ICSP2014 Proceedings ISAR imaging with random missing observations based on non-iterative signal reconstruction algorithm Yang Wang, Jian Kang, Runbin Zhang
Research Institute of Electronic Engineering Technology, Harbin Institute of Technology, Harbin, China
[email protected],
[email protected],
[email protected]
ABSTRACT Random missing observations in real-world inverse synthetic aperture radar (lSAR) imaging may appear due to the instability of the radar system. Under this circumstance, the signal of one range bin over the slow-time has limited samples. Using traditional range-Doppler algorithm, high-quality ISAR images for random missing observations cannot be obtained. Recently, a new non-iterative algorithm based on the combined robust statistics and compressive sensing (CS) theory has been proposed to efficiently recover a complete signal from a small random set of samples, showing robustness in the presence of noise [16]. It is also important to emphasize that non-iterative method is computationally simpler than the iterative signal reconstruction solutions, and thus more amenable to practical applications. Therefore, based on the non-iterative robust signal reconstruction method, a new algorithm for ISAR imaging with random missing observations is proposed in this paper. The efficiency of the proposed approach is demonstrated on the examples with simulated and real data. Terms-Compressive sensing, Non-iterative algorithm, Signal reconstruction, Random observations, TSAR imaging
the L-estimation is applied [16]. This method is a non-iterative algorithm which decreases the computational complexity comparing with the existing iterative solutions (greedy algorithms). In this paper, the non-iterative signal reconstruction is applied in ISAR imaging with random missing observations, showing its efficiency when applied to simulated and real data. The paper is organized as follows. Tn Section 2, returned radar signal model is introduced. Section 3 discusses the phenomenon of random missing observations and demonstrates the performance of the proposed algorithm for ISAR imaging with random missing observations. Tn Section 4, signal simulation is provided, effectiveness and robustness of this algorithm are tested on simulated and real data. Conclusion is given in Section 5. 2.
Assume that the rotational motion of the target is stable and the motion compensation, such as range alignment and phase adjustment has been completed. Over the slow-time t in a range cell, the phase function can be written in the form [17]:
Index
1.
INTRODUCTION
TSAR imaging for moving targets can be widely applied in military and civilian fields [1-4]. By transmitting wideband signals and processing coherent azimuth echoes, high range and cross-range resolution can be obtained. In real-world ISAR imaging, the phenomenon of random missing observations always exists because of the problems that appear in sensing devices. Tn this situation, traditional TSAR imaging, using range-Doppler algorithm, cannot provide high-quality images due to the interference induced by missing echoes. Tn the past few years, CS has attracted significant attention of research community. The CS principle allows the reconstruction of signal from a very limited number of randomly chosen measurements [5-7]. Recently, the CS-based signal reconstruction approaches have been widely used in various radar applications [8-9]. For instance, the CS theory has been applied in SAR and TSAR imaging with missing observations, such as sparse aperture (SA) [10-12]. However, in many real scenarios, the circumstance of missing observations always exists due to the instability of the radar system. Currently, a new signal recovery method based on robust statistics theory [13-15] and CS has been introduced for reconstruction of complete signal from a small set of samples that remain after
978-1-4799-2186-7114/$31.00 ©2014 IEEE
RETURNED RADAR SIGNAL MODEL
z(t)=
� O";exp (j 4:fr (R(t)+x;sine(t)+y;cose(t)) J (1)
K-j
°
where fa is the carrier frequency, c is the velocity of light, 0"; is the magnitude of the i-th scatter, K represents the whole number of point scatterers in one range cell, R(t) is the uncompensated translational residual and e(t) is the rotational angle. (x;, yJ is the coordinate of the i-th scatter. During the imaging interval, we can assume small angle approximation for the target, such as x;sin e(t) � x;e(t) and y;cose(t) � y;. R(t) can be expanded as R(t) � Ro +vt , where Ro is the initial range. Therefore, over the slow-time in a range cell, the signal has the form of the sinusoidal signal: K-j (2) z(t)= L O"; exp(j(ao;+ajJ)) i=O
where ao; denotes the initial phase term and aj; denotes the angular frequency.
1876
3.2. New algorithm/or ISAR imaging with random missing observations
According to the sinusoidal signal recovery method [16], we propose the new algorithm for ISAR imaging with random missing observations. It can be summarized through the following steps: Step 1. Compress the limited received signals in the range domain applying the Fourier transform after motion compensation. Step 2. For each range bin, the signal with limited echoes has the form of (3). Compute the following equation for each nq , where nq nl'nz, ..., nQ e(nq, k) z (nq ) exp(-j27rknq / N ) (4) Step 3. Compute the sum of general deviations for each frequency k 0,1, ... , N -1 with the expression: =
=
Radar
=
GD(k)
Fig. 1. ISAR imaging geometry
1
PHENOMENON OF RANDOM MISSING OBSERVATlONS AND NEW ALGORITHM FOR ISAR IMAGING WITH RANDOM MISSING OBSERVATIONS
3.
