XIONG Xuying, LI Gen, MA Yanheng, and CHU Lina

Institute of UAV Engineering, Army Engingeering University, Shijiazhuang 050003, China

Abstract: Strong spatial variance of the imaging parameters and serious geometric distortion of the image are induced by the acceleration and vertical velocity in a high-squint synthetic aperture radar (SAR) mounted on maneuvering platforms. In this paper, a frequency-domain imaging algorithm is proposed based on a novel slant range model and azimuth perturbation resampling. First, a novel slant range model is presented for mitigating the geometric distortion according to the equal squint angle curve on the ground surface. Second, the correction of azimuth-dependent range cell migration (RCM) is achieved by introducing a high-order time-domain perturbation function. Third,an azimuth perturbation resampling method is proposed for azimuth compression. The azimuth resampling and the time-domain perturbation are used for correcting first-order and high-order azimuthal spatial-variant components, respectively. Experimental results illustrate that the proposed algorithm can improve the focusing quality and the geometric distortion correction accuracy of the imaging scene effectively.

Keywords: synthetic aperture radar (SAR) imaging, maneuvering platform, high-squint, azimuth perturbation resampling.

1. Introduction

Synthetic aperture radar (SAR) can improve the information acquisition ability. High-squint SAR imaging systems mounted on maneuvering platforms can further improve the ability of field reconnaissance, precision strike,and self-survival. The corresponding imaging algorithms have attracted great attention recently [1-3].

The slant range model is the basis of SAR imaging algorithms. It determines the application range of the algorithm, the complexity of geometric distortion correction, and the development of subsequent algorithms. The maneuvering platform moves along a curvilinear trajectory in three-dimensional (3-D) space. This nonlinear trajectory induces different slant range histories for different ground targets in the range-Doppler domain. Therefore, the traditional hyperbolic model [4] based on the data recording parameters (slant range and azimuth time)is non-applicable for maneuvering SAR imaging. Although a precise slant range model can be easily obtained by using the target coordinates, it is difficult to analyze and correct the spatial variance of the imaging parameters [5-7]. In [8], the slant range model represented by the vector is adopted for designing the maneuvering SAR imaging algorithm in the wavenumber domain.However, it is difficult to correct the spatial variance of the imaging parameters since the target coordinates are needed for calculating the vector. Based on the equivalent squint model used for high-squint diving SAR [9], Li et al. [10] proposed a slant range model with the data recording parameters. Based on this model, the spatial variance of the imaging parameters can be well analyzed and corrected. However, Li’s model cannot depict accurately the slant range history of the ground target for the platform with a large acceleration.

Up to now, some approaches have been proposed to improve the focusing quality of high-squint SAR imaging. An et al. [11,12] proposed the method of azimuth nonlinear chirp scaling (ANCS), which presents a good way to correct the azimuth dependence of Doppler parameters. To further extend the focus depth, Li et al. [13]studied an improved imaging algorithm based on 2-D range cell migration correction (RCMC) considering the range dependence of the squint angle. However, those algorithms are only applicable to SAR imaging with the parallel track. Based on the 3-D acceleration model, Zeng et al. [14] proposed an improved imaging algorithm that adopts the method of Doppler domain blocking to correct the azimuth space variance. However, complex azimuth blocking and splicing operations limit the practical application of the algorithm. A modified Omega-K algorithm with constant acceleration was proposed in [15],but it ignores the high-order dependence of Doppler parameters. The method of the ANCS is widely adopted for high-squint maneuvering SAR imaging [7,16,17], but it is difficult to determine an optimal value of the scaling factor under different accelerations. Besides, uncorrected high-order azimuth spatially variant components of Doppler parameters limit the focus depth of the imaging scene.

In this paper, a novel frequency-domain imaging algorithm is proposed for highly squinted maneuvering SAR. First, according to the equal-squint-angle curve on the ground plane, a novel slant range model based on the data recording parameters is constructed to describe the slant range history of the spatially variant scattering point.And then, a high-order time-domain perturbation function is constructed by separating the azimuth spatially variant component of the linear RCM (LRCM) to correct the azimuth dependence of RCM. In the process of azimuth compression, a perturbation resampling method is proposed to eliminate high-order azimuth spatially variant components of the Doppler parameters. Finally, based on the proposed slant range model, an accurate geometric distortion correction method is presented.

