CN104483671B - Sparse representation theory-based synthetic aperture radar imaging method - Google Patents

Abstract

The invention discloses a sparse representation theory-based synthetic aperture radar imaging method which is mainly used for solving the problem that the traditional sparse representation-based imaging method cannot process complicated scenes and has large loss. The sparse representation theory-based synthetic aperture radar imaging method comprises the following steps: 1, obtaining a ground scene echo matrix Y; 2, constructing a ground scene azimuth matrix R and a ground scene range basis matrix G; 3, obtaining a matrix relation according to the matrixes in the steps 1 and 2, Gaussian noise N and a scattering coefficient matrix F: Y=RFG+N; 4, constructing an optimization function f (F, V, Dx, Dy) under a total variation frame according to the matrix relation, an intermediate variable matrix V, a horizontal total variation matrix Dx, a vertical total variation matrix Dy and the matrixes in the steps 1 and 2; and 5, solving the optimization function to obtain the image of the ground scene. The sparse representation theory-based synthetic aperture radar imaging method is capable of realizing the high-resolution imaging of synthetic aperture radars, is high in imaging speed, and can be used for large-area topographic mapping.

Description

Synthetic aperture radar image-forming method based on sparse representation theory

Technical field

The invention belongs to Radar Technology field, the synthesis of specifically a kind of large scene that can effectively process complexity Aperture radar imaging method, can be used for large-scale terrain mapping, cartography, round-the-clock global reconnaissance.

Background technology

Synthetic aperture radar SAR is a kind of imaging radar with high resolution, has round-the-clock, round-the-clock imaging energy Power, is widely used in terms of military and civilian.The high-resolution of SAR, relies on broadband signal on radial distance, hundreds of Megahertz frequency band Range resolution unit can be narrowed down to sub-meter grade；Orientation then rely on radar platform move, equally in sky Between form very long linear array, and each echo storage is made the ARRAY PROCESSING synthesizing.

In SAR system, radar is along course line earthward scene transmitting pulse, and receives from the scattering of this ground scene Echo, then utilize focusing algorithm, such as Range Doppler, Chirp Scaling processes echo, reconstructs ground scene. Although this method is more effective, need echo to be digitized locate using the sample frequency more than nyquist frequency Reason, higher sample frequency is undoubtedly a huge challenge for the A/D converter of receiving terminal；Sample the big of acquisition simultaneously Amount sampled data also increases storage and the burden of transmission；The resolution limitations of the SAR image that traditional method reconstructs out are in SAR The bandwidth of system signal, can only improve the bandwidth of signal, so also increase the complexity of hardware, Er Qiexiang to improve resolution Dry spot noise, and the impact to picture quality for the secondary lobe artifact is also very serious.

The sparse representation theory emerged in large numbers in recent years is pointed out, if certain high dimensional signal is sparse or in itself under certain transform domain It is sparse, this signal can be projected on lower dimensional space with the incoherent observing matrix of transform domain by one, and pass through Optimization method realizes high probability signal reconstruction.The proposition of sparse representation theory is that the high-resolution imaging of synthetic aperture radar provides Probability.Theoretical according to this, Vishal M.Patel et al. is in document " Compressed Synthetic Aperture Radar.IEEE JournalofSelected Topicsin Signal Processing,Vol.4,NO.2,April 2010. " propose a kind of imaging method based on total variation rarefaction representation, by scene vectorization and construct observing matrix, then profit The method being reconstructed with reception echo.Although this imaging method can be become under less than Nyquist sampling frequency Picture, but generally require to devote a tremendous amount of time it is impossible to reach the imaging processing of real-time, and be imaged effectiveness poor so that It can only process less scene.

Content of the invention

Present invention aims to the deficiency of above-mentioned prior art, a kind of synthesis based on sparse representation theory is proposed Aperture radar imaging method, to realize directly ground complicated two-dimensional scene being reconstructed, subtracts while keeping high-resolution Few imaging time.

