An image formation algorithm for missileborne circularscanning SAR
 Yesheng Gao^{1}Email author,
 Kaizhi Wang^{1} and
 Xingzhao Liu^{1}
https://doi.org/10.1186/1687618020132
© Gao et al.; licensee Springer. 2013
Received: 11 August 2012
Accepted: 7 December 2012
Published: 2 January 2013
Abstract
Circularscanning SAR is an imaging mode with its antenna beam rotating continuously with respect to the vertical axis. An image formation algorithm for the missileborne circularscanning SAR is proposed in this article. Based on the principle of the polar format algorithm, the focus algorithm is generalized to form each subimage when the antenna beam scans at an arbitrary position. By calculating the 2D position of each calibration point between the scatterers and the subimages, a method is presented to correct the geometric distortion of each subimage. This method is able to correct the geometric distortion even in the case of high maneuvering. These subimages are then mosaicked together to form a circular image. The simulation results under three different maneuvering trajectories are given, the subimages are formed by the focusing algorithm, and then the final circular image can be formed by mosaicking 71 subimages, each of which is after geometric distortion correction. The simulations validate the proposed image formation algorithm, and the results satisfy system design requirements.
1 Introduction
Synthetic aperture radar (SAR) is a form of radar system to provide high resolution images with the use of the relative motion between the target region and the antenna, which is usually mounted on a moving platform [1–3]. The conventional platform includes aircraft, spacecraft, and satellite. The radar can also be mounted on a missile for military applications [4].
SAR system usually operates in three modes: stripmap, spotlight, and scan [1]. Circularscanning SAR is different from these three modes, with its antenna beam rotating continuously with respect to the vertical axis [5, 6]. It can provide SAR image of both sides of the flight path, and can also extend imaged area during a single pass with the same antenna. Missileborne circularscanning SAR suffers from complicated imaging problems: high speed, high squint angle, and high maneuvering. Sun discussed the properties of the circularscanning SAR signal and presented an image formation algorithm based on the extended chirp scaling algorithm (ECSA) [5]. It is nature to increase the sampling rate and the memory storage for the ECSA as the squint angle increases. Li proposed a geometricdistortion correction algorithm for the successive subimages formed by the linear rangeDoppler algorithm (LRDA) [6]. LRDA is an efficient image formation algorithm, but it takes extra computations to compensate for motion errors due to the high maneuvering.
We concentrate on the image formation during the missile descending stage. An image formation algorithm for the missileborne circularscanning SAR is proposed. A processed aperture time is defined as the time during which the antenna beam rotates 360 degrees. The processed aperture is divided into many subapertures, the signal of which is processed by using the principle of the polar format algorithm (PFA) [1, 2]. Each subaperture is used to form a SAR image, named subimage. The focus algorithm is generalized to form each subimage when the antenna beam scans at an arbitrary position. A geometricdistortion correction method is proposed. It corrects target locations by using 2D interpolation in image domain. This method can work even in the case of high maneuvering. The successive subimages are mosaicked together to form a circular image. Compared with the existing algorithms of the circularscanning SAR, the proposed algorithm does not need to increase the sampling rate and memory storage when the squint angle increases. Meanwhile, there is no extra computation to motion compensation since it is included in the subimage focusing algorithm.
The remainder of the article is organized as follows. In Section 2, the imaging geometry of the missileborne circularscanning SAR is introduced. In Section 3, the image formation algorithm is detailed. In Section 4, simulation results under three different maneuvering trajectories are described. Section 5 presents our conclusions.
2 Imaging geometry of the missileborne circularscanning SAR