2. Slant range model for maneuvering SAR platforms

Small-aperture SAR imaging is more convenient to realize motion compensation and real-time imaging processing. The small-aperture data is a part of full-aperture data. Using the small-aperture data, a sub-region of the acquired scene can be focused. Then, we can get the focused image of the whole scene by the mosaicking technology. We mainly study small-aperture SAR imaging in this paper. As the small-aperture time is short, targets in the beam scope can be approximately regarded as the entire illumination during the small-aperture time. The work mode of the SAR in this paper can be regarded as a kind of spotlight mode.

The geometric model of high-squint small-aperture SAR imaging with a curved trajectory is shown in Fig.1.During a synthetic aperture time, the SAR platform maneuvers along the curveABDwith 3-D constant accelerationAt azimuth timeta=0, the platform position is at pointBwhich is the synthetic aperture center.The 3-D velocity of the platform at pointBisv=(vx,0,vz),wherevxandvzdenote horizontal and vertical velocities,respectively. PointPirradiated by the beam center is the scene center. The initial slant range of pointPis denoted byRref, which is used as the referenced slant range. The squint angle θAis defined as the angle betweenBPand theYOZplane. Aftertn, the platform moves from pointBto pointC. AssumingQis an arbitrary point on the imaging region,R0denotes the instantaneous slant range of pointQwhen the platform is at the pointC.

Fig. 1 Geometric model of highly squinted SAR with a curved trajectory

As shown in Fig.1(a), denoting the coordinates of pointQas (x,y), the instantaneous slant range of the pointQcan be expressed as

wherehis the platform height at pointB.

Although this is an accurate and concise slant range model used in [6,18,19], it is difficult to analyze and correct the spatial variance of the imaging parameters.

Based on the equivalent squint model used for highsquint diving SAR [9], Li et. al. proposed a slant range model [10,16]. In [10] and [16], the data recording parametersR0andtnare used to depict the instantaneous slant range of the pointQ. The geometric model is shown in Fig. 1(b). The specific expression of the slant range model appeared in (2)-(15) in [10], given by

In Li’s model, the equidistant curveA0B0D0is parallel to the platform trajectoryABD. It means thatQwould be out of the ground plane for the platform with vertical velocity and acceleration. Therefore, the imaging plane depicted by (2) is actually a curved surface parallel to the trajectory of the platform. Therefore, (2) cannot accurately represent the slant range history of the ground scattering points under a 3-D acceleration. Accordingly, the accuracy of azimuth compression and geometric distortion correction will be affected.

To describe accurately the slant range history of the ground targets, a modified slant range model is proposed in this paper using the data recording parametersR0andtn, shown by Fig. 1(c). As is shown in Fig. 1(c), when the platform is at any positionCwith the azimuth timetn, an equal-squint-angle curveEFon the ground plane can be found. Each pointQinEFsatisfies that the angle betweenCQand the YOZ plane is equal to θA. Therefore,the coordinates (x0,y0) of pointQin the Cartesian frame can be expressed by

Based on (1) and (3), the proposed slant range model can be written as

Being different from Li’s model, the equidistant curveGHin Fig. 1(c) is on the ground plane, so the proposed slant range model can accurately depict the slant range history of the ground scattering points. To simplify the slant range model, (4) is expanded atta=tnby theNthorder Taylor series, given by

whereki(R0,tn) represents theith-order coefficient of the Taylor series expansion, given by

For the process of the RCMC, the approximation error of the slant range model should be less than 1/4 range resolution unit (RRU). To determine an appropriate expansion order, the expansion errors of the 2nd-, 3rd- and 4thorder models in (5) are analyzed in Fig. 2 under the simulation parameters in Table 1.