For achieving the above object, technical scheme comprises the steps：

(1) according to radar earthward scene transmitting pulseObtain the echo-signal of ground scene

In formula,For fast time, t_aFor slow time, f_cFor carrier frequency, γ is chirp rate, and v is carrier aircraft speed, and C is electromagnetic wave Spread speed, w_a() is orientation window function, w_r() is chirp pulse signal time window function, and P is ground scene in side Position to discrete grid block number, Q be ground scene distance to discrete grid block number, p be ground scene orientation pth Individual discrete grid block, q be ground scene distance to q-th discrete grid block, f_pqScattering for scattering point at (p, q) individual grid Coefficient, R₀For the vertical oblique distance of radar to ground scene center, R_qFor vertical oblique distance from radar to distance to q-th discrete grid block, x_pX-axis coordinate for p-th discrete grid block of orientation；

(2) to echo-signalCarry out two-dimensional discrete sampling, obtain following echo matrix Y：

Wherein, M is the exomonental number of orientation, and N is that in each pulse, distance is to sampling number；For , in the sample value of (m, n) individual sampling instant, its expression is as follows for echo-signal：

In formula, λ is carrier wavelength, t_a,mFor m-th pulse,For fast n-th sampling instant value of time；

(3) construct the orientation basic matrix R of ground scene：

Wherein,

(4) distance of construction ground scene is to basic matrix G：

Wherein,

(5) according to echo matrix Y, orientation basic matrix R, distance to basic matrix G and ground scene scattering coefficient matrix F, builds following matrix relationship formula：

Y=RFG+N,

In formula, N is the noise matrix with echo matrix Y with dimension, and the form of scattering coefficient matrix F is as follows：

Wherein, p=1,2 ..., P, q=1,2 ..., Q；

(6) the scattering coefficient matrix F according to ground scene, obtains intermediate variable matrix V, horizontal total variation matrix D_x, hang down Straight total variation matrix D_y：

V=F

D_x=D_hF,

D_y=FD_v

In formula, D_hFor horizontal total variation operator matrix, D_vFor vertical total variation operator matrix；

(7) parameter that the matrix relationship formula of step (5) and step (6) obtain, the optimization under construction total variation framework are utilized Function f (F, V, D_x,D_y)：

WhereinRepresent the operator finding a function minima, λ is total variation penalty factor, ν penalizes for scene scatters coefficient The factor, μ is echo penalty factor, | | ||_FRepresent the Frobenius norm seeking matrix, || | |₁Represent to matrix each Element is added after taking absolute value again；

(8) use Split-Bregman method to majorized function f (F, V, the D in step (7)_x,D_y) be iterated solving, obtain Final iteration result F to ground scene scattering coefficient matrix F^*；

(9) step (8) is obtained with the final iteration result F of ground scene scattering coefficient matrix F^*Modulus of access, output obtains The image of ground scene.

The present invention compared with prior art has the advantage that：

First, the present invention is processed by scene is done with total variation, then utilizes Split-Bregman method imaging processing, So that the method based on rarefaction representation processes complicated scene becomes possibility.

Second, the present invention is directly reconstructed imaging to two-dimentional ground scene image, it is to avoid scene image vectorization The observing matrix that reconstructing method produces is excessive, and the problem that amount of calculation sharply increases has large scene fast imaging, little scene is real-time The advantage of imaging, has obvious advantage to the application scenario that image taking speed has high demands.

Brief description

Fig. 1 is the flowchart of the present invention；

Fig. 2 is the image of the Harbor scene that present invention emulation uses；

Fig. 3 is the image of the Harbor scene being reconstructed with the inventive method；

Fig. 4 is the image of the terraced fields scene that present invention emulation uses.

Specific implementation method

Below in conjunction with the accompanying drawings the present invention is described in further detail.

With reference to Fig. 1, the specific implementation step of the present invention is as follows：

Step 1：Obtain discrete two-dimensional echo matrix Y.

(1a) carrier aircraft moves ahead along course, radar earthward scene transmitting chirp：

In formula, f_cFor carrier frequency, γ is chirp rate, and t is full-time,For fast time, t_aFor slow time, these three times Relation be Represent the time window function of chirp pulse signal, T_rIt is pulse persistance Time；

(1b) receive the echo impulse of this scene, obtaining echo-signal is

In formula, v is carrier aircraft speed, and C is propagation velocity of electromagnetic wave, w_a() is orientation window function, w_r() is linear frequency modulation Time pulse signal window function, P be ground scene orientation discrete grid block number, Q be ground scene distance to from Scattered meshes number, p be ground scene orientation p-th discrete grid block, q be ground scene distance to q-th discrete Grid, f_pqFor the scattering coefficient of scattering point at (p, q) individual grid, R₀For the vertical oblique distance of radar to ground scene center, R_q For vertical oblique distance from radar to distance to q-th discrete grid block, x_pX-axis coordinate for p-th discrete grid block of orientation；