The imaging geometry of missileborne circularscanning SAR is illustrated in Figure 1. Symbols are listed as follows. For clarity of the illustration, some symbols are not labeled in Figure 1.
v _{a}: velocity of the missile. Its horizontal component coincides with the positive xaxis.
H: altitude of the missile.
θ _{d}: dive angle which identifies the direction of the missile velocity relative to the horizontal direction.
p _{m}: the position of the antenna phase center (APC).
${p}_{\text{m}}^{{}^{\prime}}$: the corresponding nadir point of p _{m}.
Ω: rotating speed of the antenna beam. It is considered positive when rotating counterclockwise.
θ _{r}: the angle that goes counterclockwise from the positive xaxis to the ground beam orientation.
ψ _{a}: incidence angle.
β _{r}: twoway range beamwidth.
β _{a}: twoway azimuth beamwidth.
φ _{g}: projection of β _{a}onto the xy plane.
α: Doppler cone angle.
θ _{gs}: ground squint angle.
Here, assume that θ _{r0}=180 ^{∘}, when t=0.
where r _{m}(t) is the footprint radius (distance between ${p}_{\text{m}}^{\prime}$ and O _{temp}), r _{l}(t) and r _{u}(t) is the lower and upper limit radius, respectively.
In particular, [x _{sc}(0),y _{sc}(0)]=[0,0]according to the aforementioned assumption.
3 Image formation algorithm for the missileborne circularscanning SAR
The image formation algorithm for the missileborne circularscanning SAR is comprised of three steps: (1) subimage formation, (2) geometricdistortion correction, (3) image mosaicking. Subimages after correcting geometric distortion are mosaicked together to get the final circular image. These three steps are discussed in detail in the following sections.
3.1 Subimage formation
where T _{sub}is the subaperture time, N is the total number of the subimages. In a single subaperture time (${T}_{\text{sub}}/2\le \widehat{t}<{T}_{\text{sub}}/2$), the radar platform moves from the start of the subaperture to the end of the subaperture, θ _{r} changes from θ _{r}(t _{n}) to θ _{r}(t _{n} + T _{sub}).
The overlap region of the two neighboring footprints is considered a spotlight imaging, the center angle of the sector Δ θ _{r} can be approximated as Δ θ _{r}≈θ _{r}(t _{n} + T _{sub})−θ _{r}(t _{n}). The lower limit of Δ θ _{r}is determined by the required crossrange resolution of the subimages [6].
3.2 Geometricdistortion correction
The geometric distortion is inevitable, its effects become more evident as the resolution and the squint angle increase [9]. It is necessary to correct geometric distortion before imaging mosaicking. Because the geometric distortion is spatialvariant, its correction must be implemented by calculating the 2D position of each calibration point between the scatterers and the subimages [6].
where R _{t} is the distance from the APC to the calibration point, R _{a}is the distance from the APC to the temporary scene center.
Here, the higher order terms are not emphasized.
$\frac{\text{d}t}{\text{d}{\theta}_{\text{a}}}{}_{\text{c}}$ and ${\left(\frac{{R}_{\Delta}}{sin{\psi}_{\text{a}}}\right)}^{\prime}{}_{\text{c}}$ in Equation (18) can be calculated as follows.
Here, R→_{t} is the APC position vector relative to the calibration point, R→_{a}is the APC position vector relative to the temporary scene center, r→_{t}is the calibration point position vector relative to the temporary scene center. V→_{ac} is the velocity vector in the X _{t}Y _{t}Z _{t}coordinate system, V _{Xc}and V _{Yc} are its components in the X _{t} and Y _{t}axes. The subscript c refers to the values of the corresponding variables at the subaperture center.
where ${p}_{\text{r}}=\frac{c}{2{B}_{\text{r}}}\frac{{N}_{\text{rout}}}{{N}_{\text{rFFT}}}$, ${p}_{\text{a}}=\frac{c}{2{B}_{\text{a}}}\frac{{N}_{\text{aout}}}{{N}_{\text{aFFT}}}$. B _{r} and B _{a} are range and azimuth output bandwidth, N _{xout}is the sample number of the interpolation output in the range/azimuth direction, N _{xFFT} is the FFT size.
Here, R _{a}is the distance between the APC and the temporary scene center. r _{t}is the distance between the temporary scene center and the calibration point. T _{X} is the coordinate of the calibration point in the X _{t}axis, V _{Z} is the component of v _{a}in the Z _{t}axis, V _{X} in the X _{t}axis, these variables are evaluated in the X _{t}Y _{t}coordinate system. θ _{ac}, ψ _{ac}, V _{xc}, V _{zc}, and R _{ac} are the values of the corresponding variables at the subaperture center.
The geometric distortion can be corrected as follows.