Table 1 Simulation parameters

Fig. 2 Taylor expansion error analysis of Rp(ta;R0,tn) at ta=tn

Assuming that (xc,yc) are the scene center coordinates,a boundary point with coordinates (xc+750,yc) is used for the simulation analysis in Fig. 2. It can be seen that only the 4th-order Taylor expansion error is far less than 1/4 RRU (0.187 m). Therefore, the 4th-order Taylor model is adopted for the RCMC processing in this paper.

Supposing that the transmitted signal is linearly modulated in frequency, the received echo signal from the targetQin the range frequency domain can be expressed by

whereKrrepresents the range chirp rate, c is the speed of light,frandfcare the range frequency and the carrier frequency, respectively. And the range and azimuth window functions are omitted for brevity.

3. Range dimension processing

Linear RCMC (LRCMC) is widely adopted to eliminate strong range-Doppler coupling for high-squint SAR imaging. However, for SAR imaging mounted on maneuvering platforms, LRCM has a strong 2-D spatial variability due to the acceleration and descending velocity. It impacts on the RCMC and azimuth compression and needs to be corrected.

First, spatial-invariant LRCMC referenced to the scene center point is performed, given by

wherek10=k1(Rref,0) is the referenced LRCM.

After spatial-invariant LRCMC, the signal can be written as

wherekres(Rlrcm,tn)=k1(Rlrcm,tn)-k10denotes the spatialvariant LRCM term. It is often ignored in conventional algorithms [14,16,20].

In (9), the beam center slant range changes fromR0toRlrcm=R0-k10tn, which means the targets lying in the same range cellRlrcmhave different initial slant rangesR0. This tory and Doppler parameters. The parameterskres(Rlrcm,tn) andki(Rlrcm,tn) vary with range and azimuth positionsRlrcmandtn, making the unified processing unsatisfactory further increases the spatial variabilities of RCM trajec-(i.e., RCMC and azimuth compression).

To correct azimuth spatial-variant RCM, an extended time-domain perturbation function is constructed as follows by introducing an additional time-domain secondorder modulation phase.

whereq2,q3andq4are undetermined time-domain perturbation coefficients. In (10),q2is used to correct the azimuth dependence of the LRCM. The proposed timedomain perturbation function is different from the nonlinear chirp scaling (NCS) factor in [16]. In [16], only the 3rd- and 4th-order modulation phases are considered.

The 1st-order component is the dominant component of the azimuth-dependent RCMs. Thus,kres(Rlrcm,tn),k2(Rlrcm,tn) andk3(Rlrcm,tn) are expanded attn=0 in Taylor’s seriesand kept up to the 1st-order term, given by

The spatial variance of the 4th-order coefficientk4(Rlrcm,tn)has little influence on the imaging process. It can be replaced by the referenced parameter with

After the time-domain perturbation correction (TDPC),the signal can be expressed by

whereknewi(i=1,2,3,4) denotes theith-order RCM coefficients after TDPC.

In (13), the time-domain perturbation coefficients are coupled withtnin the 1st-, 2nd- and 3rd-order RCM coefficients, so their azimuth dependence can be easily solved by making the linear terms oftnzero. These can be expressed as

In (13), additional azimuth-dependent termsare very weak, and their influences on RCMC can be ignored. The RCM coefficients after the TDPC can be written as

As can be seen from (15), the TDPC successfully corrects the azimuth dependence of the RCM coefficients,but additional azimuth-dependent range offsetThe range offset affects azimuth compression and will be considered in our subsequent process.

Transforming (12) into 2-D frequency domain using the series reversion method [21,22] and expanding the phase term atfr=0, we can get

where

In (17),Br_0,Br_1andBr_2are the azimuth frequency domain modulation phase, the unified RCM, and the secondary range compression, respectively. The range filtering function constructed in the 2-D frequency domain is

Multiplying (16) by (18) and transforming the result into 2-D time domain, we can get the range compression result, given by

whereRnewis the focused range position and φadenotes the phase history in the azimuth time domain. The plane expressed byRnewandtnis defined as the focused plane.

