An improved motion compensation method for ... - Semantic Scholar
SCIENCE CHINA Information Sciences
. RESEARCH PAPER .
December 2014, Vol. 57 122301:1–122301:13 doi: 10.1007/s11432-014-5189-2
An improved motion compensation method for high resolution UAV SAR imaging FAN BangKui1,2 , DING ZeGang1 * , GAO WenBin1 & LONG Teng1 1School
of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China; 2Beijing Institute of Information Technology, Beijing 102401, China Received May 1, 2014; accepted July 2, 2014
Abstract The polynomial and sinusoidal motion errors always exist in the unmanned aero vehicle (UAV) SAR due to the small size and low velocity of the platform, causing serious spectrum compressing/stretching and significant spectral replicas of the azimuth signal. The motion errors induce serious blurring of the SAR image and “ghost targets”, and can hardly be precisely estimated by the conventional motion compensation (MOCO) method. In this paper, an improved MOCO method is proposed to estimate and eliminate the motion errors in the high resolution UAV SAR without high precision inertial navigation system (INS) data. The time domain range walk correction (RWC) operation in the coarse phase error estimation process of the proposed MOCO method is the key operation that ensures the estimation accuracy of the whole MOCO method. Finally, the validity of the improved MOCO method is verified by computer simulations and real UAV SAR data processing. Keywords motion error, unmanned aero vehicle (UAV) SAR, motion compensation (MOCO), range walk correction (RWC), high resolution
Citation Fan B K, Ding Z G, Gao W B, et al. An improved motion compensation method for high resolution UAV SAR imaging. Sci China Inf Sci, 2014, 57: 122301(13), doi: 10.1007/s11432-014-5189-2
1
Introduction
Due to the improvement of hardware and MOCO methods, SAR has been greatly developed [1,2] and the resolution of the SAR image is progressively improved in the last decades. As highlighted in [3], the platform motion is the solution and also the problem for SAR imaging. Motion error of the SAR platform from the ideal track causes the phase error in the Doppler history of the illuminated target, which induces the blurring of the final SAR image. The influence of the phase error on SAR imaging is particularly serious in the UAV SAR system because the platform is quite small and gets seriously affected by the atmospheric turbulences. In general, the accuracy of navigation systems is not enough to achieve a high accuracy MOCO. Moreover, the high precision navigation system cannot be equipped onboard the UAV SAR due to the weight and cubage limitation. Hence, the autofocus technique estimating the phase error from the raw data is a must for the UAV SAR imaging. Lately, many MOCO methods have been proposed to estimate the motion error from the raw data in UAV SAR. In [4], a 3-D MOCO method based on the instantaneous Doppler rate (DR) estimation is * Corresponding author (email: [email protected])
c Science China Press and Springer-Verlag Berlin Heidelberg 2014
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proposed. This method extracts the necessary motion parameters, i.e., forward velocity and displacement in line-of-sight direction, from the raw data via the map-drift techniques [5–7]. The DRs from different range blocks and azimuth sub-apertures are estimated and the double integral of the DRs are used to obtain the motion error. For the UAV SAR imaging, a number of iterations are needed to reach an acceptable accuracy due to the instability of the platform, resulting in cumbersome computation. Another MOCO method based on the phase gradient autofocus algorithm (PGA) [8–12] is also proposed. In [11], the weighted phase estimation kernel is proposed for the stripmap mode SAR. However, it only accounts for the range-invariant motion error; hence it provides poor estimation results for the UAV SAR due to the range variation of the motion error. Therefore, the MOCO method accounting for the range variation of the motion error are proposed based on PGA. In [13], the weighted total least square (WTLS) estimation kernel is proposed to estimate the motion error varying with the look angle. In [14], the raw data is split both in range and azimuth directions, and a weighted and local maximum-likelihood (LML) PGA kernel is proposed to estimate the range-variant motion error. In the MOCO methods described above, the influence of the non-systematic range cell migration (NRCM) [14] on the estimation accuracy of the autofocus algorithms is neglected in most of the cases. In [14], the influence of the NRCM on the estimation accuracy of the autofocus algorithms is ignored by a range down-sampling process, and the influence of the NRCM is ignored directly in [13]. However, in UAV SAR, due to the low flight velocity, the motion error seriously affects the signal’s spectrum. The NRCM after range-Doppler domain RCMC is quite large and spans over a large number of range bins. Therefore, after the range-Doppler domain RCMC [15–18] and a reasonable range down-sampling, the target’s energy still spills in many range bins such that the target can hardly get precisely focused by the conventional MOCO method. Moreover, the sinusoidal motion error [19] produces spectra replicas of the azimuth signal in the azimuth-frequency domain. The spectra replicas are shifted into different range bins after range-Doppler domain RCMC, which induces the “ghost targets” in different range bins in the final image. Thus, the sinusoidal motion error can hardly be estimated by the conventional MOCO method. In this paper, RCMC stands for the range-Doppler domain RCMC where no other alternative is mentioned. In this paper, the influences of the polynomial and sinusoidal motion errors are analysed in detail, and an improved MOCO method is proposed for the high resolution UAV SAR imaging. In order to account for the aforementioned difficulties encountered in the high resolution UAV SAR imaging, a twodimensional (2-D) time domain RWC is used and a range down-sampling process is performed to ensure that the residual RCM after RWC is not larger than a range bin. The improved MOCO method firstly estimates the range-invariant phase error by the coarse phase error estimation process, and then the range-variant phase error by the fine phase error estimation process. The RWC operation in the coarse phase error estimation process guaranteed the estimation accuracy of the range-invariant phase error, which subsequently ensures the accurate estimation of the range-variant phase error. The paper is organized as follows. Section 2 is dedicated to a detailed analyse of the influences of the polynomial and sinusoidal motion errors on the high resolution UAV SAR imaging. Section 3 discusses the improved MOCO method for high resolution UAV SAR imaging, with its flow chart. Section 4 presents the computer simulations and the experiments based on real data, and Section 5 concludes this paper briefly.
2
The influence of motion error on UAV SAR imaging
The motion geometry illustration of UAV SAR is shown in Figure 1. X-axis denotes the nominal flight direction, and Y -Z plane denotes the normal plane of the ideal trajectory. The black point in the ideal track are the ideal uniform sampling points and these in the actual track are the real sampling points. The vector from the ideal sampling point to the real sampling point is the motion error. The motion error is decomposed into three components which are ∆x, ∆l and ∆s. ∆x is the along-track component, ∆l is the polynomial cross-track component, and ∆s is the sinusoidal cross-track component.
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Z
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Actual track X (ta) Ideal track
Y
Azimuth block ∆s
Rs
ϕ
∆x
Quadric track
P Figure 1
2.1
Actual track
∆l
The motion geometry illustration of UAV SAR.
Figure 2
polynomial motion error.