(1c) by the echo of step (1b) ground sceneCarry out two-dimensional discrete sampling, obtain echo matrix Y：

Wherein M is the exomonental number of orientation, and N is distance in each pulse to sampling number；Table Show the sample value of (m, n) individual sampling instant of echo, its expression is as follows：

In formula, λ is carrier wavelength, t_a,mFor m-th pulse,For fast n-th sampling instant value of time.

Step 2：The construction orientation basic matrix R of ground scene and distance are to basic matrix G.

(2a) according to step 1 parameter, the orientation gene polyadenylation signal r (t of construction ground scene_a,m,p)：

Wherein, t_a,mFor m-th pulse, p=1,2 ..., P, m=1,2 ..., M, w_a() is orientation window function, and v is to carry Motor speed, λ is carrier wavelength, R₀For the vertical oblique distance of radar to ground scene center, x_pX for p-th discrete grid block of orientation Axial coordinate；

(2b) according to orientation gene polyadenylation signal r (t_a,m, p), obtain orientation basic matrix R：

In formula, M is the exomonental number of orientation, and P is the discrete grid block number of ground scene orientation；

(2c) according to step 1 parameter, the distance of construction ground scene is to gene polyadenylation signal

In formula,For fast n-th sampling instant value of time, q=1,2 ..., Q, n=1,2 ..., N, γ are chirp rate, w_r() is chirp time window function, and C is propagation velocity of electromagnetic wave, and λ is carrier wavelength, R_qFor radar to distance to The vertical oblique distance of q-th discrete grid block；

(2d) according to distance to gene polyadenylation signalObtain distance to basic matrix G：

Wherein, N be distance in each pulse to sampling number, Q be ground scene D distance to discrete grid block Number.

Step 3：Echo matrix Y that obtained according to step 1 and 2, orientation basic matrix R, distance are to basic matrix G and ground field The scattering coefficient matrix F of scape, builds following matrix relationship formula：

Y=RFG+N

In formula, N is the noise matrix with echo matrix Y with dimension, and the form of scattering coefficient matrix F is as follows：

Wherein, p=1,2 ..., P, q=1,2 ..., Q.

Step 4：Matrix relationship formula construction total variation framework in the matrix variables being obtained according to step 1 and 2 and step 3 Under majorized function f (F, V, D_x,D_y).

(4a) the scattering coefficient matrix F according to ground scene, obtains intermediate variable matrix V, horizontal total variation matrix D_x, hang down Straight total variation matrix D_y：

V=F

D_x=D_hF,

D_y=FD_v

In formula, D_hFor horizontal total variation operator matrix, D_vFor vertical total variation operator matrix；

(4b) majorized function f (F, V, the D according to the matrix variables in step (4a), under construction total variation framework_x,D_y)：

Wherein,Represent the operator finding a function minima, λ is total variation penalty factor, ν penalizes for scene scatters coefficient The factor, μ is echo penalty factor, || | |_FRepresent the Frobenius norm seeking matrix, | | | |₁Represent to matrix each Element is added after taking absolute value again.

Step 5：With Split-Bregman method to majorized function f (F, V, the D in step 4_x,D_y) be iterated solving, The image of the ground scene after being imaged.

(5a) according to majorized function f (F, V_,D_x,D_y), obtain majorized function L (F, V, the D of Split-Bregman form_x, D_y)：

In formula, B_xFor horizontal total variation matrix auxiliary variable, B_yFor vertical total variation matrix auxiliary variable, W is scattering coefficient Matrix auxiliary variable；

(5b) initialize：Iterative steps k=0, scattering coefficient matrix F after kth time iteration^kFor all 1's matrix, kth time iteration Intermediate variable matrix V afterwards^kFor all 1's matrix, horizontal total variation matrix after kth time iterationFor all 1's matrix, hang down after kth time iteration Straight total variation matrixFor all 1's matrix；Scattering coefficient matrix auxiliary variable W after kth time iteration^kFor full 0 matrix, kth time iteration Horizontal total variation matrix auxiliary variable afterwardsFor full 0 matrix, vertical total variation matrix auxiliary variable after kth time iterationFor complete 0 matrix；Setting total variation penalty factor λ>0, scene scatters coefficient penalty factor ν>0, echo penalty factor μ>0, stopping criterion for iteration ε= 5×10^-4；