4. Azimuth dimension processing and geometric distortion correction

4.1 Azimuth dimension processing

In (19), the solving precision of φahas an impact on azimuth compression. In [16] and [23], φawas obtained by the series reversion method based on the azimuth frequency-domain modulation phase. The solving process contains many approximations and ignores the influence of the range offset on Doppler parameters. To achieve more accurate azimuth compression, a precise solving methodforφaispresented as follows.muthpositiontn,its slant range historyafter spatial-in-

Fora point target with initial slant rangeand azivariant LRCMC and TDPC can be represented by

After RCMC and range compression, the focused range position changes fromR0toRnew, and the relation betweenR0andRnewcan be expressed by

Substituting (20) into (21), the exact slant range history represented byRnewandtncan be written as

For the process of azimuth compression, the approximation error of the slant range model should be far less than 1/8 wavelength. Under the simulation parameters in Table 1 with the wavelength 0.017 m, the approximation errors of the 2nd- and 3rd-order Taylor expansion models ofRp(ta;R0(Rnew,tn),tn) atta=0 are analyzed. Because the Taylor expansion accuracy is independent on the target position, the scene center point is used for the analysis. The results are given in Fig.3.

It can be found that the expansion error of the 3rd-order model is far less than 1/8 wavelength (0.002 m),meeting the requirement for azimuth compression. Therefore,can be represented by

Substituting (23) into (22), the exact azimuth time domain modulation phase φarepresented byRnewandtncan be written as

where

Transforming (24) into the azimuth frequency domain,the phase term Φacan be obtained as

whereBidenotes theith-order phase coefficient in the azimuth frequency domain. We can find Φahas complex spatial variance in range and azimuth. Theith-order maximum spatial-variant phase can be expressed by

wherei=2,3,4,5,facandBsubdenote the Doppler center and the Doppler bandwidth of the point target (Rnew,tn) on the focused plane, respectively.

Using simulation parameters in Table 1,ΔΦi(Rnew,tn)is analyzed withRnew=Rrefand the results are shown in Fig. 4. We can find the 2nd-, 3rd- and 4th-order maximum spatial-variant phases are larger than π/4 with the increase of the azimuth position, but the 5th-order spatialvariant phase is far less than π/4. Therefore, in the process of azimuth compression, the 2nd-, 3rd- and 4th-order spatial-variant phase should be corrected.

Fig. 4 Analysis of the maximum spatial-variant phase

The method of ANCS has been widely used for azimuth compression in high-squint SAR imaging. In the ANCS, the scaling factor α has an influence on imaging results [11,12,16]. The value of α needs to be determined by experience and a small amount of trial-and-error testing according to the specific SAR parameters [12]. However, in maneuvering SAR imaging, it is difficult to determine an optimal value of α under different accelerations. Besides, the ANCS method can only correct the 2nd-order azimuth variability of the Doppler rate. It limits the focus depth of the imaging scene. In this paper, a perturbation resampling method for azimuth compression is proposed. The first- and high-order spatial variance of the Doppler parameters are corrected by azimuth resampling and high-order perturbation functions, respectively. Without the scaling factor α, the proposed method is more suitable for SAR imaging with maneuvers.The specific derivation process is as follows.

For convenient analysis of azimuth spatial variance,theith-order Doppler parameterbi(Rnew,tn) in (24) is expanded attn=0 by the Taylor series and kept up to the 3rd-order term, given by

To migrate the spatial variance of the Doppler spectrum, the spatial variance of the Doppler centerb1(Rnew,tn)is removed in the azimuth time domain by constructing the time-domain phase compensation function

wherep1is used to remove the azimuth-independent Doppler center,p2andp3are used to remove the 1st- and the 2nd-order azimuthal spatial-variant components of the Doppler center, respectively. These coefficients can be expressed by

The solution method ofp2andp3is the same as that ofqiin (14).

After removing the spatial-variant Doppler center, a high-order time-domain perturbation function is constructed for correcting high-order spatial-variant components of the Doppler parameters, given by

wherep4andp5are undetermined high-order perturbation coefficients.