The influence of polynomial motion error on imaging
The polynomial motion error in the normal plane of the ideal trajectory is illustrated in Figure 2. For a long azimuth exposure period, the order of the motion error is usually high due to the instability of the platform, which is shown as the black line. However, for a short azimuth exposure period, the motion error is supposed to be quadric, which is induced by a constant radial acceleration (RA). Regarding an azimuth sub-aperture, it could be rationally assumed to have a constant velocity and RA. The instantaneous slant range R (ta ; Rs ) from a target P to the platform with the presence of a RA aR is expressed as q 2 R (ta ; Rs ) = (Rs · sin φ − V · ta )2 + Rs · cos φ + 12 aR · t2a p (1) ≈ Rs2 − 2Rs · V · ta · sin φ + (V 2 + aR · Rs · cos φ) t2a ,
where Rs is the slant range at beam center crossing time, ta is the slow time (azimuth time), φ is the squint angle, V is the flight velocity, and aR is the RA. The Doppler rate of the target is influenced by the RA, which can be obtained from (1) and is expressed as 2 2V 2 cos2 φ − aR cos φ. (2) fdr e = − λRs λ In ideal case, the azimuth Doppler rate is expressed as fdr = −
2V 2 cos2 φ . λRs
(3)
The spectrum compression ratio εa is defined as the ratio of the azimuth bandwidth with the presence of RA aR to the ideal azimuth bandwidth, which is expressed as εa =
fdr e aR Rs =1+ 2 . fdr V cos φ
(4)
For a short azimuth exposure period, the influence of aR on R (ta ; Rs ) is quite small since aR ≪ Rs . However, the azimuth spectrum in the presence of the RA is seriously either compressed or stretched since the velocity is quite small. Comparing with the ideal RCM represented by the black solid line in Figure 3, the maximum and minimum of the RCM have small changes in the presence of a RA, but the RCM distribution in rangeDoppler domain changes significantly; thus the RCM distortion occurs, which is shown by the black dashed and dashdot lines in Figure 3. In Figure 3, Ba0 is the ideal bandwidth, while Ba1 and Ba2 are the compressed and stretched bandwidths, respectively. Due to the RCM distortion induced by the RA, a residual RCM appears when the RCMC is implemented, which is expressed as Rs cos φ 2 , λfa 1 − 2V
∆RaR (fa ; Rs ) = Rrd (fa , aR ; Rs ) − r
(5)
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where fa is the azimuth frequency, and Rrd (fa , aR ; Rs ) is the RCM in the range-Doppler domain in the presence of the RA aR ,which is expressed as v u u 1 − ( √ 2 V sin φ )2 V +aR Rs cos φ u . (6) Rrd (fa , aR ; Rs ) = Rs t )2 1 − ( √ 2 λfa 2
V +aR Rs cos φ
When only the low order terms are considered, the ideal RCM is expressed as R0 (fa ; Rs ) = Rs
sin φ 1+ cos2 φ
2 ! λfa 1 + 2 sin2 φ λfa − sin φ + − sin φ 2V cos4 φ 2V
= Rs + Rlinear (fa ; Rs ) + Rquad (fa ; Rs ) , where Rlinear (fa ) and Rquad (fa ) are the linear and quadratic components of the ideal RCM, respectively. Similarly, the residual RCM is expressed as 1 − 1 Rlinear (fa ; Rs ) ∆RaR (fa ; Rs ) = εa 1 1 1 1 Rquad (fa ; Rs ) . (7) − 2 −1 + + ε2a εa 1 + 3 tan2 φ εa Since the azimuth bandwidth is compressed/stretched by a factor of εa , the maximum residual RCM in the range-Doppler domain is expressed as εa − 1 2 ∆Rmax (fa ; Rs ) = (1 − εa ) Rlinear max (fa ; Rs ) + 1 − εa + Rquad max (fa ; Rs ) , (8) 1 + 3 tan2 φ
where Rlinear max (fa ; Rs ) and Rquad max (fa ; Rs ) are the maximum of the linear and quadratic components of the ideal RCM, respectively. Thus, the maximum residual RCM after RCMC is expressed as ∆RaR
max
= (1 − εa ) V sin φ · Ta sub 2 V · 1 + 2 sin2 φ εa − 1 2 + 1 − εa + · (Ta − Ta sub ) · Ta sub , Rs 1 + 3 tan2 φ
(9)
where Ta sub is the time length of the azimuth sub-aperture and Ta is the synthetic aperture time. For a high resolution UAV SAR equipped over a small platform, the spectrum compression ratio is always much greater or smaller than one, thus the residual RCM spans over a large number of range bins. In the conventional MOCO method, a range down-sampling process is utilized to remove the influence of the residual RCM on the precise motion error estimation of the subsequent autofocus algorithms, and a range down-resolution process always accompanies the range down-sampling process since the typical range oversampling rate is just 1.2, which inherently has an energy loss of the signal. Therefore, the range down-sampling rate should be close to negligible. However, the residual RCM is so large that it still spans over many range bins even if a reasonable range down-sampling process is operated, which significantly degrades the motion error estimation accuracies of the subsequent autofocus algorithms. Thus, the SAR image can hardly be precisely focused by the conventional MOCO method. 2.2
The influence of sinusoidal motion error on imaging
Due to the small size of the light UAV platform, high frequency motion error is induced by air turbulence and mechanical vibration. Suppose that the high frequency motion error can be described by a sinusoidal function which is expressed as ∆r (ta ) = ae sin (2πfe ta ) , (10) where ae is the amplitude, and fe is the frequency of the motion error.
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Ideal RCM after RCMC Replica RCM I
Ba2
Ideal RCM
R
fa
Ba0
Ideal RCMC line Replica RCM after RCMC
Ba1 fa
R
Figure 3 Comparison of RCMs with the presence of different RAs.
Figure 4 The residual RCM after RCMC with the presence of the sinusoidal motion error.
As described in [19], the aberration term induced by the motion errors after the range Fourier transform is expressed as E (fr , ta ) = exp {−j4π (f0 + fr ) ∆r (ta ) /c} =
+∞ X
k=−∞
Ck j k Jk (A (fr )) cos (2πkfe ta − kπ/2) ,
(11)
where C0 = 1, Ck = 1/2 (k 6= 0), Jk (A (fr )) is the kth-order first kind Bessel function, f0 is the carrier frequency, fr is the range frequency, and A (fr ) = (f0 + fr ) ae . In general, the dependence of A (·) on fr can be neglected. The signal spectrum in the range-Doppler domain is expressed as Srd e (tr , fa ; Rs ) =
+∞ X
k=−∞
Ck′ j|k|−k J|k| (Ac ) Srd (tr , fa − kfe ) ,
(12)
where Srd (tr , fa ) is the spectrum with respect to the ideal trajectory. Eq. (12) states that, with the presence of the sinusoidal motion error, the raw data spectrum in the range-Doppler domain consists of a summation of shifted versions of the spectrum with respect to the ideal trajectory. Therefore, if the received raw data is assessed by the standard imaging algorithm, either by RD algorithm or CS algorithm, the paired echoes will appear in the azimuth. Amplitudes of the echoes depend upon the weighting factors J (Ac ), and spatial separation of that is related to the frequency of the sinusoidal deviation. After range compression (RC), RCMC and second range compression (SRC), a shifted version of the ideal spectrum is expressed as cos φ 2Rs cos φ Srd k (tr , fa ; Rs ) = A′1 Sinc B tr − − +1 c D (fa − kfe , V ) D (fa , V ) 4πRs cos φD (fa − kfe , V ) f0 , (13) ·Wa (fa − kfe ) · exp −j c where D (fa , V ) =
s
1−
c2 fa2 , 4V 2 f02
sinc(x) = sin(πx)/(πx),
(14) (15)
and B is the bandwidth of the range pulse. Comparing with the ideal spectrum, aberration arises in the range envelope, which is expressed as 1 1 . (16) − ∆Re = Rs cos φ · D (fa − kfe ) D (fa )
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It can be inferred from (16) that the range envelope aberration ∆Re of the spectral replica increases with fe and varies with fa . The residual RCM with the presence of the sinusoidal motion error after RCMC is illustrated in Figure 4. The black solid lines in Figure 4 represent the ideal RCMs before and after RCMC, while the black dashed lines represent the RCMs of the replica before and after RCMC. It should be noted that the black dashdot line represents the ideal RCMC line, which is symmetrical with the ideal RCM line. It can be found from Figure 4 that the spectral replica are shifted into different range bins after RCMC, which is considered by a range down-sampling operation since the range shift of the spectral replica varies with fa .