(5c) obtain scattering coefficient matrix F after+1 iteration of kth^k+1；

(5c1) according to orientation basic matrix R, distance to basic matrix G, obtains orientation symmetrical matrix Ψ, and distance is to symmetrical Matrix Φ：

Ψ=R^TR,

Φ=GG^T

Wherein, ()^TRepresenting matrix transposition；

(5c2) according to feature decomposition orientation symmetrical matrix Ψ, distance, to symmetrical matrix Φ, obtains orientation orthogonal matrix U_R, orientation diagonal matrix L_R, distance is to orthogonal matrix U_G, distance is to diagonal matrix L_G；

(5c3) according to echo matrix Y, orientation basic matrix R, distance is to basic matrix G, intermediate variable square after kth time iteration Battle array V^k, scattering coefficient matrix auxiliary variable W after kth time iteration^k, orientation orthogonal matrix U_R, distance is to orthogonal matrix U_G, orientation To diagonal matrix L_R, distance is to diagonal matrix L_G, obtain scattering coefficient companion matrix after+1 iteration of kth according to equation below

In formula, p=1,2 ..., P, q=1,2 ..., Q；

(5c4) according to orientation orthogonal matrix U_R, distance is to orthogonal matrix U_G, scattering coefficient auxiliary moment after+1 iteration of kth Battle arrayObtain scattering coefficient matrix F after+1 iteration of kth according to equation below^k+1：

(5d) obtain intermediate variable matrix V after+1 iteration of kth^k+1；

(5d1) according to horizontal total variation operator matrix D_h, vertical total variation operator matrix D_v, obtain horizontal total variation matrix E, vertical total variation matrix Z：

(5d2) according to feature decomposition horizontal total variation matrix E, vertical total variation matrix Z, obtain horizontal total variation orthogonal moment Battle array U_x, horizontal total variation diagonal matrix L_x, vertical total variation orthogonal matrix U_y, vertical total variation diagonal matrix L_y；

(5d3) according to scattering coefficient matrix F after+1 iteration of kth^k+1, horizontal total variation operator matrix D_h, vertical total variation Operator matrix D_v, scattering coefficient matrix auxiliary variable W after kth time iteration^k, horizontal total variation matrix auxiliary variable after kth time iterationVertical total variation matrix auxiliary variable after kth time iterationHorizontal total variation matrix after kth time iterationKth time is repeatedly For rear vertical total variation matrixHorizontal total variation orthogonal matrix U_x, vertical total variation orthogonal matrix U_y, horizontal total variation is diagonal Matrix L_x, vertical total variation diagonal matrix L_y, obtain intermediate variable companion matrix after+1 iteration of kth according to equation below

In formula, p=1,2 ..., P, q=1,2 ..., Q；

(5d4) according to horizontal total variation orthogonal matrix U_x, vertical total variation orthogonal matrix U_y, middle anaplasia after+1 iteration of kth Amount companion matrixObtain intermediate variable matrix V after+1 iteration of kth according to equation below^k+1：

(5e) according to intermediate variable matrix V after+1 iteration of kth^k+1, after kth time iteration, horizontal total variation matrix auxiliary becomes AmountVertical total variation matrix auxiliary variable after kth time iterationObtain level after+1 iteration of kth according to equation below complete Variation matrixVertical total variation matrix after+1 iteration of kth

In formula, p=1,2 ..., P, q=1,2 ..., Q, D_hFor horizontal total variation operator matrix, D_vFor vertical total variation operator Matrix, sign () is sign function；

(5f) according to intermediate variable matrix V after above-mentioned+1 iteration of the kth having obtained^k+1, scattering system after+1 iteration of kth Matrix number F^k+1, horizontal total variation matrix after+1 iteration of kthVertical total variation matrix after+1 iteration of kthKth Horizontal total variation matrix auxiliary variable after secondary iterationVertical total variation matrix auxiliary variable after kth time iterationKth time is repeatedly For rear scattering coefficient matrix auxiliary variable W^k, obtain scattering coefficient matrix auxiliary variable W after+1 iteration of kth according to equation below^{k +1}, horizontal total variation matrix auxiliary variable after+1 iteration of kthVertical total variation matrix auxiliary variable after+1 iteration of kth