Step 1: Set calibration grid, as shown in Figure 5.

Step 2: Calculate the position of each calibration point in the subimage, (N _{r},N _{a}). These calibration points are constrained within the instantaneous imaging area.

Step 3: Obtain the intensity of the corresponding position via 2D interpolation.
3.3 Image mosaicking
Subimages are focused as discussed in Section 3.1. The method proposed in Section 3.2 can correct their geometric distortion. The final circular image is formed by mosaicking together these subimages. The image mosaicking can be described as follows.

Step 1: Calculate the size of the final output image, allocate a matrix to store the image. The complete illuminated area (0 ^{°}=? _{r}<360 ^{°}) can be calculated according to the radar system parameters. Thus, the size of the output image can be determined when p _{r}and p _{a}are known. It should be noted that each pixel in the output image corresponds to a calibration point.

Step 2: Correct the geometric distortion of each calibration point in the temporary illuminated area. The temporary illuminated area can be calculated using the radar system parameters, the instantaneous radar platform position and the value of ? _{r}. The geometric distortion of each calibration point in this area can be corrected by using the method described in Section 3.2. Then the subimage without geometric distortion is stored into the positions where the calibration points are located.

Step 3: If all the subimages are formed (in the later simulations, the number of subimages to constitute a circular image is 71), output the final circular image. Otherwise, return to Step 2 to form the next subimage. Thus, the final circular image can be formed by adding successive subimages.
Computational consideration
where L _{out} is the output length of the interpolation, L _{f} is the length of the filter, r _{DS} is the downsampling ratio, M _{k} is a constant and its typical value is 1. 25[1]. The interpolation can also be implemented by chirp ztransform [9]. The lengths of IFFT operation are N _{rout} and N _{aout} in the range and crossrange direction, respectively. After position calculation, the geometricdistortion correction is implemented by interpolation operation, the computational complexity can be determined by Equation (27). Image mosaicking is separated as the third step, but its realization is included in geometricdistortion correction and no extra computation is needed.
4 Simulation results
Simulation parameters
Parameter  Value 

Horizontal velocity v _{x}  956 m/s 
Platform altitude H  >10 km 
Antenna rotating speed Ω  293 ^{∘}/s 
Dive angle θ _{d}  45.5 ^{∘}/s 
Bandwidth B  32 MHz 
Range beamwidth β _{r}  21 ^{∘} 
Azimuth beamwidth β _{a}  3.9 ^{∘} 
Sector center angle Δ θ _{r}  10 ^{∘} 
Scene extent W _{x}×W _{y}  20 km × 20 km 
Pixel resolution p _{r}×p _{a}  10 m × 10 m 
In Section 4.1, the vertical velocity v _{z} does not change with time, and the horizontal velocity v _{x}is also constant. To validate the image formation algorithm in subimage formation and geometricdistortion correction, the simulated point targets are placed as a rectangle with the size of 11×15, the distance between two adjacent targets is 200 m. The characteristics of the imaged area (footprint radius, and coordinates of the temporary scene center) and the crossrange resolution of successive subimages are analyzed. Three subimages and their geometricdistortion correction results are given. Quantitative analysis of the result is also given.
In Section 4.2, the horizontal velocity is still constant, the vertical velocity is uniformly accelerated. The point targets are placed the same as those in Section 4.1. The characteristics of imaged area, the crossrange resolution of successive subimages, and three typical subimages are given. Quantitative analysis of the result is also given.
In Section 4.3, the horizontal velocity is still constant, the vertical velocity changes with time in sinusoidal form. In this section, the final circular image is give. The ground point targets are placed uniformly with the interval of 200 m × 200 m, the total number of the targets is 95×95. The crossrange resolution of successive subimages are illustrated.
The major computations involved in the image formation algorithm are interpolation and IFFT, both operations are applied in the range and azimuth directions. In the range direction, the value of L _{out} is 2048, L _{f} is 8, r _{DS} is 1. In the azimuth direction, L _{out}is 340, L _{f} is 8, r _{DS} is 1.51. The IFFT size in range is 2048, azimuth 512.
4.1 Constant velocity in the vertical direction
PSLRs analysis of the IPRs (vertical velocity is constant)
PSLR of range IPR  PSLR of crossrange IPR  

Nearest target  −12.04 dB  −13.03 dB 
Middle target  −13.28 dB  −13.23 dB 
Farthest target  −12.14 dB  −12.01 dB 
4.2 Uniformly acceleration in the vertical direction
PSLRs analysis of the IPRs (vertical velocity is uniformly accelerated)
PSLR of range IPR  PSLR of crossrange IPR  

Nearest target  −13.11 dB  −11.54 dB 
Middle target  −13.25 dB  −13.13 dB 
Farthest target  −13.19 dB  −12.00 dB 
4.3 Vertical velocity changes with time in sinusoidal form
5 Conclusion
An image formation algorithm for the missileborne circularscanning SAR is proposed in this article. The final circular image is formed through subimage formation, geometricdistortion correction, and image mosaicking. Simulation results under three different maneuvering motions are given, each point target in the illuminated area is well focused, the geometric distortion is corrected using the method presented in Section 3.2, and the final circular image can be generated through image mosaicking. From the simulation results done so far, we are confident that the image formation algorithm is stable and effective for missileborne circularscanning SAR, even under highly maneuvering conditions. Our further study will focus on the impact of assuming the wrong calibration point height, and more precise geometricdistortion correction method by using an external DEM. Motion errors and factors which deteriorate the image quality will also be our further study.
Declarations
Acknowledgements
The authors wish to thank Dr. Daiyin Zhu from Nanjing University of Aeronautics and Astronautics for helpful technical discussions.
Authors’ Affiliations
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