3
The improved motion compensation method for UAV SAR imaging
The phase error and NRCM are induced into the azimuth signal by the motion error of UAV SAR. The phase error is always estimated by the PGA-based autofocus algorithms due to the instability of the platform, while a partial NRCM is removed using the measurements of the navigation system, and the residual NRCM is ignored directly or by a range down-sampling process. However, the high precision navigation system can neither be equipped onboard the UAV SAR due to the weight and cubage limitation, nor can an available coarse information from the navigation system be obtained. Thus, a large residual RCM is induced after RCMC, which is analyzed in Section 2. Considering the inherent drawback of energy loss of the signal, the residual RCM still spans over many range bins after a reasonable range down-sampling process, and significantly influences the estimation accuracy of the PGA-based autofocus algorithms. Therefore, an improved MOCO method is proposed to ensure the motion error estimation accuracy. The flowchart of the proposed method is shown in Figure 5. 3.1
Coarse phase error estimation
The coarse phase error estimation process consists of 3 main operations, namely RWC, range downsampling and range-invariant phase error estimation. It is the first two operations that ensure the estimation accuracy of the range-invariant phase error. 3.1.1 Range walk correction It is demonstrated in Section 2 that the range envelope aberration induced by the motion error is small, while the range-Doppler domain signal spectrum is seriously compressed or stretched in the presence of the motion errors. Therefore, a 2-D time domain RWC operation is utilized to replace the rangeDoppler domain RCMC operation in the coarse phase error estimation process, which induces much smaller residual RCM that can be ignored by a slight range down-sampling operation. 3.1.2 Range down-sampling The instantaneous slant range expressed by (1) can be expanded as 1 V2 + aR cos φ t2a . R (ta ; Rs ) ≈ Rs − V sin φ · ta + 2 Rs
(17)
Since the effective squint angles of the targets change with the azimuth positions, the residual RCMs of the targets at different azimuth positions are different after the time domain RWC, which is shown by the black solid lines in Figure 6. Therefore, the maximum residual RCM after RWC is expressed as 2 2 β V cos2 φ Ta sub − sin φ + + aR cos φ , (18) ∆Rmax (Rs , φ) = V Ta sub sin φ + 2 Rs 8 where β is the azimuth beamwidth.
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Raw data
Range compression
Azimuth blocking Range envelope correction and Full-aperture coarse phase error correction
Rrange envelope correction and coarse phase correction
RWC
RCMC
RCMC
Range down-sampling Range blocking Azimuth de-chirping
Full-aperture fine phase error correction
Azimuth de-chirping WPGA LML-WPGA
Azimuth compression Fine phase error
Coarse phase error
Final well-focused SAR image
Full-aperture phase fusion
Full-aperture coarse and fine phase errors Figure 5
Flowchart of the improved motion compensation method for UAV SAR imaging.
V Equidistance
Ta_sub
Rs
P2
P1
P3
RCM Figure 6
The residual RCMs of the targets at different azimuth positions after RWC.
It should be noted that the time length of the azimuth sub-aperture Ta sub is considerably short and εa is much smaller or larger than one in UAV SAR data processing due to the instability of the platform. The residual RCM described by (9) is rewritten as follows: ∆RaR
max
= (1 − εa ) V sin φ · Ta sub 2 V · 1 + 2 sin2 φ εa − 1 · (Ta − Ta sub ) Ta sub . + 1 − ε2a + Rs 1 + 3 tan2 φ
(19)
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In the low-squint case, the residual RCM is dominated by the quadratic component. It is obvious that the quadratic component of the residual RCM described by (19) is always larger than that of (18). In the high-squint case, the residual RCM is dominated by the linear component and sin φ > sin(φ + β2 ) − sin φ. It should also be noted that the value of sin φ increases quickly with the squint angle φ, unlike that of the value of sin(φ + β2 ) − sin φ. Hence the residual RCM described by (19) is much larger than that of (18). Therefore, the residual RCM described by (19) is always larger than that of (18). It becomes necessary to ensure that the maximum residual RCM after RWC is not greater than a range bin, considering which the maximum residual RCM should meet the condition that ∆RaR
max
= (1 − εa ) V sin φ · Ta sub 2 V · 1 + 2 sin2 φ εa − 1 + 1 − ε2a + · (Ta − Ta sub ) Ta sub . Rs 1 + 3 tan2 φ
(20)
In the high resolution UAV SAR, the residual RCM after RWC is evidently larger than a range bin due to the large azimuth beam width and high range resolution. Therefore, the range down-sampling is needed. It must be noted that the range down-sampling rate in the proposed MOCO method is much less than that of the conventional MOCO method since the residual RCM after RWC is much smaller than that after RCMC, which is demonstrated above, thus causing much less energy loss of the signal and ensuring the estimation accuracy of the subsequent PGA-based autofocus algorithms. 3.1.3 Range-invariant phase error estimation After RWC and range down-sampling, the energy of the same target is assembled in the same range bin, and the range-invariant phase error can be estimated by the WPGA. The azimuth signal is de-chirped before the estimation. After the estimation, the range-invariant phase errors estimated from the subapertures should be fused into the full-aperture coarse phase error, which is used to correct the range envelope aberration and compensate for the range-invariant phase error in full-aperture SAR imaging. In order to obtain the phase error in full aperture precisely, there should be an overlap between the adjective sub-apertures. The WPGA kernel [12] is given by b˙ WML ϕ (h) = arg e
K X wk · [conj (s (k, h)) · s (k, h + 1)] , PK j=1 wj k=1
h = 1, 2, . . . , J,
(21)
b˙ e is the estimated phase gradient, K is the number of selected range bins, J + 1 is the azimuth where ϕ length of samples, conj (·) is the conjugate operator, wk is the weight in the k th range bin, and s (k, :) is the signal of the kth selected range bin. 3.2
Fine phase error estimation
When the range-invariant phase error in the sub-aperture is obtained by WPGA, the range envelope aberration can be corrected and the range-invariant phase error can be compensated for, which ensures that NRCM is absent and the energy of the same target can be assembled into the same range bin after RCMC. The estimation accuracy of the range-variant phase error is also guaranteed since the rangeinvariant phase error is compensated for before the azimuth de-chirping. The range-variant phase error is estimated by LML-WPGA. The data after RCMC is split into B blocks in range, and G range bins b˙ ′e (b, :). The invariant phase are selected in each range block to estimate the invariant phase gradient ϕ gradient is given by G X mb,g · [conj (sb (g, h)) · sb (g, h + 1)] b˙ ′e (b, h) = arg , (22) ϕ PG g=1 j=1 mb,j
where mb,g denotes the signal-to-clutter ratio (SCR) weight of the gth sample bin in the bth range block.
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Simulation parameters
Table 1 Parameter
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Symbol
Value
Wave band
Unit
Ku PRT
1
ms
Beam width
β
2
deg
Squint angle
θ
10
deg
Bandwidth
B
1
GHz
Velocity
V
14
m/s
Range sampling rate
fs
1.4
GHz
200
200
220
220
240
240 Azimuth (Hz)
Azimuth (Hz)
Pulse repeat interval
260 280
260 280
300
300
320
320
340
−600 −400 −200 0 200 Range (MHz)
400
600
340
−600 −400 −200 0 200 Range (MHz)
(a) Figure 7
400
600
(b) 2
2-D spectrums influenced by different RAs. (a) The −0.2 m/s RA; (b) the 0 m/s2 RA.
When the invariant phase gradients in each range blocks are obtained, the range-variant phase error is estimated by the LML-WPGA kernel. Details for the same can be found in [14]. It should be noted that the RWC operation in the coarse phase error estimation process guarantees the estimation accuracy of the range-invariant phase error, which subsequently ensures the estimation accuracy of the range-variant phase error. On comparison with the conventional MOCO method, it is found that the RWC operation in the coarse phase error estimation process is the main improvement of the improvised MOCO method, which guarantees the estimation accuracy of the whole MOCO method.
