Special Section on Onboard Compression and Processing for Space Data Systems

Imaging method for highly squinted synthetic aperture radar with under-sampled echo data

[+] Author Affiliations
Fu-fei Gu, Yong-an Chen

Air Force Engineering University, Institute of Information and Navigation, No. 5 Department, Xi’an 710077, China

Qun Zhang

Air Force Engineering University, Institute of Information and Navigation, No. 5 Department, Xi’an 710077, China

Fudan University, Key Laboratory for Information Science of Electromagnetic Waves (Ministry of Education), Shanghai 200433, China

Collaborative Innovation Center of Information Sensing and Understanding, Xi’an 710077, China

Ying Luo

Collaborative Innovation Center of Information Sensing and Understanding, Xi’an 710077, China

Xidian University, The National Laboratory of Radar Signal Processing, Xi’an 710071, China

J. Appl. Remote Sens. 9(1), 097495 (Jun 19, 2015). doi:10.1117/1.JRS.9.097495
History: Received November 9, 2014; Accepted May 15, 2015
Text Size: A A A

Open Access Open Access

Abstract.  The amount of echo data is very large in highly squinted synthetic aperture radar (SAR) imaging with high resolution. To solve this problem, an imaging method for highly squinted SAR with under-sampled echo data based on compressed sensing (CS) is put forward. First, the echo signal model of highly squinted SAR is analyzed and a nonlinear chirp scaling (NCS) imaging method with the Nyquist-sampled echo data is proposed, with which the range walk is corrected and the range-azimuth coupling is mitigated. Based on the imaging method, the NCS operator is established. Combining the NCS operator and CS theory, a highly squinted SAR imaging scheme is formulated. The modified iterative thresholding algorithm is utilized to solve the imaging scheme, which forms a highly squinted SAR imaging method. With the proposed method, just a small amount of imaging data is required for highly squinted SAR imaging. Finally, the effectiveness of the proposed method is proven by the simulations.

Figures in this Article

Synthetic aperture radar (SAR) can realize imaging for ground targets all day and under all weather conditions, which can achieve a high resolution in the range direction due to the use of high transmitted-pulse bandwidth and in the azimuth direction due to the storage of data over a certain observation time.1,2 In the conventional SAR imaging mode, the pointing direction of the antenna is nearly perpendicular to the flight path. However, on some occasions, in order to point at an angle from the broadside, squint mode is adopted to increase the flexibility of SAR.3,4 Compared to the broadside SAR, squint SAR can give repeat observations on the same region during its whole flight track.5 With this prominent advantage, the highly squinted SAR is becoming the research focus.36 The conventional imaging method for highly squinted SAR is based on the matched filter (MF) algorithm, which needs Nyquist samples of the echoes. However, along with the improvement of range resolution, the SAR imaging system requires increasing measurements, storage, and downlink bandwidth.7 Hence, we intend to find an imaging method for highly squinted SAR, which can not only guarantee the quality of the imaging results but also dramatically reduce the amount of required echo data.

In the last few years, the compressed sensing (CS) theory was introduced in Refs. 8 and 9, which indicates that one can stably and accurately reconstruct nearly sparse signals from dramatically under-sampled data in an incoherent domain. With this prominent advantage, the CS theory has been used in SAR data processing to reduce the data amount and enhance the system performance.1017 A compressive radar imaging scheme based on the CS theory was first reported by Baraniuk and Steeghs,11 with which the pulse compression MF is no longer needed. In 12, the SAR raw data are compressed and the imaging result is reconstructed by using the CS framework with real wavelets. References 13 and 14 convert the two-dimensional (2-D) SAR imaging problem into a subsignal collection problem, which could dramatically reduce the amount of raw data. However, high computation and memory costs are required to decompose and recover the 2-D subsignals. The CS processing is performed after the conventional range compression in Refs. 15 and 16, which could acquire a high-resolution azimuth profile. In 17, a 2-D sparse SAR imaging scheme with stepped-frequency waveform is put forward. The advantage of this method is that only a small number of frequencies and echo data are needed to reconstruct the image of targets. However, all of these works are suitable only for broadside SAR. There are few reference works devoted specifically to highly squinted SAR imaging combining with CS theory. Due to the special imaging geometry, the highly squinted SAR brings new complexities and challenges to the imaging process. Thus, the aforementioned CS-SAR imaging methods cannot be utilized in highly squinted SAR imaging with the under-sampled echo data.

In this paper, an imaging method for highly squinted SAR based on the CS theory is put forward. First, an imaging method for highly squinted SAR with the Nyquist-sampled data is proposed. The range walk removal function is constructed to correct the linear range walk and the nonlinear chirp scaling (NCS) method is applied to mitigate the range-azimuth coupling. Second, in order to achieve the highly squinted SAR imaging with the under-sampled echo data, the NCS operator and CS imaging scheme are formulated based on the aforementioned imaging method. And then, the imaging result is obtained by using the modified iterative thresholding algorithm (ITA). The main contributions of the present work can be concluded as follows:

  1. In order to facilitate the analysis of imaging method with the under-sampled echo data, a new NCS imaging method with Nyquist-sampled echo data is put forward, with which the range walk can be corrected and the range-azimuth coupling can be mitigated.
  2. The NCS operator and CS imaging scheme based on the Nyquist-sampled imaging method are established. The modified ITA is utilized to solve the imaging scheme, which forms a highly squinted SAR imaging method. By using the method, just a small amount of imaging data is required for highly squinted SAR imaging.