Multiplying (19) by (28) and (30) and transforming the result into the azimuth frequency domain, the phase term can be expressed by

In (32),Bai_kdenotes thekth-order Taylor expansion coefficient ofBaiattn=0. The 2nd- and 3rd-order spatialvariant coefficients ofBa2cannot be ignored for extended scene imaging. By makingBa2_2andBa2_3zero,p4andp5can be obtained as follows:

According to (32), the following unified azimuth filtering function can be constructed:

After high-order time-domain perturbation and unified azimuth filtering, the residual frequency-domain azimuth modulation phase can be expressed by

To eliminate the azimuth dependence ofthe method of azimuth frequency domain resampling is presented, given by

wherefa_newdenotes the new azimuth frequency.

After azimuth resampling, the azimuth inverse Fourier transform can be applied to achieve 2-D compression,and the result is given by

In the process of azimuth compression, the residual maximal non-linear phase error iswhich can be used to calculate the effective imaging region. In general, the maximum phase error should not exceed π/4.Therefore, the effective imaging region satisfies

According to (38), using the simulation parameters in Table 1, the effective imaging region shown in Fig. 5 can be obtained.InFig. 5, the value range ofRnewisRref-500≤Rnew≤Rref+500.

Fig. 5 Effective imaging regions on the focused plane and the ground plane

It can be seen that with the increase of the slant rangeRnew, the effective azimuth imaging region gradually becomes larger. The azimuth width of the effective imaging area on the ground plane is about 1.2 km, and the range width is about 1.5 km. The range width is mainly dependent on the value range ofRnew. Here, the whole imaging process of high-squint SAR mounted on maneuvering platforms has been completed and the imaging flow chart is shown in Fig. 6, where FFT represents fast Fourier transform, and IFFT means inverse FFT.

Fig. 6 Flow chart of imaging algorithm for high-squint SAR mounted on maneuvering platforms

4.2 Geometric distortion correction

To obtain the ground plane image and realize the sub-region mosaicking, geometric distortion correction processing of the focused SAR image is required. Here, an accurate geometric distortion correction method for SAR imaging mounted on maneuvering platforms is presented.

It is assumed that pointQwith the coordinates(x0,y0)is any point in the ground plane. First, we need to calculate its projection positions (R0,tn) on the imaging plane.

According to (3), the following equation can be obtained:

Expanding (39) into the polynomial form oftn, we can get the following quartic equation:

where

As the linear termb1tnin (40) is dominant, the root closest to -b0/b1among all the four roots of (40) is the only valid root.

After gettingtn,R0can be calculated by

Then, the projection position (Rnew,tn) of pointQon the focused plane can be obtained according to (21) and(37). Finally, the scattering information of the target point(x0,y0)can be obtained by the nearest neighbor interpolation or 2-D sinc interpolation.

5. Simulation analysis

In this section, simulation results are presented to show the performance of the proposed algorithm. The simulation parameters are shown in Table 1. The cascaded TNCS algorithm [16] is chosen as the reference algorithm.

As shown in Fig. 7, a point target array is used for the simulation. The target array is arranged on the ground plane along the projection direction of the radar line of sight (RLOS) and its vertical direction. The size of the point target array is 1 km×1 km in both range and azimuth directions, which is inside the effective imaging region shown in Fig. 5(b). In the simulations, three point targets marked by 1-3 are extracted for detailed analysis.Point target 2 located at the scene center has no spatial variance, and point targets 1 and 3 located at the scene edge have intense spatial variance. The simulation results are presented in three parts. Part I shows the effectiveness of the proposed slant range model and azimuth perturbation resampling method. Part II verifies the validity of the proposed geometric distortion correction method.And Part III exhibits the imaging results of the simulated scene.

Fig. 7 Targets distribution diagram for maneuvering SAR imaging

Part I To demonstrate the effectiveness of the proposed slant range model, the imaging results of the proposed imaging method adopting the reference slant range modelR2(ta;R0,tn) are also presented. The contour maps of the focused point targets 1, 2 and 3 are presented in Fig. 8. Since there is no spatial variance and slant range model error at the scene center, target 2 can be focused well under both the proposed method and the reference method. Due to the existence of slant range model error and uncorrected high-order spatial variance, point targets 1 and 3 have obvious defocusing phenomena shown in Fig. 8 (a). It can be seen from Fig. 8 (c) that the focusing effects of point targets 1 and 3 are improved a lot by using the proposed method, but the defocusing phenomena still exist in azimuth direction affected by the slant range model error. By contrast, Fig. 8 (b) shows that the proposed method can focus the point targets 1 and 3 well under the proposed accurate slant range model. The simulation results in Fig. 8 indicate that the proposed accurate slant range model and the imaging method can effectively improve the focusing effect.