4
Computer simulations and experiments based on real data
To demonstrate the analyses on the above, computer simulations and experiments based on real data are performed. The simulation parameters are summarized in Table 1. 4.1
Simulation for influence of constant RA on UAV SAR imaging
The simulated echo is received from a point target at a center slant range of 890.8 m. The added constant 2 2 RA is −0.2 m/s . The influence of the −0.2 m/s RA on the 2-D spectrum is illustrated in Figure 7. It 2 is clear that the signal spectrum with the presence of the −0.2 m/s RA is severely compressed. The spectrums after RCMC are shown in Figure 8. It is clear that the energy of the signal spectrum with the presence of the RA of −0.2 m/s2 spills in many range bins after RCMC. The residual RCM is 4.8m and spans over 48 range bins, which is in accordance to the result calculated from (10). 2 The imaging results are shown in Figure 9. The imaging result with the presence of a −0.2 m/s RA is shown in Figure 9(a), which is poorly focused. The energy of the point target spills in many range bins, which corresponds to the residual RCM after RCMC in the rang-Doppler domain.
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200
200
220
220
240
240 Azimuth (Hz)
Azimuth (Hz)
Fan B K, et al.
260 280
260 280
300
300
320
320
340 887 888 889 890 891 892 893 894 895 Range (m)
340 887 888 889 890 891 892 893 894 895 Range (m)
(a)
(b)
Figure 8 The range-Doppler domain spectrums influenced by different RAs after RCMC. (a) The −0.2 m/s2 RA; (b) the 0 m/s2 RA.
−0.5 −0.4 −0.10
−0.2
−0.05 Azimuth (Hz)
Azimuth (Hz)
−0.3
−0.1 0 0.1 0.2
0 0.05
0.3 0.10
0.4 887 888 889 890 891 892 893 894 895 Range (m)
(a) Figure 9
4.2
887 888 889 890 891 892 893 894 895 Range (m)
(b)
Simulation of SAR imaging influenced by different RAs. (a) The −0.2 m/s2 RA; (b) the 0 m/s2 RA.
Simulation for influence of sinusoidal motion error on UAV SAR imaging
The simulated echo is received from a point target at a center slant range of 890.8 m. The amplitude and frequency of the simulated sinusoidal motion error are 0.002 m and 25 Hz, respectively. The echo after range compression is shown in Figure 10(a), the motion error cannot be seen because its amplitude is much smaller than the range bin. However, the spectral replicas appear after the azimuth Fourier transform, which is shown in Figure 10(b). The spectral replicas are shifted into different range bins after RCMC, as is illustrated by Figure 10(c). The shifted spectral replicas produce the “ghost targets” in the final image, as shown in Figure 10(d). 4.3
Experiments based on real data
Experiments based on real data are also performed to validate the improved MOCO method. The raw data is recorded by a high resolution UAV SAR with a quite slow flight velocity. The velocity of the UAV is about 15 m/s, the bandwidth of the transmitted signal is 1 GHz, and the flight height is about 300 m. During the processing, the time length of each azimuth sub-aperture is 0.5 s, and a 50% overlap is taken. After RWC, the range resolution and range sampling rate are reduced with a factor of 5 simultaneously. The phase gradients estimated from the sub-apertures are fused to compensate for the motion error of the full-aperture data, which is shown in Figure 11. The phase gradient estimated by the improved MOCO method is represented by the blue line, while the one estimated by the conventional MOCO
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50
0.8
45
0.6
40
0.4
35
0.2
350
−0.2 −0.4
25
−0.6
15
20
−0.8
900
300
30
0
1100 1000
Azimuth (Hz)
1.0
Azimuth (s)
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800 700
250
600 500 400
200
300
10
−1.0 887 888 889 890 891 892 893 894 895 Range (m)
200
5
150 887 888 889 890 891 892 893 894 895 Range (m)
(a)
100
(b)
350
2500
×104 2.5
0.20 0.15
2000
250
0.10 Azimuth (s)
Azimuth (Hz)
300
1500
200
2.0
0.05 0
1.5
−0.05 −0.10
1000
1.0
−0.15 −0.20
150 500 887 888 889 890 891 892 893 894 895 Range (m)
887 888 889 890 891 892 893 894 895 Range (m)
(c)
0.5
(d)
Phase gradient (rad)
Figure 10 Simulation of SAR imaging influence by the sinusoidal motion error. (a) The signal after the range compression; (b) the range-Doppler domain spectrum before RCMC; (c) the range-Doppler domain spectrum after RCMC; (d) the imaging result.
Figure 11
1.0 0.8 0.6 0.4 0.2 0 −0.2 −0.4 −0.6 −0.8 −1.0
The improved method The conventional method
0
1
2
3 Time (s)
4
5
6
The estimated phase gradients with different MOCO methods.
method is represented by the red line. It can be found that the improved MOCO method can precisely estimate the sinusoidal motion error, unlike the conventional one. After RCMC, the range compressed signals in the 2-D time domain that obtained by the conventional MOCO method and the improved MOCO method are shown in Figure 12. It can be found that the quite large residual RCM in Figure 12(a) can be accurately corrected by the improved MOCO method, which is shown in Figure 12(b).
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−1.5
−1.5
−1.0
−1.0
−0.5
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Azimuth (s)
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0 0.5
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1.0
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735
740 Range (m)
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745
(a)
740 Range (m)
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(b)
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Azimuth
Figure 12 The RCMC results obtained by different MOCO methods. (a) The conventional MOCO method; (b) the improved MOCO method.
Range
Range (a)
(b)
Figure 13 The imaging result obtained by different MOCO methods. (a) The conventional MOCO method; (b) the improved MOCO method. Table 2 Method
Performance comparison Normalized amplitude of the “ghost target” (dB)
Conventional MOCO method
−14.97
Improved MOCO method
−32.57
The imaging result obtained by the conventional MOCO method is shown in Figure 13(a), in which the SAR image is a little out of focus, and “ghost targets” appear in the azimuth. The imaging result obtained by the improved MOCO method is shown in Figure 13(b), in which the SAR image is well focused and the “ghost targets” are suppressed to the level of the background clutter. The method proposed in [14] is the representation of the conventional MOCO method used for real data processing in this section. The performance comparison is given by Table 2.
5
Conclusion
In this paper, the influences of the polynomial and sinusoidal motion errors on UAV SAR imaging are analysed in detail, and an improved MOCO method is proposed for the high resolution UAV SAR imaging without the requirement of high precision INS data. The improved MOCO method firstly estimates the range-invariant phase error by the coarse phase error estimation process, and then finds the range-variant
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phase error by the fine phase error estimation process. The RWC operation in the coarse phase error estimation process is the main improvement in the proposed method, which guarantees the estimation accuracy of the whole MOCO method. Computer simulations and experiments based on real data have demonstrated that the improved MOCO method can obtain an enhanced precision in the estimation of the motion error than the conventional MOCO method with respect to the UAV SAR imaging.
Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos. 61370017, 61032009, 61225005, 61120106004).