The rest of the paper is organized as follows. Section 2 gives the signal and geometry model of the highly squinted SAR. An NCS imaging method based on range walk removal is also introduced in this section. In Sec. 3, the NCS operator and CS imaging scheme are constructed. Moreover, the modified ITA is utilized to solve the imaging scheme. Simulation results are presented in Sec. 4 to validate the effectiveness of proposed approach. Finally, we make some conclusions in Sec. 5.

There are some papers that have put forward highly squinted SAR imaging methods with Nyquist-sampled data.3,4 However, it is difficult to obtain the highly squinted imaging operator from these methods. The imaging operator is the key factor in CS SAR imaging method. In order to facilitate the construction of the highly squinted imaging operator, we first focus on establishment of an imaging approach for highly squinted SAR with Nyquist-sampled data in this section.

Echo Signal Model for Highly Squinted Synthetic Aperture Radar

The squinted SAR geometry is shown in Fig. 1. An SAR sensor travels along a straight-line flight path during a synthetic aperture length L. The velocity and height of platform are v and h, respectively. During the data acquisition, the radar beam directs at the target with a squinted angle θs. Symbol R0 denotes the vertical distance between the scene center and the flight path.

Graphic Jump Location
Fig. 1
F1 :

Imaging model of highly squinted synthetic aperture radar (SAR): (a) flow chart of highly squinted SAR; (b) geometric model of flight path and point scatterer.

The geometric model of flight path and point scatterer is shown in Fig. 1(b). The distance between the point scatterer P and the flight path is Rp. During the data acquisition, the transmitted chirp signal is Display Formula

s(t^,tm)=rect(t^Tp)exp[j2πf0t+jπγt^2],(1)
where rect(t^/Tp)=1 is the rectangular window with a width of Tp, Tp is the pulse repetition interval. Term t^ denotes range fast time, t=t^+tm, and tm is the azimuth slow time. Term f0 is the carrier frequency, and γ is the chirp rate. After downconversion to the baseband, the received signal from the point scatterer P is given by Display Formula
sr(t^,tm;Rp)=rect[t^2R(tm;Rp)/cTp]exp{jπγ[t^2R(tm;Rp)c]2}exp[j4πR(tm;Rp)λ],(2)
where R(tm;Rp) is the instantaneous range from the point scatterer P to radar, and it can be expressed as Display Formula
R(tm;Rp)=Rp2+(vtmXp)22Rp(vtmXp)sinθs.(3)

Expanding the distance R(tm;Rp) in Taylor’s series and neglecting the higher-order terms, R(tm;Rp) can then be expressed approximately as follows: Display Formula

R(tm;Rp)=Rp(vtmXp)sinθs+(vtmXp)2cos2θs2Rp=Rp2+cos2θs(vtmXp)2(vtmXp)sinθs.(4)

Nonlinear Chirp Scaling Imaging Method Based on Range Walk Removal

Applying Fourier transform (FT) with respect to t^, the signal sr(t^,tm;Rp) is transferred into range Doppler domain (i.e., range time and azimuth frequency domain), it yields Display Formula

Sr(fr,tm;Rp)=exp(jπfr2γ)exp[j4πR(tm;Rp)(fr+f0)c].(5)

Submitting Eq. (4) into Eq. (5), we can obtain Display Formula

Sr(fr,tm;Rp)=exp[j4πXpsinθs(fr+f0)c]exp[j4πvtmsinθs(fr+f0)c]exp(jπfr2γ)exp[j4πRp2+cos2θs(vtmXp)(fr+f0)c].(6)

As can be seen from Eq. (6), the second exponential term of Sr(fr,tm;Rp) is the linear range walk term. Therefore, the range walk removal function H1(fr,tm) can be written as Display Formula

H1(fr,tm)=exp[j4πvtmsinθs(fr+f0)c].(7)

To obtain the 2-D spectrum, we apply FT in azimuth to the aforementioned compensated signal, i.e., Display Formula

S(fr,fa;Rp)=exp(jπfr2γ)exp[j4π(Xpsinθs)(fr+f0)c]exp(j2πfaXpv)exp[jΞ(fr,fa;Rp)],(8)
where Ξ(fr,fa;Rp)=(4πRpf0/c)(1+fr/fa)2(faλ/2vcosθs)2. We can find that Ξ(fr,fa;Rp) is the coupling term of the range frequency fr and azimuth Doppler fa. It has the characteristic of spatial variance. In order to obtain the focused imaging result, Ξ(fr,fa;Rp) should be accurately compensated. Expanding it in Taylor’s series, we can obtain Display Formula
Ξ(fr,fa;Rp)=ϕ0(fa)+ϕ1(fa)fr+ϕ2(fa)fr2+ϕ3(fa)fr3+o(fa,fr4),(9)
where Display Formula
ϕ0(fa)=4πRpf0cB(fa);ϕ1(fa)=4πRpcB(fa);ϕ2(fa)=2πRpcfc1B2(fa)B3(fa);
Display Formula
ϕ3(fa)=2πRpcfc21B2(fa)B5(fa);B(fa)=1(faλ2vcosθs)2.

The chirp rate γe(fa,Rp) of S(fr,fa;Rp) is equivalent to the following expression: Display Formula

1γe(fa,Rp)=1γ+2Rpcfc1B2(fa)B3(fa).(10)

Inspecting Eq. (10), γe(fa,Rp) is dependent on the Doppler frequency fa and the distance Rp, which means the signal S(fr,fa;Rp) has the characteristic of spatial variance. However, the spatial variance is not considered in the conventional chirp scaling algorithm (CSA). Thus, if using the conventional CSA, the imaging edge would be defocused. The NCS algorithm has considered the spatial variance. Therefore, we adopt it for the following imaging processing.

