (IFFT) on the distance direction to obtain an echo after walking correction:

wherein, the fourth term in the formula

A constant term related to the initial slope distance is shown, a fifth term represents Doppler phase modulation generated by the relative motion process, when the target is positioned in the forward looking range of the airborne platform,

the approximation is:

locate in omega of airspace

Has a target backscattering coefficient of

R ₀ ,θ ₀ ,

Respectively representing objects

The total echo of the detection airspace is written as:

equation (16) reduces to the echo model for a single range bin:

wherein,

is equivalent to a two-dimensional convolution kernel,

representing the target azimuth pitch scattering distribution of the corresponding range unit;

considering additive noise, there are:

wherein, denotes a two-dimensional convolution,

is equivalent to a two-dimensional convolution kernel,

representing additive gaussian noise;

discretization of the two-dimensional convolution model (18) is:

Y＝AXB ^T +N (19)

wherein,

an echo matrix representing a slice at a certain distance,

is a backscattering coefficient matrix of the target, K ₁ 、K ₂ Respectively representing the position of the backscattering coefficient matrix and the number of pitching sampling points,

in order to be additive to the noise,

representing a real space, T represents the transpose of a matrix,

and

the azimuth and elevation antenna pattern modulation matrices are represented separately, written specifically as:

wherein, [ a (θ) ] _L1 ),a(θ _L1-1 ),…,a(θ ₁ )]The azimuth dimension antenna pattern sampling points are represented,

representing the antenna pattern sample point, L, in elevation ₁ And L ₂ Respectively representing the number of sampling points of an azimuth dimension antenna directional diagram and a pitching dimension antenna directional diagram, and the number of the sampling points M, N and K ₁ ，K ₂ ，L ₁ ，L ₂ Satisfies the relationship:

s3, calculating an autocorrelation matrix R;

scattering result x from the target ^(q+1) Writing as follows:

x ^(q+1) ＝P ^(q) F ^H (R ^(q) ) ^-1 y (23)

whereinQ denotes the number of iterations, R ^(q) ＝F ^H P ^(q) F is P by the q-th iteration ^(q) Is obtained by calculation, P ^(q) ＝diag(x ^(q) ) Diag (·) denotes constructing a diagonal matrix from a certain vector, H denotes a conjugate transpose operation, Y = vec (Y) is a one-dimensional vectorization form of the echo Y, and R is diagonally loaded:

R＝FPF ^H +μI (24)

where μ represents a regularization parameter, I represents a unit matrix, and R is reduced by the CG algorithm ^-1 The computational complexity of (2);

s4, updating U in an iterative manner;

defining variables

The variable u is updated by the following formula:

u _l+1 ＝u _l +α _l d _l (25)

wherein l represents the number of CG iterations, d _l Denotes the direction of conjugation, α, with respect to R _l Represents a step size, satisfies

ρ _l Representing an intermediate variable, a weight vector w _l Satisfies the following conditions:

w _l ＝Rd _l+1 (26)

equation (25) is written in two-dimensional form:

wherein, U _l 、D _l Represents u _l And d _l In two-dimensional form of (1), satisfy u _l ＝vec(U _l )，d _l ＝vec(D _l ) Thus, there are:

U _l+1 ＝U _l +α _l D _l (28)

wherein,

wherein an indicates a conjugate operation, an indicates a Hadamard product of the matrix,

and

respectively represent vectors d _l+1 And w _l The (i) th element of (a),

to represent

The full 1 vector of row 1 and column,

represents N _θ All 1 vectors, w, of row 1 column _l Satisfy w _l ＝vec(W _l )；

S5, iteratively updating W;

equation (26) is written in two dimensions:

wherein, the sigma is the target scattering power, and the requirement of sigma is satisfied _ij ＝|X _ij | ² The relation P = diag { vec (Σ) }, Σ is satisfied with the matrix P _ij I row j column element representing the target scattering power matrix Σ, i =1 ₁ ，j＝1,...,K ₂ ，X _ij The ith row and the jth column of the matrix target X are represented, P is a diagonal matrix, and P is converted into sigma-sum (A) ^H D _l+1 B ^* ) Hadamard product relationship of (a), thus:

W _l ＝A(Σ⊙(A ^H D _l+1 B ^* ))B ^T +μD _l+1 (31)

s6, iteratively updating D;

d _l the one-dimensional update expression is written as:

d _l+1 ＝g _l +β _l d _l (32)

wherein, g _l Representing gradient vectors, intermediate variable beta _l+1 ＝ρ _l+1 /ρ _l ；

Writing equation (32) in two dimensions:

wherein, G _l Denotes g _l In two-dimensional form of (b), satisfies g _l ＝vec(G _l ) Thus, there are:

D _l+1 ＝G _l +β _l D _l (35)

wherein, beta _l+1 The update method of (2) is the same as formula (32), and

||·|| _F representing a matrix Frobenius norm;

s7, iteratively updating G;

in the formula (32), the gradient g _l The update formula of (2) is as follows:

g _l+1 ＝g _l -α _l w _l (36)

converting the above formula to a two-dimensional form:

obtaining:

G _l+1 ＝G _l -α _l W _l (38)

when it satisfies

Or stopping CG iteration after reaching the specified iteration times, wherein epsilon represents a small positive number, and outputting a two-dimensional CG iteration result

S8, iterating a two-dimensional form of an estimation result;

according to equation (23), the two-dimensional representation of the scattering of the target is written as:

obtaining a two-dimensional target scattering iterative formula:

X ^(q+1) ＝Σ ^(q) ⊙(A ^H U ^(q) B ^* ) (40)

wherein, U ^(q) Satisfy vec (U) ^(q) )＝R ^-1 y, at the start of the target scatter iteration, the variable Σ is initialized to: sigma-shaped ⁽⁰⁾ ＝A ^H YB ^H At the start of a CG iteration, variables are initialized to: u shape ₀ ＝0，D ₀ ＝0，β ₀ ＝0，G ₀ ＝Y，

By repeated iterations of two-dimensional object scatter (equation (40)) and CG (equations (28), (29), (31), (35), and (38)), X in model equation (19) can be solved directly.

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