References 1 Lin Y G, Zhang B C, Jiang H, et al. Multi-channel SAR imaging based on distributed compressive sensing. Sci China Inf Sci, 2012, 55: 245–259 2 Liu Y, Wu Q S, Sun G C, et al. Parameter estimation of moving targets in the SAR system with a low PRF sampling rate. Sci China Inf Sci, 2012, 55: 337–347 3 Carrara W G, Goodman R S, Majewski R M. Spotlight Synthetic Aperture Radar: Signal Processing Algorithm. Boston: Artech House, 1995 4 Xing M D, Jiang X W, Wu R B, et al. Motion compensation for UAV SAR based on raw radar data. IEEE Trans Geosci Remot Sens, 2009, 47: 2870–2883 5 Oliver C, Quegan S. Understanding Synthetic Aperture Radar Images. Norwood: Artech House, 1999 6 Franceschetti G, Lanari R. Synthetic Aperture Radar Processing. Boca Raton: CRC Press, 1999 7 Samczynski P, Pietrzyk G, Gorzelanczyk A. Coherent map drift technique. IEEE Trans Geosci Remot Sens, 2010, 48: 1505–1517 8 Gonz´ alez-Partida J T, Almorox-Gonz´ alez P, Burgos-Garcia M. SAR system for UAV operation with motion error compensation beyond the resolution cell. Sensors, 2008, 8: 3384–3405 9 Wahl D E, Eichel P H, Ghiglia D C, et al. Phase gradient autofocus-a robust tool for high resolution phase correction. IEEE Trans Aerosp Electron Syst, 1994, 30: 827–835 10 Chan H L, Yeo T S. Noniterative quality phase-gradient autofocus (QPGA) algorithm for spotlight SAR imagery. IEEE Trans Geosci Remot Sens, 1998, 36: 1531–1539 11 Ye W, Yeo T S, Bao Z. Weighted least-squares estimation of phase errors for SAR/ISAR autofocus. IEEE Trans Geosci Remot Sens, 1999, 37: 2487–2494 12 Cˆ amara de Macedo K A, Scheiber R, Moreira A. An autofocus approach for residual motion errors with application to airborne repeat-pass SAR interferometry. IEEE Trans Geosci Remot Sens, 2008, 46: 3151–3162 13 Li Y K, Liu C, Wang Y F, et al. A robust motion error estimation method based on raw data. IEEE Trans Geosci Remot Sens, 2012, 50: 2780–2790 14 Zhang L, Qiao Z J, Xing M D, et al. A robust motion compensation approach for UAV SAR imagery. IEEE Trans Geosci Remot Sens, 2012, 50: 3202–3218 15 Prats P, Cˆ amara de Macedo K A, Reigber A, et al. Comparison of topography- and aperture-dependent motion compensation algorithms for airborne SAR. IEEE Geosci Remot Sens Lett, 2007, 4: 349–353 16 Cumming I G, Wong F H. Digital Processing of Synthetic Aperture Radar Data: Algorithms and Implementation. Boston: Artech House, 2005. 169–221 17 Zeng T, Liu L S, Ding Z G. Improved stepped-frequency SAR imaging algorithm with the range spectral-length extension strategy. IEEE J Sel Top Appl Earth Obs Remot Sens, 2012, 5: 1483–1494 18 Raney R K, Runge H, Bamler R, et al. Precision SAR processing using chirp scaling. IEEE Trans Geosci Remot Sens, 1994, 32: 786–799 19 Fornaro G. Trajectory deviations in airborne SAR: analysis and compensation. IEEE Trans Aerosp Electron Syst, 1999, 35: 997–1005
. RESEARCH PAPER .
December 2014, Vol. 57 122301:1–122301:13 doi: 10.1007/s11432-014-5189-2
An improved motion compensation method for high resolution UAV SAR imaging FAN BangKui1,2 , DING ZeGang1 * , GAO WenBin1 & LONG Teng1 1School
of Information and Electronics, Beijing Institute of Technology, Beijing 100081, China; 2Beijing Institute of Information Technology, Beijing 102401, China Received May 1, 2014; accepted July 2, 2014
Abstract The polynomial and sinusoidal motion errors always exist in the unmanned aero vehicle (UAV) SAR due to the small size and low velocity of the platform, causing serious spectrum compressing/stretching and significant spectral replicas of the azimuth signal. The motion errors induce serious blurring of the SAR image and “ghost targets”, and can hardly be precisely estimated by the conventional motion compensation (MOCO) method. In this paper, an improved MOCO method is proposed to estimate and eliminate the motion errors in the high resolution UAV SAR without high precision inertial navigation system (INS) data. The time domain range walk correction (RWC) operation in the coarse phase error estimation process of the proposed MOCO method is the key operation that ensures the estimation accuracy of the whole MOCO method. Finally, the validity of the improved MOCO method is verified by computer simulations and real UAV SAR data processing. Keywords motion error, unmanned aero vehicle (UAV) SAR, motion compensation (MOCO), range walk correction (RWC), high resolution
Citation Fan B K, Ding Z G, Gao W B, et al. An improved motion compensation method for high resolution UAV SAR imaging. Sci China Inf Sci, 2014, 57: 122301(13), doi: 10.1007/s11432-014-5189-2
1
Introduction
Due to the improvement of hardware and MOCO methods, SAR has been greatly developed [1,2] and the resolution of the SAR image is progressively improved in the last decades. As highlighted in [3], the platform motion is the solution and also the problem for SAR imaging. Motion error of the SAR platform from the ideal track causes the phase error in the Doppler history of the illuminated target, which induces the blurring of the final SAR image. The influence of the phase error on SAR imaging is particularly serious in the UAV SAR system because the platform is quite small and gets seriously affected by the atmospheric turbulences. In general, the accuracy of navigation systems is not enough to achieve a high accuracy MOCO. Moreover, the high precision navigation system cannot be equipped onboard the UAV SAR due to the weight and cubage limitation. Hence, the autofocus technique estimating the phase error from the raw data is a must for the UAV SAR imaging. Lately, many MOCO methods have been proposed to estimate the motion error from the raw data in UAV SAR. In [4], a 3-D MOCO method based on the instantaneous Doppler rate (DR) estimation is * Corresponding author (email: [email protected])
c Science China Press and Springer-Verlag Berlin Heidelberg 2014
info.scichina.com
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proposed. This method extracts the necessary motion parameters, i.e., forward velocity and displacement in line-of-sight direction, from the raw data via the map-drift techniques [5–7]. The DRs from different range blocks and azimuth sub-apertures are estimated and the double integral of the DRs are used to obtain the motion error. For the UAV SAR imaging, a number of iterations are needed to reach an acceptable accuracy due to the instability of the platform, resulting in cumbersome computation. Another MOCO method based on the phase gradient autofocus algorithm (PGA) [8–12] is also proposed. In [11], the weighted phase estimation kernel is proposed for the stripmap mode SAR. However, it only accounts for the range-invariant motion error; hence it provides poor estimation results for the UAV SAR due to the range variation of the motion error. Therefore, the MOCO method accounting for the range variation of the motion error are proposed based on PGA. In [13], the weighted total least square (WTLS) estimation kernel is proposed to estimate the motion error varying with the look angle. In [14], the raw data is split both in range and azimuth directions, and a weighted and local maximum-likelihood (LML) PGA kernel is proposed to estimate the range-variant motion error. In the MOCO methods described above, the influence of the non-systematic range cell migration (NRCM) [14] on the estimation accuracy of the autofocus algorithms is neglected in most of the cases. In [14], the influence of the NRCM on the estimation accuracy of the autofocus algorithms is ignored by a range down-sampling process, and the influence of the NRCM is ignored directly in [13]. However, in UAV SAR, due to the low flight velocity, the motion error seriously affects the signal’s spectrum. The NRCM after range-Doppler domain RCMC is quite large and spans over a large number of range bins. Therefore, after the range-Doppler domain RCMC [15–18] and a reasonable range down-sampling, the target’s energy still spills in many range bins such that the target can hardly get precisely focused by the conventional MOCO method. Moreover, the sinusoidal motion error [19] produces spectra replicas of the azimuth signal in the azimuth-frequency domain. The spectra replicas are shifted into different range bins after range-Doppler domain RCMC, which induces the “ghost targets” in different range bins in the final image. Thus, the sinusoidal motion error can hardly be estimated by the conventional MOCO method. In this paper, RCMC stands for the range-Doppler domain RCMC where no other alternative is mentioned. In this paper, the influences of the polynomial and sinusoidal motion errors are analysed in detail, and an improved MOCO method is proposed for the high resolution UAV SAR imaging. In order to account for the aforementioned difficulties encountered in the high resolution UAV SAR imaging, a twodimensional (2-D) time domain RWC is used and a range down-sampling process is performed to ensure that the residual RCM after RWC is not larger than a range bin. The improved MOCO method firstly estimates the range-invariant phase error by the coarse phase error estimation process, and then the range-variant phase error by the fine phase error estimation process. The RWC operation in the coarse phase error estimation process guaranteed the estimation accuracy of the range-invariant phase error, which subsequently ensures the accurate estimation of the range-variant phase error. The paper is organized as follows. Section 2 is dedicated to a detailed analyse of the influences of the polynomial and sinusoidal motion errors on the high resolution UAV SAR imaging. Section 3 discusses the improved MOCO method for high resolution UAV SAR imaging, with its flow chart. Section 4 presents the computer simulations and the experiments based on real data, and Section 5 concludes this paper briefly.