Before the NCS processing, the third-order term of S(fr,fa;Rp) should be filtered. According to the above analysis, the filtered function H2(fr) is given by Display Formula

H2(fr)=exp[jϕ3(fa)fr3].(11)

Multiplying Eq. (11) with Eq. (8) and transforming the result into the range Doppler domain by the range inverse FT (IFT), we can obtain Display Formula

s(t^,fa;Rp)=exp(j4πXpsinθsf0c)exp(j2πfaXpv)exp[j4πRpλB(fa)]exp{jπγe(fa,Rp)[t^2Xpsinθsc2Rpc1B(fa)]2}.(12)

According to the NCS algorithm, the NCS operation function can be written as Display Formula

H3(t^,fa;R0)=exp{jπq2(fa)[t^τd(fa,R0)]2},(13)
where q2(fa) is the modulation rate of H3(t^,fa;R0), which is abbreviated as q2. Multiplying Eq. (13) with Eq. (12) and applying the principle of stationary phase to the result, we can obtain the result in the 2-D frequency domain as Display Formula
S(fr,fa;Rp)=exp(j4πXpsinθsf0c)exp(j2πfaXpv)exp[j4πR0λB(fa)]exp[jZ(fr,fa,Δτ)],(14)
where Δτ=τd(fa,R0)τd(fa,Rp). There are four exponential terms in Eq. (14). The first exponential term determines the range displacement, which is dependent on target azimuth position and squint angle; the second term determines the azimuth position of the target; the third term is the azimuth-independent phase modulation term; the fourth exponential term is the cross-coupling term. Expanding the fourth exponential term into a power series of fr, we can obtain Display Formula
Z(fr,fa,Δτ)=C(q2,fa,fr,fr2)+D(q2,fa)Δτfr.(15)

The first term of Eq. (15) is independent from the range distance, and the second term is the main factor that causes linear shift in the range direction. In order to eliminate the geometric distortion induced by the spatial variance, we set Display Formula

D(q2,fa)=1α(fa),α(fa)=1B(fa).(16)

Solving Eq. (16), the modulation rate q2 can be written as Display Formula

q2=γe(fa,Rp)[α(fa)1].(17)

Submitting Eq. (17) into Eq. (14), we can obtain Display Formula

S(fr,fa;Rp)=exp(j4πXpsinθsf0c)exp(j2πfaXpv)exp[j4πR0λB(fa)]exp[j2πτd(fa,Rp)fr]exp[jπα(fa)γe(fa,Rp)fr2]exp[jM(fa,Rp)],(18)
where Display Formula
{M(fa,Rp)=4πγe(fa,Rp)a(fa)[a(fa)1]Δτ2τd(fa,R0)=2{Xpsinθs+Rp+[α(fa)1]R0}c.

From Eq. (18), the range compression function and range cell migration correction (RCMC) function can be given by Display Formula

H4(fr,fa;Rp)=exp{j4π[a(fa)1]R0cfr}exp[jπfr2α(fa)γe(fa;Rp)].(19)

Multiplying Eq. (19) with Eq. (18), we can complete range compression and RCMC. In order to implement the azimuth processing, we transform the result into the range Doppler domain by range IFT. The azimuth compression and residual phase compensated functions H5(fa,Δτ;Rp) are written as Display Formula

H5(fa,Δτ;Rp)=exp[j4πR0λB(fa)]exp[jM(fa,Rp)].(20)

Then azimuth IFT is applied to the aforementioned result back into the slow-time domain. The imaging result of highly squinted SAR with the Nyquist samples is obtained. The focused imaging result is with some degree of geometric distortion in range and azimuth directions. The detailed geometric correction method is not discussed in this paper, which can be found in 4.

The imaging method for highly squinted SAR with Nyquist-sampled data is proposed in Sec. 2.2. However, the amount of echo data is huge with high-resolution imaging. Therefore, in order to reduce the amount of echo data, the sampled ratio is set lower than that of the Nyquist requires. A sensing method for direct sampling and compressing analog signals is analog-to-information conversion (AIC).10 We adopt the AIC framework with random under-sampling scheme in range direction. The essence of this scheme is to nonuniformly under-sample the echo signal. As for the under-sampled echo data, the aforementioned proposed method with Nyquist-sampled data becomes invalid. An imaging algorithm should be proposed.

The superiority of CS lies in that it merges sensing and compressing together, and a small number of “random” measurements can carry enough information to reconstruct the original signal. Thus, the CS theory is introduced to the highly squinted SAR imaging to reduce the amount of echo data. The CS imaging method is put forward with three sequenced steps: First, the NCS operator is constructed based on the imaging method proposed in Sec. 2.2. Second, CS imaging scheme based on the NCS operator is established in details. Third, the modified ITA is utilized to solve the imaging scheme. The relationship between the imaging method with Nyquist samples and with under samples is shown in Fig. 2.

Graphic Jump Location
Fig. 2
F2 :

Relationship between imaging method with Nyquist samples and with under-samples.

Compressed Sensing Model for Highly Squinted Synthetic Aperture Radar

In order to facilitate the following analysis, the NCS imaging method based on range walk removal in Sec. 2.2 can be expressed as follows: Display Formula

Θ=Γ(sr)=ωa·{[({[({ω^a·[(sr·ωr)·H1]}·H2)·ω^r]·H3}·ωr)·H4]·ω^r·H5},(21)
where (·) denotes the Hadamard product. Terms ω and ω^, respectively, are the discrete FT (DFT) matrix and inverse DFT matrix (in practice, they are both implemented by FFT) to perform, the subscripts a and r denote the direction of azimuth and range along which the FFT performs. Term sr is the echo signal matrix, the size of which is Na×Nr. Term Na is the sampled number in azimuth direction and Nr is the number of range cells. Term Θ is the imaging result for highly squinted SAR. Term Γ(·) is the NCS operator, which is the key factor in CS imaging model.