W^k+1=W^k+V^k+1-F^k+1

(5g) according to scattering coefficient matrix F after+1 iteration of kth^k+1, scattering coefficient matrix F after kth time iteration^k, by as follows Formula obtains mean square error M：

In formula, || ||_FRepresent the Frobenius norm seeking matrix；

(5h) judge whether mean square error M≤ε sets up, if setting up execution step (5i)；Otherwise make k=k+1, return to step (5c) continue iteration to run；

(5i) make F^*=F^k+1, obtain the final iteration result F of ground scene scattering coefficient matrix F^*；

(5j) to final iteration result F^*Modulus of access, output obtains the image of ground scene.

The effect of the present invention can be illustrated by following emulation experiments：

1. simulated conditions

(1a) operation platform configuration：

CPU：Intel(R)Core(TM)2Duo CPU E4500@2.20GHz；Internal memory：2GB (hundred million energy DDR2 667MHZ)；

Operating system：Windows 32 SP1 operating systems of 7 Ultimate；Simulation software：MATLAB R(2011b).

(1b) simulation parameter setting

Transmission signal adopts linear FM signal, transmission signal parameters and experiment simulation parameter setting as shown in table 1.

Table 1 transmission signal parameters and experiment simulation parameter

Parameter	Value
		Carrier frequency	fc=10GHz
The linear FM signal persistent period	Tr=5 μ s
		Linear FM signal modulating bandwidth	Br=37.5MHz
Carrier aircraft platform speed	V=150m/s
		Antenna equivalent aperture	D=4m
The vertical oblique distance of carrier aircraft and scene center	R0=20Km
		Pulse recurrence frequency	PRF=300Hz
Sample frequency	fs=225MHz

2. emulation content and result

Emulation 1, according to the simulation parameter of table 1, is 687 × 803 harbour with synthetic aperture radar to the size shown in Fig. 2 Scene, is detected and is obtained echo, reconstructs the image of Harbor scene with the inventive method, and result is as shown in Figure 3.Image table Bright：The present invention enables the high-resolution reconstruct to ground complexity large scene.

Emulation 2, according to the simulation parameter of table 1, emulation obtains the echo of different size terraced fields scene as shown in Figure 4, its Middle Fig. 4 (a) is 64 × 64 terraced fields scene, and Fig. 4 (b) is 128 × 128 terraced fields scene, and Fig. 4 (c) is 256 × 256 terraced fields Scene, Fig. 4 (d) is 512 × 512 terraced fields scene.

According to Fig. 4 (a), the echo of Fig. 4 (b), Fig. 4 (c), Fig. 4 (d), with the inventive method respectively to Fig. 4 (a), Fig. 4 (b), Fig. 4 (c), Fig. 4 (d) is reconstructed, and records corresponding reconstitution time loss, as shown in table 2.

Table 2 the inventive method reconstructs the time loss of different size terraced fields scene

Terraced fields scene	Fig. 4 (a)	Fig. 4 (b)	Fig. 4 (c)	Fig. 4 (d)
					The size of scene	64×64	128×128	256×256	512×512
Time loss (s)	0.126	0.369	1.592	14.739

Be can be seen that with respect to existing by the time loss that the inventive method of table 2 reconstructs different size terraced fields scene By the rarefaction representation algorithm of two-dimensional scene vectorization, the method for the present invention greatly accelerated the imaging time to scene it is achieved that Little scene realtime imaging, the purpose of large scene fast imaging, there is significant practicality.