2
The influence of motion error on UAV SAR imaging
The motion geometry illustration of UAV SAR is shown in Figure 1. X-axis denotes the nominal flight direction, and Y -Z plane denotes the normal plane of the ideal trajectory. The black point in the ideal track are the ideal uniform sampling points and these in the actual track are the real sampling points. The vector from the ideal sampling point to the real sampling point is the motion error. The motion error is decomposed into three components which are ∆x, ∆l and ∆s. ∆x is the along-track component, ∆l is the polynomial cross-track component, and ∆s is the sinusoidal cross-track component.
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Z
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Actual track X (ta) Ideal track
Y
Azimuth block ∆s
Rs
ϕ
∆x
Quadric track
P Figure 1
2.1
Actual track
∆l
The motion geometry illustration of UAV SAR.
Figure 2
polynomial motion error.
The influence of polynomial motion error on imaging
The polynomial motion error in the normal plane of the ideal trajectory is illustrated in Figure 2. For a long azimuth exposure period, the order of the motion error is usually high due to the instability of the platform, which is shown as the black line. However, for a short azimuth exposure period, the motion error is supposed to be quadric, which is induced by a constant radial acceleration (RA). Regarding an azimuth sub-aperture, it could be rationally assumed to have a constant velocity and RA. The instantaneous slant range R (ta ; Rs ) from a target P to the platform with the presence of a RA aR is expressed as q 2 R (ta ; Rs ) = (Rs · sin φ − V · ta )2 + Rs · cos φ + 12 aR · t2a p (1) ≈ Rs2 − 2Rs · V · ta · sin φ + (V 2 + aR · Rs · cos φ) t2a ,
where Rs is the slant range at beam center crossing time, ta is the slow time (azimuth time), φ is the squint angle, V is the flight velocity, and aR is the RA. The Doppler rate of the target is influenced by the RA, which can be obtained from (1) and is expressed as 2 2V 2 cos2 φ − aR cos φ. (2) fdr e = − λRs λ In ideal case, the azimuth Doppler rate is expressed as fdr = −
2V 2 cos2 φ . λRs
(3)
The spectrum compression ratio εa is defined as the ratio of the azimuth bandwidth with the presence of RA aR to the ideal azimuth bandwidth, which is expressed as εa =
fdr e aR Rs =1+ 2 . fdr V cos φ
(4)
For a short azimuth exposure period, the influence of aR on R (ta ; Rs ) is quite small since aR ≪ Rs . However, the azimuth spectrum in the presence of the RA is seriously either compressed or stretched since the velocity is quite small. Comparing with the ideal RCM represented by the black solid line in Figure 3, the maximum and minimum of the RCM have small changes in the presence of a RA, but the RCM distribution in rangeDoppler domain changes significantly; thus the RCM distortion occurs, which is shown by the black dashed and dashdot lines in Figure 3. In Figure 3, Ba0 is the ideal bandwidth, while Ba1 and Ba2 are the compressed and stretched bandwidths, respectively. Due to the RCM distortion induced by the RA, a residual RCM appears when the RCMC is implemented, which is expressed as Rs cos φ 2 , λfa 1 − 2V
∆RaR (fa ; Rs ) = Rrd (fa , aR ; Rs ) − r
(5)
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where fa is the azimuth frequency, and Rrd (fa , aR ; Rs ) is the RCM in the range-Doppler domain in the presence of the RA aR ,which is expressed as v u u 1 − ( √ 2 V sin φ )2 V +aR Rs cos φ u . (6) Rrd (fa , aR ; Rs ) = Rs t )2 1 − ( √ 2 λfa 2
V +aR Rs cos φ
When only the low order terms are considered, the ideal RCM is expressed as R0 (fa ; Rs ) = Rs
sin φ 1+ cos2 φ
2 ! λfa 1 + 2 sin2 φ λfa − sin φ + − sin φ 2V cos4 φ 2V
= Rs + Rlinear (fa ; Rs ) + Rquad (fa ; Rs ) , where Rlinear (fa ) and Rquad (fa ) are the linear and quadratic components of the ideal RCM, respectively. Similarly, the residual RCM is expressed as 1 − 1 Rlinear (fa ; Rs ) ∆RaR (fa ; Rs ) = εa 1 1 1 1 Rquad (fa ; Rs ) . (7) − 2 −1 + + ε2a εa 1 + 3 tan2 φ εa Since the azimuth bandwidth is compressed/stretched by a factor of εa , the maximum residual RCM in the range-Doppler domain is expressed as εa − 1 2 ∆Rmax (fa ; Rs ) = (1 − εa ) Rlinear max (fa ; Rs ) + 1 − εa + Rquad max (fa ; Rs ) , (8) 1 + 3 tan2 φ
where Rlinear max (fa ; Rs ) and Rquad max (fa ; Rs ) are the maximum of the linear and quadratic components of the ideal RCM, respectively. Thus, the maximum residual RCM after RCMC is expressed as ∆RaR
max
= (1 − εa ) V sin φ · Ta sub 2 V · 1 + 2 sin2 φ εa − 1 2 + 1 − εa + · (Ta − Ta sub ) · Ta sub , Rs 1 + 3 tan2 φ
(9)
where Ta sub is the time length of the azimuth sub-aperture and Ta is the synthetic aperture time. For a high resolution UAV SAR equipped over a small platform, the spectrum compression ratio is always much greater or smaller than one, thus the residual RCM spans over a large number of range bins. In the conventional MOCO method, a range down-sampling process is utilized to remove the influence of the residual RCM on the precise motion error estimation of the subsequent autofocus algorithms, and a range down-resolution process always accompanies the range down-sampling process since the typical range oversampling rate is just 1.2, which inherently has an energy loss of the signal. Therefore, the range down-sampling rate should be close to negligible. However, the residual RCM is so large that it still spans over many range bins even if a reasonable range down-sampling process is operated, which significantly degrades the motion error estimation accuracies of the subsequent autofocus algorithms. Thus, the SAR image can hardly be precisely focused by the conventional MOCO method. 2.2
The influence of sinusoidal motion error on imaging
Due to the small size of the light UAV platform, high frequency motion error is induced by air turbulence and mechanical vibration. Suppose that the high frequency motion error can be described by a sinusoidal function which is expressed as ∆r (ta ) = ae sin (2πfe ta ) , (10) where ae is the amplitude, and fe is the frequency of the motion error.