Equation (21) is a reversible process. If we input the complex image data, the echo signal can be obtained by implementing the inverse process of Eq. (21). The inverse process can be expressed as Display Formula

sr=Γ1(Θ)={ωa·[({[({[(ω^a·Θ)·H5*]·ωr}·H4*)·ω^r]·H3*}·ωr)·H2*]}·H1*·ω^r,(22)
where (·)* denotes conjugation. From the CS theory point of view, the under-sampled signal can be viewed as a low-dimensional measurement of Nyquist-sampled signal. The low-dimensional measurement matrix Φ={ϕc,d} is a Nr-by-N^r random partial unit matrix,18 where N^r is the under-sampled number in range direction, and Display Formula
ϕc,d={1,c=md0,othersd=1,,N^r;md1,,Nr;.(23)

The elements in each row vector of Φ are 0, other than the mdth element, where md is the random number. Thus, we can obtain Display Formula

srcom=sr·Φ.(24)

From Eq. (24), we can find that the size of under-sampled signal srcom is Na×N^r. The under-sampled ratio of echo signal is defined by η=N^r/Nr. Submitting Eq. (24) into Eq. (22), we can obtain Display Formula

srcom=Γ1(Θ)·Φ.(25)

When Γ1(·)·Φ satisfies the restricted isometry property (RIP), the imaging result Θ can be obtained by solving the following optimization problem: Display Formula

minΘ0subject tosrcom=Γ1(Θ)·Φ,(26)
where ·0 denotes l0 norm and min(·) denotes the minimization. That Γ1(·)·Φ satisfies the RIP is proven in detail in the 1.

Modified Iterative Thresholding Algorithm

How to solve the optimization problem of Eq. (26) is an important aspect in CS theory. Two majors are usually called “L1-minimization” and “greedy pursuit” and the others are the “nonconvex optimization” and the “Bayesian framework.”19 Basis pursuit is one kind of the L1-minimization algorithm, which is based on the interior point.20 ITA is another kind of the L1-minimization algorithm, which is known as a first-order algorithm.21 By the ITA, Eq. (26) can be transferred into the following expression: Display Formula

minΘ{12srcomΓ1(Θ)·Φ22+λΘ1},(27)
where λ is a regularization parameter, which is utilized to balance the precision and sparsity of the reconstructed result. Equation (27) can be effectively solved by ITA.21 The conventional ITA generates a sequence of approximates according to Display Formula
Θ^k=soft[Θ^k1+μAH(srcomA·Θ^k1),λ],(28)
where soft(x,λ)=sign(x)·max(|x|λ,0), A is the measurement matrix, and (·)H denotes the conjugate transposition. Term μ controls the convergence of the ITA. However, A of Eq. (28) cannot be obtained. Thus, Eq. (28) should be modified to the following expression: Display Formula
Θ^k=soft{Θ^k1+μ·Γ[srcomΓ1(Θ^k1)·Φ]ΦH,λ}.(29)

There are two parameters λ and μ that need to be set. According to Refs. 7, 22, and 23, λ and μ can be set as Display Formula

μk=Γ1(ΔΘ^k)·Φ22ΔΘ^k22,λk=|Θ^k+uk·ΔΘ^k|Iuk,(30)
where |·|I is its I’th largest component in magnitude. The detailed steps of the modified ITA can be outlined as follows:

  • Under-sampled echo data srcom, NCS imaging operator Γ(·) and inverse imaging operator Γ1(·), low-dimensional measurement matrix Φ.
  • The imaging result Θ of highly squinted SAR.
  • Initialization: Θ0=0, the residual error p0=srcom, the maximum iteration Kmax;
  • For k=0 to Kmax do the following steps;
  • Matched filter on residual error: ΔΘ^k=Γ(pk·ΦH);
  • Compute the current estimation Θ^k+1: Θ^k+1=soft(Θ^k+uk·ΔΘ^k,λ);
  • Update the residual error: pk+1=srcomΓ1(Θ^k+1)·Φ;
  • Check the stopping criterion: if |Θ^k+1Θ^k|2/|Θ^k|2>ρ, then set k=k+1 and go to Step 3; otherwise end.

Then Θ^(k) can be obtained, which is the imaging result for highly squinted SAR with under-sampled echo data.

In this section, some simulations are conducted to demonstrate the effectiveness and feasibility of the proposed method. The experiments with simulated data using airborne SAR parameters shown in Table 1 are carried out.

Table Grahic Jump Location
Table 1Synthetic aperture radar parameters.

The simulation uses an array of three targets, which are located in a 5km×5km grid in the slant range plane in azimuth/range, as shown in Fig. 3. Three targets have the same azimuth position and a distance of 2 km in the range direction. In the imaging methods, target P2 located at the center of observed scene is selected as the reference target.

Graphic Jump Location
Fig. 3
F3 :

Flight geometry and target distribution in the slant plane for the simulation.

First, the simulated echo data are sampled with the Nyquist theory. The proposed imaging algorithm in Sec. 2.2 is applied to process the simulated data. Figures 4(a) and 4(b) show the images of three targets when considering and not considering the spatial variance. It can be noted that the imaging result of target P2 is not affected by the spatial variance, which is focused well with both two methods. However, when not considering the spatial variance, the imaging results of targets P1 and P3 cannot be focused as shown in Fig. 4(b). By using the proposed method in Sec. 2.2, the targets P1 and P3 are focused quite well as shown in Fig. 4(a). Therefore, the proposed method in Sec. 2.2 is valid and feasible.