Claims (4)

1. a kind of synthetic aperture radar image-forming method based on sparse representation theory, comprises the steps：

(1) according to radar earthward scene transmitting pulseObtain the echo-signal of ground scene

$y(t_a,\hat{t})=\underset{p=1}{\overset{P}{\Σ}}\underset{q=1}{\overset{Q}{\Σ}}{f}_{pq}{w}_a\left(t_a-\frac{x_p}{v_0}\right)\exp \left(-j\frac{2{\mathrm{\πv}}^2}{{\mathrm{\λR}}_0}{\left(t_a-\frac{x_p}{v_0}\right)}^2\right)w_r\left(\hat{t}-\frac{2R_q}{C}\right)\exp \left(j\mathrm{\π}\mathrm{\γ}{\left(\hat{t}-\frac{2R_q}{C}\right)}^2\right)\exp \left(-j\frac{4{\mathrm{\πf}}_c}{C}{R}_q\right)$

In formula,For fast time, t_aFor slow time, f_cFor carrier frequency, γ is chirp rate, v₀For carrier aircraft speed, C is electromagnetic wave propagation Speed, w_a() is orientation window function, w_r() is chirp pulse signal time window function, and P is ground scene in orientation Discrete grid block number, Q be ground scene distance to discrete grid block number, p be ground scene p-th of orientation from Scattered grid, q be ground scene distance to q-th discrete grid block, f_pqScattering system for scattering point at (p, q) individual grid Number, R₀For the vertical oblique distance of radar to ground scene center, R_qFor vertical oblique distance from radar to distance to q-th discrete grid block, x_p X-axis coordinate for p-th discrete grid block of orientation；

(2) to echo-signalCarry out two-dimensional discrete sampling, obtain following echo matrix Y：

Wherein, M is the exomonental number of orientation, and N is that in each pulse, distance is to sampling number；For echo letter Number (m, n) individual sampling instant sample value, its expression is as follows：

$y(t_{a,m},{\hat{t}}_n)=\underset{p=1}{\overset{P}{\Σ}}\underset{q=1}{\overset{Q}{\Σ}}{f}_{pq}{w}_a\left(t_{a,m}-\frac{x_p}{v_0}\right)\exp \left(-j\frac{2{\mathrm{\πv}}^2}{{\mathrm{\λR}}_0}{\left(t_{a,m}-\frac{x_p}{v_0}\right)}^2\right)w_r\left({\hat{t}}_n-\frac{2R_q}{C}\right)\exp \left(j\mathrm{\π}\mathrm{\γ}{\left({\hat{t}}_n-\frac{2R_q}{C}\right)}^2\right)\exp \left(-j\frac{4{\mathrm{\πR}}_q}{{\mathrm{\λ}}_c}\right)$

In formula, λ_c=C/f_cFor carrier wavelength, t_a,mFor m-th impulse ejection moment,For fast n-th sampling instant value of time；

(3) construct the orientation basic matrix R of ground scene：

Wherein,

(4) distance of construction ground scene is to basic matrix G：

Wherein,

(5) according to echo matrix Y, orientation basic matrix R, distance to basic matrix G and ground scene scattering coefficient matrix F, structure Build following matrix relationship formula：

Y=RFG+N,

In formula, N is the noise matrix with echo matrix Y with dimension, and the form of scattering coefficient matrix F is as follows：

Wherein, p=1,2 ..., P, q=1,2 ..., Q；

(6) the scattering coefficient matrix F according to ground scene, obtains intermediate variable matrix V, horizontal total variation matrix D_x, vertically entirely become Difference matrix D_y：

V=F

D_x=D_hF,

D_y=FD_v

In formula, D_hFor horizontal total variation operator matrix, D_vFor vertical total variation operator matrix；

(7) parameter that the matrix relationship formula of step (5) and step (6) obtain, the majorized function f under construction total variation framework are utilized (F,V,D_x,D_y)：

$\underset{F,V,D_x,D_y}{\min }f(F,V,D_x,D_y)=\left\{\left|\right|D_x|{|}_1+|\left|D_y\right|{|}_1+\frac{\mathrm{\λ}}{2}\left|\right|D_{h}V-D_x|{|}_F^2+\frac{\mathrm{\λ}}{2}||{\mathrm{VD}}_v-D_y|{|}_F^2+\frac{v}{2}\left|\right|V-F|{|}_F^2+\frac{\mathrm{\μ}}{2}||RFG-Y|{|}_F^2\right\},$

WhereinRepresent the operator finding a function minima, λ is total variation penalty factor, ν is scene scatters coefficient penalty factor, μ For echo penalty factor, | | | |_FRepresent the Frobenius norm seeking matrix, | | | |₁Represent that each element to matrix takes It is added after absolute value again；

(8) use Split-Bregman method to majorized function f (F, V, the D in step (7)_x,D_y) be iterated solving, obtain ground The final iteration result F of face scene scatters coefficient matrix F^*：