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Ideal RCM after RCMC Replica RCM I
Ba2
Ideal RCM
R
fa
Ba0
Ideal RCMC line Replica RCM after RCMC
Ba1 fa
R
Figure 3 Comparison of RCMs with the presence of different RAs.
Figure 4 The residual RCM after RCMC with the presence of the sinusoidal motion error.
As described in [19], the aberration term induced by the motion errors after the range Fourier transform is expressed as E (fr , ta ) = exp {−j4π (f0 + fr ) ∆r (ta ) /c} =
+∞ X
k=−∞
Ck j k Jk (A (fr )) cos (2πkfe ta − kπ/2) ,
(11)
where C0 = 1, Ck = 1/2 (k 6= 0), Jk (A (fr )) is the kth-order first kind Bessel function, f0 is the carrier frequency, fr is the range frequency, and A (fr ) = (f0 + fr ) ae . In general, the dependence of A (·) on fr can be neglected. The signal spectrum in the range-Doppler domain is expressed as Srd e (tr , fa ; Rs ) =
+∞ X
k=−∞
Ck′ j|k|−k J|k| (Ac ) Srd (tr , fa − kfe ) ,
(12)
where Srd (tr , fa ) is the spectrum with respect to the ideal trajectory. Eq. (12) states that, with the presence of the sinusoidal motion error, the raw data spectrum in the range-Doppler domain consists of a summation of shifted versions of the spectrum with respect to the ideal trajectory. Therefore, if the received raw data is assessed by the standard imaging algorithm, either by RD algorithm or CS algorithm, the paired echoes will appear in the azimuth. Amplitudes of the echoes depend upon the weighting factors J (Ac ), and spatial separation of that is related to the frequency of the sinusoidal deviation. After range compression (RC), RCMC and second range compression (SRC), a shifted version of the ideal spectrum is expressed as cos φ 2Rs cos φ Srd k (tr , fa ; Rs ) = A′1 Sinc B tr − − +1 c D (fa − kfe , V ) D (fa , V ) 4πRs cos φD (fa − kfe , V ) f0 , (13) ·Wa (fa − kfe ) · exp −j c where D (fa , V ) =
s
1−
c2 fa2 , 4V 2 f02
sinc(x) = sin(πx)/(πx),
(14) (15)
and B is the bandwidth of the range pulse. Comparing with the ideal spectrum, aberration arises in the range envelope, which is expressed as 1 1 . (16) − ∆Re = Rs cos φ · D (fa − kfe ) D (fa )
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It can be inferred from (16) that the range envelope aberration ∆Re of the spectral replica increases with fe and varies with fa . The residual RCM with the presence of the sinusoidal motion error after RCMC is illustrated in Figure 4. The black solid lines in Figure 4 represent the ideal RCMs before and after RCMC, while the black dashed lines represent the RCMs of the replica before and after RCMC. It should be noted that the black dashdot line represents the ideal RCMC line, which is symmetrical with the ideal RCM line. It can be found from Figure 4 that the spectral replica are shifted into different range bins after RCMC, which is considered by a range down-sampling operation since the range shift of the spectral replica varies with fa .
3
The improved motion compensation method for UAV SAR imaging
The phase error and NRCM are induced into the azimuth signal by the motion error of UAV SAR. The phase error is always estimated by the PGA-based autofocus algorithms due to the instability of the platform, while a partial NRCM is removed using the measurements of the navigation system, and the residual NRCM is ignored directly or by a range down-sampling process. However, the high precision navigation system can neither be equipped onboard the UAV SAR due to the weight and cubage limitation, nor can an available coarse information from the navigation system be obtained. Thus, a large residual RCM is induced after RCMC, which is analyzed in Section 2. Considering the inherent drawback of energy loss of the signal, the residual RCM still spans over many range bins after a reasonable range down-sampling process, and significantly influences the estimation accuracy of the PGA-based autofocus algorithms. Therefore, an improved MOCO method is proposed to ensure the motion error estimation accuracy. The flowchart of the proposed method is shown in Figure 5. 3.1
Coarse phase error estimation
The coarse phase error estimation process consists of 3 main operations, namely RWC, range downsampling and range-invariant phase error estimation. It is the first two operations that ensure the estimation accuracy of the range-invariant phase error. 3.1.1 Range walk correction It is demonstrated in Section 2 that the range envelope aberration induced by the motion error is small, while the range-Doppler domain signal spectrum is seriously compressed or stretched in the presence of the motion errors. Therefore, a 2-D time domain RWC operation is utilized to replace the rangeDoppler domain RCMC operation in the coarse phase error estimation process, which induces much smaller residual RCM that can be ignored by a slight range down-sampling operation. 3.1.2 Range down-sampling The instantaneous slant range expressed by (1) can be expanded as 1 V2 + aR cos φ t2a . R (ta ; Rs ) ≈ Rs − V sin φ · ta + 2 Rs
(17)
Since the effective squint angles of the targets change with the azimuth positions, the residual RCMs of the targets at different azimuth positions are different after the time domain RWC, which is shown by the black solid lines in Figure 6. Therefore, the maximum residual RCM after RWC is expressed as 2 2 β V cos2 φ Ta sub − sin φ + + aR cos φ , (18) ∆Rmax (Rs , φ) = V Ta sub sin φ + 2 Rs 8 where β is the azimuth beamwidth.
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Raw data
Range compression
Azimuth blocking Range envelope correction and Full-aperture coarse phase error correction
Rrange envelope correction and coarse phase correction
RWC
RCMC
RCMC
Range down-sampling Range blocking Azimuth de-chirping
Full-aperture fine phase error correction
Azimuth de-chirping WPGA LML-WPGA
Azimuth compression Fine phase error
Coarse phase error
Final well-focused SAR image
Full-aperture phase fusion
Full-aperture coarse and fine phase errors Figure 5
Flowchart of the improved motion compensation method for UAV SAR imaging.
V Equidistance
Ta_sub
Rs
P2
P1
P3
RCM Figure 6
The residual RCMs of the targets at different azimuth positions after RWC.