Graphic Jump Location
Fig. 4
F4 :

Imaging results under different situations: (a) imaging result with spatial variance considered; (b) imaging result with spatial variance not considered.

Next, in order to validate the proposed method in Sec. 3, the echo data are under-sampled by the random scheme. Let the under-sampled ratios η be 1/2 and 1/3. The reconstructed imaging results by using the CS imaging method are presented in Fig. 5. Figures 5(a) and 5(b) are the imaging results when η is 1/2 and 1/3, respectively. Comparing with Fig. 4(a), we can find that the sidelobes of Fig. 5 are lower and the quality of the resultant image is good enough. Therefore, by using the CS imaging method in Sec. 3, just a small amount of imaging data is required for highly squinted SAR imaging.

Graphic Jump Location
Fig. 5
F5 :

Results of compressed sensing (CS) imaging method with different under-sampled ratios: (a) imaging result with η=1/2; (b) imaging result with η=1/3.

In addition, to further evaluate the performance of the proposed algorithm, the measured parameters peak sidelobe ratios (PSLRs) of target P2 in range and azimuth directions are calculated. PSLRs of target P2 in Fig. 4(a) are 13.21 and 13.01dB, respectively, which are nearly the theoretical values. However, PSLRs of target P2 in Fig. 5(a) are 16.12 and 15.09dB, respectively, which are much smaller than that of Fig. 4(a). It means that the quality of CS imaging method is better than that of imaging method proposed in Sec. 2.2.

In order to further validate our method, the simulated scene data are utilized in the following analysis. The simulation parameters are set the same as that of the above point target simulation. The observed scene is shown in Fig. 6. Figure 7(a) shows the imaging result without correcting geometric distortion by Nyquist-sampled imaging method, while Fig. 7(b) is the imaging result with geometric distortion corrected. Let the under-sampled ratio η be 3/4. The respective imaging results obtained by Nyquist-sampled imaging method and the proposed CS method are shown in Fig. 7(c) and 7(d). We can find that the Nyquist-sampled imaging method can only obtain image when samples are fully adopted. While η is 3/4, the imaging quality is poor. However, the proposed CS method can perfectly recover the image, with a much reduced sidelobe. Figures 7(e) and 7(f) show the imaging results by the proposed CS imaging method with η=1/2 and 1/4, respectively. When η is 1/2, the CS method can recover the image. However, when the under-sampled ratio is 1/4, the quality of the image is not very good; only the main information of observed scene can be obtained. It is easy to be comprehended and also according with the CS reconstructing theory. Therefore, the proposed method is valid and feasible.

Graphic Jump Location
Fig. 7
F7 :

Imaging results of area targets with different methods and different under-sampled ratios: (a) imaging result of Nyquist samples with geometric distortion not corrected; (b) imaging result of Nyquist samples with geometric distortion corrected; (c) imaging result by Nyquist samples imaging method with η=1/2; (d-f) the imaging results by the proposed compressed sensing method with η=3/4, 1/2, and 1/4, respectively.

Next, the peak signal-to-noise ratio (PSNR) and CPU times are utilized to compare the quality and efficiency of the proposed methods. The experiment is run in MATLAB 2010a on a computer with an Intel Pentium 3.2 GHz Dual-Core processor and 3GB memory. The comparison results are shown in Table 2. As can be observed from the Table, PSNR of η=3/4 is higher than that of η=1/2. It means that the quality of η=3/4 is better, which is accordingly with the imaging results shown in Figs. 7(d) and 7(e). The CPU time comparison result shows that the Nyquist-sampled method is faster than the CS imaging method. However, the CPU time of the two methods is at the same level of magnitude. And even considering the multiple iterations, the CPU time will still not exceed two orders of magnitude.

Table Grahic Jump Location
Table 2Comparison of peak signal-to-noise ratio (PSNR) and CPU time.

In this paper, an imaging method for highly squinted SAR with under-sampled echo data based on CS theory is proposed, which can solve the problem of the huge data amount in highly squinted SAR system. At first, the imaging method with the Nyquist-sampled echo data is proposed. Based on the method, the CS imaging method with the under-sampled echo data is then put forward. By using the proposed method, the highly squinted SAR imaging result can be obtained with a small amount of echo data.

However, the sampling rate of the proposed method is mainly determined by the sparsity of observed scene, which cannot be strictly defined. Even in some specific conditions, there are still many small targets in the background. In practice, we assume them to be zeros. As a result, the neglected small scatterers may affect the understanding of the imaging result. In order to obtain the small scatterers, the sampling rate of the proposed method cannot be reduced, which should be 100% Nyquist sampling rate. How to use our method for the nonsparse scene with under-sampled echo data is our current research.

From Eq. (22), Γ1(Θ) can be explicitly expressed by Display Formula

Γ1(Θ)={ωa·[({[({[(ωa·Θ)·H5*]·ωr}·H4*)·ωr]·H3*}·ωr)·H2*]}·H1*·ωr.(31)

Let θ denotes the vector form of Θ, namely, θ=vec(Θ). According to 7, Γ1(Θ) can be written as matrices, and we have Display Formula

vec[Γ1(X)]=G·x=ω^r·H^1*·ω^a·H^2*·ω^r·H^3*·ω^r·H4*·ω^r·H^5*·ω^a·x,(32)
where Display Formula
ω^a=INrωa,ω^r=ωrINaH^1*=diag[vec(H1*)];H^2*=diag[vec(H2*)];H^3*=diag[vec(H3*)];H^4*=diag[vec(H4*)];H^5*=diag[vec(H5*)].

