An image formation algorithm for missile-borne circular-scanning SAR
© Gao et al.; licensee Springer. 2013
Received: 11 August 2012
Accepted: 7 December 2012
Published: 2 January 2013
Circular-scanning SAR is an imaging mode with its antenna beam rotating continuously with respect to the vertical axis. An image formation algorithm for the missile-borne circular-scanning 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 2-D 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.
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 .
SAR system usually operates in three modes: stripmap, spotlight, and scan . Circular-scanning 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. Missile-borne circular-scanning SAR suffers from complicated imaging problems: high speed, high squint angle, and high maneuvering. Sun discussed the properties of the circular-scanning SAR signal and presented an image formation algorithm based on the extended chirp scaling algorithm (ECSA) . It is nature to increase the sampling rate and the memory storage for the ECSA as the squint angle increases. Li proposed a geometric-distortion correction algorithm for the successive subimages formed by the linear range-Doppler algorithm (LRDA) . 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 missile-borne circular-scanning 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 geometric-distortion correction method is proposed. It corrects target locations by using 2-D 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 circular-scanning 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 missile-borne circular-scanning 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 missile-borne circular-scanning SAR
v a: velocity of the missile. Its horizontal component coincides with the positive x-axis.
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).
: 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 x-axis to the ground beam orientation.
ψ a: incidence angle.
β r: two-way range beamwidth.
β a: two-way azimuth beamwidth.
φ g: projection of β aonto the x-y 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 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 missile-borne circular-scanning SAR
The image formation algorithm for the missile-borne circular-scanning SAR is comprised of three steps: (1) subimage formation, (2) geometric-distortion 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 subis the subaperture time, N is the total number of the subimages. In a single subaperture time (), 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 ris determined by the required cross-range resolution of the subimages .
3.2 Geometric-distortion correction
The geometric distortion is inevitable, its effects become more evident as the resolution and the squint angle increase . It is necessary to correct geometric distortion before imaging mosaicking. Because the geometric distortion is spatial-variant, its correction must be implemented by calculating the 2-D position of each calibration point between the scatterers and the subimages .
where R t is the distance from the APC to the calibration point, R ais the distance from the APC to the temporary scene center.
Here, the higher order terms are not emphasized.
and in Equation (18) can be calculated as follows.
Here, R→t is the APC position vector relative to the calibration point, R→ais the APC position vector relative to the temporary scene center, r→tis the calibration point position vector relative to the temporary scene center. V→ac is the velocity vector in the X t-Y t-Z tcoordinate system, V Xcand 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 , . B r and B a are range and azimuth output bandwidth, N xoutis the sample number of the interpolation output in the range/azimuth direction, N xFFT is the FFT size.
Here, R ais the distance between the APC and the temporary scene center. r tis 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 ain the Z t-axis, V X in the X t-axis, these variables are evaluated in the X t-Y tcoordinate 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 2-D 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 rand p aare 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.
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. The interpolation can also be implemented by chirp z-transform . The lengths of IFFT operation are N rout and N aout in the range and cross-range direction, respectively. After position calculation, the geometric-distortion 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 geometric-distortion correction and no extra computation is needed.
4 Simulation results
Horizontal velocity v x
Platform altitude H
Antenna rotating speed Ω
Dive angle θ d
Range beamwidth β r
Azimuth beamwidth β a
Sector center angle Δ θ r
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 xis also constant. To validate the image formation algorithm in subimage formation and geometric-distortion 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 cross-range resolution of successive subimages are analyzed. Three subimages and their geometric-distortion 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 cross-range 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 cross-range 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 outis 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 cross-range IPR
4.2 Uniformly acceleration in the vertical direction
PSLRs analysis of the IPRs (vertical velocity is uniformly accelerated)
PSLR of range IPR
PSLR of cross-range IPR
4.3 Vertical velocity changes with time in sinusoidal form
An image formation algorithm for the missile-borne circular-scanning SAR is proposed in this article. The final circular image is formed through subimage formation, geometric-distortion 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 missile-borne circular-scanning SAR, even under highly maneuvering conditions. Our further study will focus on the impact of assuming the wrong calibration point height, and more precise geometric-distortion correction method by using an external DEM. Motion errors and factors which deteriorate the image quality will also be our further study.
The authors wish to thank Dr. Daiyin Zhu from Nanjing University of Aeronautics and Astronautics for helpful technical discussions.
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