(8a) according to majorized function f (F, V, D_x,D_y), obtain majorized function L (F, V, the D of Split-Bregman form_x,D_y)：

In formula, B_xFor horizontal total variation matrix auxiliary variable, B_yFor vertical total variation matrix auxiliary variable, W is scattering coefficient matrix Auxiliary variable；

(8b) iterative steps k=0, scattering coefficient matrix F after kth time iteration are initialized^kFor all 1's matrix, centre after kth time iteration Matrix of variables V^kFor all 1's matrix, horizontal total variation matrix after kth time iterationFor all 1's matrix, vertically entirely become after kth time iteration Difference matrixFor all 1's matrix；Scattering coefficient matrix auxiliary variable W after kth time iteration^kFor full 0 matrix, level after kth time iteration Total variation matrix auxiliary variableFor full 0 matrix, vertical total variation matrix auxiliary variable after kth time iterationFor full 0 matrix； Setting total variation penalty factor λ>0, scene scatters coefficient penalty factor ν>0, echo penalty factor μ>0, stopping criterion for iteration ε=5 × 10^-4；

(8c) obtain scattering coefficient matrix F after+1 iteration of kth^k+1；

(8d) obtain intermediate variable matrix V after+1 iteration of kth^k+1；

(8e) according to intermediate variable matrix V after+1 iteration of kth^k+1, horizontal total variation matrix auxiliary variable after kth time iteration Vertical total variation matrix auxiliary variable after kth time iterationObtain horizontal total variation square after+1 iteration of kth according to equation below Battle arrayVertical total variation matrix after+1 iteration of kth

$\begin{array}{c}{\left(D_x^{k+1}\right)}_{p,q}=sign\left({\left(D_h{V}^{k+1}+B_x^k\right)}_{p,q}\right)\max \left(\left|{\left(D_h{V}^{k+1}+B_x^k\right)}_{p,q}\right|-\frac{1}{\mathrm{\λ}},0\right)\\ {\left(D_y^{k+1}\right)}_{p,q}=sign\left({\left(V^{k+1}{D}_v+B_y^k\right)}_{p,q}\right)\max \left(\left|{\left(V^{k+1}{D}_v+B_y^k\right)}_{p,q}\right|-\frac{1}{\mathrm{\λ}},0\right)\end{array},$

In formula, p=1,2 ..., P, q=1,2 ..., Q, D_hFor horizontal total variation operator matrix, D_vFor vertical total variation Operator Moment Battle array, sign () is sign function, and λ is total variation penalty factor；

(8f) according to intermediate variable matrix V after+1 iteration of kth^k+1, scattering coefficient matrix F after+1 iteration of kth^k+1, kth+1 time Horizontal total variation matrix after iterationVertical total variation matrix after+1 iteration of kthHorizontal total variation after kth time iteration Matrix auxiliary variableVertical total variation matrix auxiliary variable after kth time iterationAfter kth time iteration, scattering coefficient matrix is auxiliary Help variable W^k, obtain scattering coefficient matrix auxiliary variable W after+1 iteration of kth according to equation below^k+1, water after+1 iteration of kth Flat total variation matrix auxiliary variableVertical total variation matrix auxiliary variable after+1 iteration of kth

$\begin{array}{c}{W}^{k+1}=W^k+V^{k+1}-F^{k+1}\\ B_x^{k+1}=B_x^k+D_h{V}^{k+1}-D_x^{k+1}\\ B_y^{k+1}=B_y^k+V^{k+1}{D}_v-D_y^{k+1}\end{array};$

(8g) according to scattering coefficient matrix F after+1 iteration of kth^k+1, scattering coefficient matrix F after kth time iteration^k, as follows Obtain mean square error：In formula, | | | |_FRepresent the Frobenius norm seeking matrix；

(8h) judge whether mean square error M≤ε sets up, if setting up execution step (8i)；Otherwise make k=k+1, return to step (8c) Continue iteration to run, wherein ε is stopping criterion for iteration；

(8i) make F^*=F^k+1, obtain the final iteration result F of ground scene scattering coefficient matrix F^*；

(9) step (8) is obtained with the final iteration result F of ground scene scattering coefficient matrix F^*Modulus of access, output obtains ground field The image of scape.