Thus, srcom=Γ1(Θ)·Φ can be written as Display Formula

S=Φ^·G·θ,(33)
where S=vec(srcom), Φ^=ΦHIna. To check whether Γ1(·)·Φ satisfies the RIP [Eq. (26)] means to check whether Φ^·G obeys the RIP. As discussed in 24, the randomly selected partial orthogonal matrix satisfies RIP. In this paper, the measurement matrix Φ^ is a random partial unit matrix. Therefore, if the imaging operator G is an orthogonal matrix, Φ^·G obeys the RIP.

The imaging operator G is an orthogonal matrix. ω^a and ω^a are the DFT matrix and IDFT matrix, which are orthogonal matrices. Thus, we have Display Formula

ω^a·(ω^a)H=Iω^a·(ω^a)H=I.(34)

Meanwhile, we also can obtain Display Formula

ω^r·(ω^r)H=Iω^r·(ω^r)H=I.(35)

H^1*, H^2*, H^3*, H^4*, and H^5* are the diagonal plural matrices, respectively. Thus, we can obtain Display Formula

H^1*·(H^1*)H=[σ1000σ2000σNrNa],H^2*·(H^2*)H=[ε1000ε2000εNrNa],(36)
Display Formula
H^3*·(H^3*)H=[ς1000ς2000ςNrNa],H^4*·(H^4*)H=[ξ1000ξ2000ξNrNa],(37)
Display Formula
H^5*·(H^5*)H=[ζ1000ζ2000ζNrNa].(38)

Furthermore, Display Formula

G·(G)H=ω^r·H^1*·ω^a·H^2*·ω^r·H^3*·ω^r·H4*·ω^r·H^5*·ω^a·(ω^r·H^1*·ω^a·H^2*·ω^r·H^3*·ω^r·H4*·ω^r·H^5*·ω^a)H=ω^r·H^1*·ω^a·H^2*·ω^r·H^3*·ω^r·H4*·ω^r·H^5*·ω^a·(ω^a)H·(H^5*)H·(ω^r)H·(H^3*)H·(ω^r)H·(H^2*)H·(ω^a)H·(H^1*)H·(ω^r)H,=[σ1·ε1·ς1·ξ1·ζ1000σ2·ε2·ς2·ξ2·ζ2000σNrNa·εNrNa·ςNrNa·ξNrNa·ζNrNa].(39)

Therefore, the imaging operator G is an orthogonal matrix. Furthermore, Φ^·G obeys the RIP. Thus, Γ1(·)·Φ satisfies the RIP.

This work was supported by the National Natural Science Foundation of China under Nos. 61201369 and 61471386.

Khwaja  A. S., and Ma  J. W., “Applications of compressed sensing for SAR moving-target velocity estimation and image compression,” IEEE Trans. Instrum. Meas.. 60, , 2848 -2860 (2011). 0018-9456 CrossRef
An  W.  et al., “Data compression for multilook polarimetric SAR Data,” IEEE Geosci. Remote Sens. Lett.. 6, (3 ), 476 –480 (2009). 1545-598X CrossRef
Zhang  L.  et al., “Wavenumber-domain autofocusing for highly squinted UAV SAR imagery,” IEEE Sens. J.. 12, (5 ), 1574 –1588 (2012). 1530-437X CrossRef
An  D. X.  et al., “Extended nonlinear chirp scaling algorithm for high-resolution highly squint SAR data focusing,” IEEE Trans. Geosci. Remote Sens.. 50, (9 ), 3595 –3609 (2012). 0196-2892 CrossRef
Vandewal  M., , Speck  R., and Süß  H., “Efficient and precise processing for squinted spotlight SAR through a modified Stolt mapping,” EURASIP J. Adv. Signal Process.. 2007, (1 ), 1 –7 (2007).CrossRef
Sun  G. C.  et al., “Focus improvement of highly squinted data based on azimuth nonlinear scaling,” IEEE Trans. Geosci. Remote Sens.. 49, (6 ), 2308 –2322 (2011). 0196-2892 CrossRef
Fang  J.  et al., “Fast compressed sensing SAR imaging based on approximated observation,” IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens.. 7, (1 ), 352 –363 (2014).
Donoho  D. L., “Compressed sensing,” IEEE Trans. Inf. Theory. 52, (4 ), 1289 –1306 (2006). 0018-9448 CrossRef
Candes  E., , Romberg  J., and Tao  T., “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory. 52,  (2 ), 489 –509 (2006). 0018-9448 CrossRef
Zhang  B. C., , Hong  W., and Wu  Y. R., “Sparse microwave imaging: principles and applications,” Sci. China Inf. Sci.. 55, (8 ), 1722 –1754 (2012). 1674-733X CrossRef
Baraniuk  R. G., and Steeghs  P., “Compressive radar imaging,” in  Proc. IEEE Radar Conference , pp. 128 –133,  Boston, Massachusetts  (2007).
Bhattacharya  S.  et al., “Synthetic aperture radar raw data encoding using compressed sensing,” in  Proc. IEEE Radar Conference ,  Rome, Italy  (2008).
Jiang  H.  et al., “Random noise imaging radar based on compressed sensing,” in  Proc. IEEE Geoscience Remote Sensing Symp. , pp. 4624 –4627,  Honolulu, Hawaii  (2011).
Vrashney  K. R.  et al., “Sparse representation in structured dictionaries with application to synthetic aperture radar,” IEEE Trans. Signal Process.. 56, (8 ), 3548 –3561 (2008). 1053-587X CrossRef
Alonso  M. T.  et al., “A novel strategy for radar imaging based on compressive sensing,” IEEE Trans. Geosci. Remote Sens.. 48, (12 ), 4285 –4295 (2010). 0196-2892 CrossRef
Patel  V. M.  et al., “Compressed synthetic aperture radar,” IEEE J. Sel. Topics Signal Process.. 4, (2 ), 244 –254 (2010).CrossRef
Gu  F. F.  et al., “Two-dimensional sparse synthetic aperture radar imaging method,” J. Appl. Remote Sens.. 9, (1 ), 096099  (2015). 1931-3195 CrossRef
Hu  L.  et al., “Compressed sensing of complex sinusoids: an approach based on dictionary refinement,” IEEE Trans. Signal Process.. 60, (7 ), 3809 –3822 (2012). 1053-587X CrossRef
Tropp  J. A., and Wright  S. J., “Computational methods for sparse solution of linear inverse problems,” Proc. IEEE. 98, (6 ), 948 –958 (2010). 0018-9219 CrossRef
Chen  S. S., , Donoho  D. L., and Saunders  M. A., “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput.. 20, (1 ), 33 –61 (1998). 1064-8275 CrossRef
Daubechies  I., , Defrise  M., and De Mol  C., “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math.. 57, (11 ), 1413 –1457 (2004). 0010-3640 CrossRef
Blumensath  T., and Davies  M. E., “Normalized iterative hard thresholding: guaranteed stability and performance,” IEEE J. Sel. Topics Signal Process.. 4, (2 ), 298 –309 (2010).CrossRef
Xu  Z. B.  et al., “L1/2 Regularization: a thresholding representation theory and a fast solver,” IEEE Trans. Neural Netw. Learn. Syst.. 23, (7 ), 1013 –1027 (2012).CrossRef
Candes  E., and Tao  T., “Near optimal signal recovery from random projections: universal encoding strategies?,” IEEE Trans. Inf. Theory. 52, (12 ), 5406 –5425 (2006). 0018-9448 CrossRef