It should be noted that the time length of the azimuth sub-aperture Ta sub is considerably short and εa is much smaller or larger than one in UAV SAR data processing due to the instability of the platform. The residual RCM described by (9) is rewritten as follows: ∆RaR
max
= (1 − εa ) V sin φ · Ta sub 2 V · 1 + 2 sin2 φ εa − 1 · (Ta − Ta sub ) Ta sub . + 1 − ε2a + Rs 1 + 3 tan2 φ
(19)
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In the low-squint case, the residual RCM is dominated by the quadratic component. It is obvious that the quadratic component of the residual RCM described by (19) is always larger than that of (18). In the high-squint case, the residual RCM is dominated by the linear component and sin φ > sin(φ + β2 ) − sin φ. It should also be noted that the value of sin φ increases quickly with the squint angle φ, unlike that of the value of sin(φ + β2 ) − sin φ. Hence the residual RCM described by (19) is much larger than that of (18). Therefore, the residual RCM described by (19) is always larger than that of (18). It becomes necessary to ensure that the maximum residual RCM after RWC is not greater than a range bin, considering which the maximum residual RCM should meet the condition that ∆RaR
max
= (1 − εa ) V sin φ · Ta sub 2 V · 1 + 2 sin2 φ εa − 1 + 1 − ε2a + · (Ta − Ta sub ) Ta sub . Rs 1 + 3 tan2 φ
(20)
In the high resolution UAV SAR, the residual RCM after RWC is evidently larger than a range bin due to the large azimuth beam width and high range resolution. Therefore, the range down-sampling is needed. It must be noted that the range down-sampling rate in the proposed MOCO method is much less than that of the conventional MOCO method since the residual RCM after RWC is much smaller than that after RCMC, which is demonstrated above, thus causing much less energy loss of the signal and ensuring the estimation accuracy of the subsequent PGA-based autofocus algorithms. 3.1.3 Range-invariant phase error estimation After RWC and range down-sampling, the energy of the same target is assembled in the same range bin, and the range-invariant phase error can be estimated by the WPGA. The azimuth signal is de-chirped before the estimation. After the estimation, the range-invariant phase errors estimated from the subapertures should be fused into the full-aperture coarse phase error, which is used to correct the range envelope aberration and compensate for the range-invariant phase error in full-aperture SAR imaging. In order to obtain the phase error in full aperture precisely, there should be an overlap between the adjective sub-apertures. The WPGA kernel [12] is given by b˙ WML ϕ (h) = arg e
K X wk · [conj (s (k, h)) · s (k, h + 1)] , PK j=1 wj k=1
h = 1, 2, . . . , J,
(21)
b˙ e is the estimated phase gradient, K is the number of selected range bins, J + 1 is the azimuth where ϕ length of samples, conj (·) is the conjugate operator, wk is the weight in the k th range bin, and s (k, :) is the signal of the kth selected range bin. 3.2
Fine phase error estimation
When the range-invariant phase error in the sub-aperture is obtained by WPGA, the range envelope aberration can be corrected and the range-invariant phase error can be compensated for, which ensures that NRCM is absent and the energy of the same target can be assembled into the same range bin after RCMC. The estimation accuracy of the range-variant phase error is also guaranteed since the rangeinvariant phase error is compensated for before the azimuth de-chirping. The range-variant phase error is estimated by LML-WPGA. The data after RCMC is split into B blocks in range, and G range bins b˙ ′e (b, :). The invariant phase are selected in each range block to estimate the invariant phase gradient ϕ gradient is given by G X mb,g · [conj (sb (g, h)) · sb (g, h + 1)] b˙ ′e (b, h) = arg , (22) ϕ PG g=1 j=1 mb,j
where mb,g denotes the signal-to-clutter ratio (SCR) weight of the gth sample bin in the bth range block.
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Simulation parameters
Table 1 Parameter
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Symbol
Value
Wave band
Unit
Ku PRT
1
ms
Beam width
β
2
deg
Squint angle
θ
10
deg
Bandwidth
B
1
GHz
Velocity
V
14
m/s
Range sampling rate
fs
1.4
GHz
200
200
220
220
240
240 Azimuth (Hz)
Azimuth (Hz)
Pulse repeat interval
260 280
260 280
300
300
320
320
340
−600 −400 −200 0 200 Range (MHz)
400
600
340
−600 −400 −200 0 200 Range (MHz)
(a) Figure 7
400
600
(b) 2
2-D spectrums influenced by different RAs. (a) The −0.2 m/s RA; (b) the 0 m/s2 RA.
When the invariant phase gradients in each range blocks are obtained, the range-variant phase error is estimated by the LML-WPGA kernel. Details for the same can be found in [14]. It should be noted that the RWC operation in the coarse phase error estimation process guarantees the estimation accuracy of the range-invariant phase error, which subsequently ensures the estimation accuracy of the range-variant phase error. On comparison with the conventional MOCO method, it is found that the RWC operation in the coarse phase error estimation process is the main improvement of the improvised MOCO method, which guarantees the estimation accuracy of the whole MOCO method.
4
Computer simulations and experiments based on real data
To demonstrate the analyses on the above, computer simulations and experiments based on real data are performed. The simulation parameters are summarized in Table 1. 4.1
Simulation for influence of constant RA on UAV SAR imaging
The simulated echo is received from a point target at a center slant range of 890.8 m. The added constant 2 2 RA is −0.2 m/s . The influence of the −0.2 m/s RA on the 2-D spectrum is illustrated in Figure 7. It 2 is clear that the signal spectrum with the presence of the −0.2 m/s RA is severely compressed. The spectrums after RCMC are shown in Figure 8. It is clear that the energy of the signal spectrum with the presence of the RA of −0.2 m/s2 spills in many range bins after RCMC. The residual RCM is 4.8m and spans over 48 range bins, which is in accordance to the result calculated from (10). 2 The imaging results are shown in Figure 9. The imaging result with the presence of a −0.2 m/s RA is shown in Figure 9(a), which is poorly focused. The energy of the point target spills in many range bins, which corresponds to the residual RCM after RCMC in the rang-Doppler domain.
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Simulation for influence of sinusoidal motion error on UAV SAR imaging
The simulated echo is received from a point target at a center slant range of 890.8 m. The amplitude and frequency of the simulated sinusoidal motion error are 0.002 m and 25 Hz, respectively. The echo after range compression is shown in Figure 10(a), the motion error cannot be seen because its amplitude is much smaller than the range bin. However, the spectral replicas appear after the azimuth Fourier transform, which is shown in Figure 10(b). The spectral replicas are shifted into different range bins after RCMC, as is illustrated by Figure 10(c). The shifted spectral replicas produce the “ghost targets” in the final image, as shown in Figure 10(d). 4.3
Experiments based on real data
Experiments based on real data are also performed to validate the improved MOCO method. The raw data is recorded by a high resolution UAV SAR with a quite slow flight velocity. The velocity of the UAV is about 15 m/s, the bandwidth of the transmitted signal is 1 GHz, and the flight height is about 300 m. During the processing, the time length of each azimuth sub-aperture is 0.5 s, and a 50% overlap is taken. After RWC, the range resolution and range sampling rate are reduced with a factor of 5 simultaneously. The phase gradients estimated from the sub-apertures are fused to compensate for the motion error of the full-aperture data, which is shown in Figure 11. The phase gradient estimated by the improved MOCO method is represented by the blue line, while the one estimated by the conventional MOCO
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Figure 10 Simulation of SAR imaging influence by the sinusoidal motion error. (a) The signal after the range compression; (b) the range-Doppler domain spectrum before RCMC; (c) the range-Doppler domain spectrum after RCMC; (d) the imaging result.
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method is represented by the red line. It can be found that the improved MOCO method can precisely estimate the sinusoidal motion error, unlike the conventional one. After RCMC, the range compressed signals in the 2-D time domain that obtained by the conventional MOCO method and the improved MOCO method are shown in Figure 12. It can be found that the quite large residual RCM in Figure 12(a) can be accurately corrected by the improved MOCO method, which is shown in Figure 12(b).
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Figure 13 The imaging result obtained by different MOCO methods. (a) The conventional MOCO method; (b) the improved MOCO method. Table 2 Method
Performance comparison Normalized amplitude of the “ghost target” (dB)
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The imaging result obtained by the conventional MOCO method is shown in Figure 13(a), in which the SAR image is a little out of focus, and “ghost targets” appear in the azimuth. The imaging result obtained by the improved MOCO method is shown in Figure 13(b), in which the SAR image is well focused and the “ghost targets” are suppressed to the level of the background clutter. The method proposed in [14] is the representation of the conventional MOCO method used for real data processing in this section. The performance comparison is given by Table 2.
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Conclusion
In this paper, the influences of the polynomial and sinusoidal motion errors on UAV SAR imaging are analysed in detail, and an improved MOCO method is proposed for the high resolution UAV SAR imaging without the requirement of high precision INS data. The improved MOCO method firstly estimates the range-invariant phase error by the coarse phase error estimation process, and then finds the range-variant
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phase error by the fine phase error estimation process. The RWC operation in the coarse phase error estimation process is the main improvement in the proposed method, which guarantees the estimation accuracy of the whole MOCO method. Computer simulations and experiments based on real data have demonstrated that the improved MOCO method can obtain an enhanced precision in the estimation of the motion error than the conventional MOCO method with respect to the UAV SAR imaging.
Acknowledgements This work was supported by National Natural Science Foundation of China (Grant Nos. 61370017, 61032009, 61225005, 61120106004).
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