2. the synthetic aperture radar image-forming method based on sparse representation theory according to claim 1, wherein said step (1) radar earthward scene transmitting pulseIt is expressed as follows：

$s(t_a,\hat{t})=w_r\left(\hat{t}\right)\exp \left(j\mathrm{\π}\mathrm{\γ}{\hat{t}}^2\right)\exp \left(j2{\mathrm{\πf}}_{c}t\right),$

In formula, f_cFor carrier frequency, γ is chirp rate, and t is full-time,For fast time, t_aFor slow time, the relation of these three times For Represent the time window function of chirp pulse signal, T_rIt is the time of pulse persistance.

3. the synthetic aperture radar image-forming method based on sparse representation theory according to claim 1, wherein said step (8c) obtain scattering coefficient matrix F after+1 iteration of kth in^k+1, carry out in accordance with the following steps：

(8c1) according to orientation basic matrix R, distance, to basic matrix G, obtains orientation symmetrical matrix Ψ, distance is to symmetrical matrix Φ：

$\begin{array}{c}\mathrm{\Ψ}=R^{T}R\\ \mathrm{\Φ}={\mathrm{GG}}^T\end{array},$

Wherein, ()^TRepresenting matrix transposition；

(8c2) according to feature decomposition orientation symmetrical matrix Ψ, distance, to symmetrical matrix Φ, obtains orientation orthogonal matrix U_R, side Position is to diagonal matrix L_R, distance is to orthogonal matrix U_G, distance is to diagonal matrix L_G；

(8c3) according to echo matrix Y, orientation basic matrix R, distance is to basic matrix G, intermediate variable matrix V after kth time iteration^k, Scattering coefficient matrix auxiliary variable W after kth time iteration^k, orientation orthogonal matrix U_R, distance is to orthogonal matrix U_G, orientation is diagonal Matrix L_R, distance is to diagonal matrix L_G, obtain scattering coefficient companion matrix after+1 iteration of kth according to equation below

In formula, ν is scene scatters coefficient penalty factor, p=1,2 ..., P, q=1,2 ..., Q；

(8c4) according to orientation orthogonal matrix U_R, distance is to orthogonal matrix U_G, scattering coefficient companion matrix after+1 iteration of kthObtain scattering coefficient matrix F after+1 iteration of kth according to equation below^k+1：

4. the synthetic aperture radar image-forming method based on sparse representation theory according to claim 1, wherein said step (8d) obtain intermediate variable matrix V after+1 iteration of kth in^k+1, carry out in accordance with the following steps：

(8d1) according to horizontal total variation operator matrix D_h, vertical total variation operator matrix D_v, obtain horizontal total variation matrix E, hang down Straight total variation matrix Z：

$\begin{array}{c}E=D_h^T{D}_h\\ Z=D_v{D}_v^T\end{array},$

Wherein, ()^TRepresenting matrix transposition；

(8d2) according to feature decomposition horizontal total variation matrix E, vertical total variation matrix Z, obtain horizontal total variation orthogonal matrix U_x, Horizontal total variation diagonal matrix L_x, vertical total variation orthogonal matrix U_y, vertical total variation diagonal matrix L_y；

(8d3) according to scattering coefficient matrix F after+1 iteration of kth^k+1, horizontal total variation operator matrix D_h, vertical total variation operator Matrix D_v, scattering coefficient matrix auxiliary variable W after kth time iteration^k, horizontal total variation matrix auxiliary variable after kth time iteration Vertical total variation matrix auxiliary variable after kth time iterationHorizontal total variation matrix after kth time iterationAfter kth time iteration Vertical total variation matrixHorizontal total variation orthogonal matrix U_x, vertical total variation orthogonal matrix U_y, horizontal total variation diagonal matrix L_x, vertical total variation diagonal matrix L_y, obtain intermediate variable companion matrix after+1 iteration of kth according to equation below

In formula, λ is total variation penalty factor, and ν is scene scatters coefficient penalty factor, p=1,2 ..., P, q=1,2 ..., Q；

(8d4) according to horizontal total variation orthogonal matrix U_x, vertical total variation orthogonal matrix U_y, after+1 iteration of kth, intermediate variable is auxiliary Help matrixObtain intermediate variable matrix V after+1 iteration of kth according to equation below^k+1：