Fu-fei Gu received his MS degree in electrical engineering from the Institute of Information and Navigation, Air Force Engineering University (AFEU), Xi’an, China, in 2011, where he is currently working toward his PhD in electrical engineering. Currently, he is with the radar and signal processing laboratory, School of Information and Navigation, AFEU. His research interests include compressed sensing (CS) theory and radar imaging.

Qun Zhang received his PhD in electrical engineering from Xidian University, Xi’an, China, in 2001. Currently, he is a professor with the School of Information and Navigation, Air Force Engineering University, and an adjunct professor with the Key Laboratory of Wave Scattering and Remote Sensing Information, Fudan University. He has published over 200 papers in journals and conferences. His main research interests include signal processing, and its application in synthetic aperture radar and ISAR.

Yong-an Chen received his bachelor’s degree in communication engineering from the Institute of Information and Navigation, Air Force Engineering University (AFEU), Xi’an, China, in 2014, where he is currently pursuing the MS degree in electrical engineering. His research interests include radar imaging and target identification.

Ying Luo received his MS degree in electrical engineering from the Institute of Telecommunication Engineering, Air Force Engineering University (AFEU), Xi’an, China, in 2008, and his PhD in electrical engineering from AFEU, in 2013. He is currently working in the Key Laboratory for Radar Signal Processing, Xidian University, as a postdoctoral fellow. His research interests include signal processing and ATR in SAR and ISAR.

© The Authors. Published by SPIE under a Creative Commons Attribution 3.0 Unported License. Distribution or reproduction of this work in whole or in part requires full attribution of the original publication, including its DOI.

Citation

Fu-fei Gu ; Qun Zhang ; Yong-an Chen and Ying Luo
"Imaging method for highly squinted synthetic aperture radar with under-sampled echo data", J. Appl. Remote Sens. 9(1), 097495 (Jun 19, 2015). ; http://dx.doi.org/10.1117/1.JRS.9.097495


Figures

Graphic Jump Location
Fig. 1
F1 :

Imaging model of highly squinted synthetic aperture radar (SAR): (a) flow chart of highly squinted SAR; (b) geometric model of flight path and point scatterer.

Graphic Jump Location
Fig. 2
F2 :

Relationship between imaging method with Nyquist samples and with under-samples.

Graphic Jump Location
Fig. 4
F4 :

Imaging results under different situations: (a) imaging result with spatial variance considered; (b) imaging result with spatial variance not considered.

Graphic Jump Location
Fig. 5
F5 :

Results of compressed sensing (CS) imaging method with different under-sampled ratios: (a) imaging result with η=1/2; (b) imaging result with η=1/3.

Graphic Jump Location
Fig. 3
F3 :

Flight geometry and target distribution in the slant plane for the simulation.

Graphic Jump Location
Fig. 7
F7 :

Imaging results of area targets with different methods and different under-sampled ratios: (a) imaging result of Nyquist samples with geometric distortion not corrected; (b) imaging result of Nyquist samples with geometric distortion corrected; (c) imaging result by Nyquist samples imaging method with η=1/2; (d-f) the imaging results by the proposed compressed sensing method with η=3/4, 1/2, and 1/4, respectively.

Tables

Table Grahic Jump Location
Table 1Synthetic aperture radar parameters.
Table Grahic Jump Location
Table 2Comparison of peak signal-to-noise ratio (PSNR) and CPU time.