Fig. 8 Contour maps of the scattering points 1,2,3

To further analyze and evaluate the focusing effects of the point targets accurately, measured parameters resolution, peak sidelobe ratio (PSLR), and integral sidelobe ratio (ISLR) are used to quantify the focusing quality in the azimuth direction. Table 2 presents the quantitative analysis results of the 2-D contour maps in Fig. 8. For a better comparison, the theoretical values of the resolution,PSLR, and ISLR are provided in Table 2. Because of the existence of high-squint angle and acceleration, the resolution of the whole imaging scene is not unified. In Table 2,0.88 m denotes the theoretical resolution of the scene center point. It can be found that the measured parameters obtained by the proposed method are closer to the theoretical values than the reference method.

Table 2 Performance analysis of the selected targets

Both the de-acceleration function of the reference algorithm and the proposed time-domain perturbation function shown by (10) will lead to the range distortion of the focused point target. The range distortion has an impact on the focus of the azimuth edge point target. However,this impact is ignored by the reference algorithm. With full consideration of the range distortion, we derive the precise azimuth time domain modulation phase in azimuth dimension processing. The impact of the range distortion on azimuth compression is shown in Fig. 9.

Fig. 9 Azimuth profiles of the focused point target 1

It can be found that after considering the influence of the range distortion, the azimuth profile of the focused point target has lower sidelobes and deeper nulls, and the azimuth focusing quality is significantly improved.

Part II To verify the effectiveness of the proposed geometric distortion correction method, a point target array of 11×11 is arranged on the ground plane along theXaxis direction andY-axis direction where the target spacing is 100 m. The nearest neighbor interpolation method is used for the operation of the geometric distortion correction. The simulation result is given by Fig. 10.

Fig. 10 Results of geometric distortion correction

As in Fig. 10 (a), due to the operation of LRCMC, the distribution of the point target array is a skew rectangle on the focused plane. Fig. 10 (b) and Fig. 10 (c) show the results of geometric distortion correction by the method in [16] and the proposed method, respectively. After the geometric distortion correction, the focused point targets should be a rectangular array parallel to theXandYaxes.However, because the reference method neglects the impact of acceleration on geometric distortion correction,the distribution of the point targets selected by the red rectangle in Fig. 10 (b) is obviously not parallel to theXaxis. It can be seen from Fig. 10 (c) that the focused point target array is parallel to theXandYaxes, indicating that the proposed method has a better geometric distortion correction effect.

Fig. 11 Imaging results of the simulated real scene

Part III Imaging results of the simulated real scene are presented in Fig. 11. Like the focused point target array in Fig. 10 (a), the image on the focused plane is also skew as shown in Fig. 11 (a). It can be seen from Fig. 11 (b)that the proposed geometric distortion correction method can effectively restore the ground-range image. As is shown in Fig. 11 (c), the isolated scattering point can be well focused by the proposed method. However, because of the slant range model error and uncompensated highorder spatial variance, the isolated scattering point focused by the reference method has an obvious defocusing phenomenon shown in Fig. 11 (d). For further comparative analysis, the azimuth profiles of the focused isolated scattering point are given in Fig. 11 (e). It is easy to find that the proposed method has a better azimuth focusing effect.

6. Conclusions

The slant range model and the imaging algorithm of highsquint SAR mounted on maneuvering platforms are studied systematically. A novel slant range model is proposed to accurately describe the slant range history of the ground scattering points. On this basis, a frequency domain imaging algorithm based on azimuth perturbation resampling is proposed. The simulation results show that the proposed slant range model is accurate and effective,and the proposed imaging method can effectively improve the focus depth and the geometric distortion correction accuracy of the imaging scene. Further considerations about the motion compensation for high-squint SAR imaging mounted on different maneuvering platforms can be discussed in the future.