References

Khwaja  A. S., and Ma  J. W., “Applications of compressed sensing for SAR moving-target velocity estimation and image compression,” IEEE Trans. Instrum. Meas.. 60, , 2848 -2860 (2011). 0018-9456 CrossRef
An  W.  et al., “Data compression for multilook polarimetric SAR Data,” IEEE Geosci. Remote Sens. Lett.. 6, (3 ), 476 –480 (2009). 1545-598X CrossRef
Zhang  L.  et al., “Wavenumber-domain autofocusing for highly squinted UAV SAR imagery,” IEEE Sens. J.. 12, (5 ), 1574 –1588 (2012). 1530-437X CrossRef
An  D. X.  et al., “Extended nonlinear chirp scaling algorithm for high-resolution highly squint SAR data focusing,” IEEE Trans. Geosci. Remote Sens.. 50, (9 ), 3595 –3609 (2012). 0196-2892 CrossRef
Vandewal  M., , Speck  R., and Süß  H., “Efficient and precise processing for squinted spotlight SAR through a modified Stolt mapping,” EURASIP J. Adv. Signal Process.. 2007, (1 ), 1 –7 (2007).CrossRef
Sun  G. C.  et al., “Focus improvement of highly squinted data based on azimuth nonlinear scaling,” IEEE Trans. Geosci. Remote Sens.. 49, (6 ), 2308 –2322 (2011). 0196-2892 CrossRef
Fang  J.  et al., “Fast compressed sensing SAR imaging based on approximated observation,” IEEE J. Sel. Topics Appl. Earth Observ. Remote Sens.. 7, (1 ), 352 –363 (2014).
Donoho  D. L., “Compressed sensing,” IEEE Trans. Inf. Theory. 52, (4 ), 1289 –1306 (2006). 0018-9448 CrossRef
Candes  E., , Romberg  J., and Tao  T., “Robust uncertainty principles: exact signal reconstruction from highly incomplete frequency information,” IEEE Trans. Inf. Theory. 52,  (2 ), 489 –509 (2006). 0018-9448 CrossRef
Zhang  B. C., , Hong  W., and Wu  Y. R., “Sparse microwave imaging: principles and applications,” Sci. China Inf. Sci.. 55, (8 ), 1722 –1754 (2012). 1674-733X CrossRef
Baraniuk  R. G., and Steeghs  P., “Compressive radar imaging,” in  Proc. IEEE Radar Conference , pp. 128 –133,  Boston, Massachusetts  (2007).
Bhattacharya  S.  et al., “Synthetic aperture radar raw data encoding using compressed sensing,” in  Proc. IEEE Radar Conference ,  Rome, Italy  (2008).
Jiang  H.  et al., “Random noise imaging radar based on compressed sensing,” in  Proc. IEEE Geoscience Remote Sensing Symp. , pp. 4624 –4627,  Honolulu, Hawaii  (2011).
Vrashney  K. R.  et al., “Sparse representation in structured dictionaries with application to synthetic aperture radar,” IEEE Trans. Signal Process.. 56, (8 ), 3548 –3561 (2008). 1053-587X CrossRef
Alonso  M. T.  et al., “A novel strategy for radar imaging based on compressive sensing,” IEEE Trans. Geosci. Remote Sens.. 48, (12 ), 4285 –4295 (2010). 0196-2892 CrossRef
Patel  V. M.  et al., “Compressed synthetic aperture radar,” IEEE J. Sel. Topics Signal Process.. 4, (2 ), 244 –254 (2010).CrossRef
Gu  F. F.  et al., “Two-dimensional sparse synthetic aperture radar imaging method,” J. Appl. Remote Sens.. 9, (1 ), 096099  (2015). 1931-3195 CrossRef
Hu  L.  et al., “Compressed sensing of complex sinusoids: an approach based on dictionary refinement,” IEEE Trans. Signal Process.. 60, (7 ), 3809 –3822 (2012). 1053-587X CrossRef
Tropp  J. A., and Wright  S. J., “Computational methods for sparse solution of linear inverse problems,” Proc. IEEE. 98, (6 ), 948 –958 (2010). 0018-9219 CrossRef
Chen  S. S., , Donoho  D. L., and Saunders  M. A., “Atomic decomposition by basis pursuit,” SIAM J. Sci. Comput.. 20, (1 ), 33 –61 (1998). 1064-8275 CrossRef
Daubechies  I., , Defrise  M., and De Mol  C., “An iterative thresholding algorithm for linear inverse problems with a sparsity constraint,” Commun. Pure Appl. Math.. 57, (11 ), 1413 –1457 (2004). 0010-3640 CrossRef
Blumensath  T., and Davies  M. E., “Normalized iterative hard thresholding: guaranteed stability and performance,” IEEE J. Sel. Topics Signal Process.. 4, (2 ), 298 –309 (2010).CrossRef
Xu  Z. B.  et al., “L1/2 Regularization: a thresholding representation theory and a fast solver,” IEEE Trans. Neural Netw. Learn. Syst.. 23, (7 ), 1013 –1027 (2012).CrossRef
Candes  E., and Tao  T., “Near optimal signal recovery from random projections: universal encoding strategies?,” IEEE Trans. Inf. Theory. 52, (12 ), 5406 –5425 (2006). 0018-9448 CrossRef

Some tools below are only available to our subscribers or users with an online account.

Related Content

Customize your page view by dragging & repositioning the boxes below.

Related Book Chapters

Topic Collections

Advertisement
  • Don't have an account?
  • Subscribe to the SPIE Digital Library
  • Create a FREE account to sign up for Digital Library content alerts and gain access to institutional subscriptions remotely.
Access This Article
Sign in or Create a personal account to Buy this article ($20 for members, $25 for non-members).
Access This Proceeding
Sign in or Create a personal account to Buy this article ($15 for members, $18 for non-members).
Access This Chapter

Access to SPIE eBooks is limited to subscribing institutions and is not available as part of a personal subscription. Print or electronic versions of individual SPIE books may be purchased via SPIE.org.