Q
=
{
dev e(n[, k), e(n2, k), ..., e(nQ'k)
l
}
=
{
- I e (np k) - mean e(n[, k), e(n2, k), ..., e(nQ, k)
}1
(5)
Q ;=[ Step 4. Tn each range bin, determine the frequency positions of the desired signal that satisfy expression: k1j arg{GD(k) < o max {GD(k)}} (6) =
3.1. Phenomenon ofrandom missing observations
After the motion compensation and range compression have been implemented, the matrix Sen, m) as the range profile with the whole pulses can be obtained. Here, n 0,1, ... , N -1 represents the pulses, while m 0,1, ... , M -1 represents the range bins. In real-world situations, there are some echoes missing and usually the missing samples are randomly chosen from the whole set. Therefore, we can obtain the range profiles with limited pulses in the matrix form S(nq, m) , where =
=
nq
=
n[, n2, ..., n(n while Q is the number of limited pulses. In
each range bin, the signal over the slow time has the following form: z(nq)
K -[ =
I i=O
0";
l
exp(j27rk i nQ / N
+
aoJ
(3)
Fig.2 shows the range profiles with limited pulses of experimental data.
where 0 is a constant threshold which can be set between 0.85 and 0.95. Step 5. Taking the detected frequency points, the signal can be recovered by the following equation: Z (A' A )-[ A' zenq ) (7) =
cs
cs
cs
where z(nq) is the vector of z(nq)' nq npn2,...,nQ in time domain, vector Z represents the Fourier transform vector corresponding to signal z and A is the CS Fourier transform matrix with Q rows corresponding to =
cs
limited pulses and K columns corresponding to the signal frequency points obtained in the step 4. Step 6. Applying the same procedure for all vectors Z (for the whole range bins), the complete ISAR image is obtained. Note that the resulting image is not affected by the effects caused by random missing observations. Additionally, we may observe that the signal recovery method is a non-iterative one, which reduces the computational cost and recovers the signal efficiently. 4.
SIMULATED AND REAL DATA RESULTS
4.1. Signal simulation
The signal simulation provides the I-D signal recovery performances of the proposed algorithm and basis pursuit (BP) method. The signal has the form: s =
Fig. 2. Range profiles of Yak-4 2 with limited pulses
1.1 exp(j327rn / N) + exp(j647rn / N) + 0.95 exp(j1287rn / N)
(8)
where N is set 128 and n is from ° to N-l. We add white Gaussian noise with SNR=-6.53db. Applying the proposed method and BP method, we recover the signal from 64 random samples and the results are demonstrated in Fig.3. From the
1877
results, the effectiveness of BP method is influenced by noise and cannot recover the signal accurately. However, the proposed method performs better comparing to BP method.
Fig. 4. Simulation data results for the proposed algorithm (a)
Range-Doppler algorithm result for the simulation data imaging with 192 observations
(b)
Proposed algorithm result for the simulation data imaging with
(c)
Range-Doppler algorithm result for the simulation data imaging
(d)
Proposed algorithm result for the simulation data imaging with
(e)
Range-Doppler algorithm result for the simulation data imaging
192 observations with 128 observations
Fig.3. I-D signal simulation results of proposed method, BP and the
128 observations
desired FFT of entire signal
with 64 observations
4.2. Simulated data
(f)
In this simulation, the proposed ISAR imaging algorithm is applied on a plane model. The radar center frequency is 5.52 GHz, the bandwidth is 400 MHz, the time width of the chirp pulse is 25.6 us, and the pulse repetition frequency (PRF) is 400 Hz. We choose 256 range samples and 256 observations during the imaging time. However, the received echoes are randomly missing and we consider three situations: high quality TSAR imaging reconstruction with 192, 128 and 64 remaining observations, respectively. Moreover, in order to test the robustness of this method, we add white Gaussian noise with SNR=-6.53db. The results achieved by the range-Doppler algorithm (left column) and by the proposed algorithm (right column) are shown in Fig. 3.
Proposed algorithm result for the simulation data imaging with 64 observations
4.3. Real data
The proposed algorithm is applied to Yak-42 plane with the 384, 256 and 128 remaining observations which are randomly chosen from the whole 512 observations, while the number of range samples is 256. According to [18], the radar parameters are listed as follows: the transmitted signal bandwidth is 400 MHz with the carrier frequency 10GHz and the pulse repetition frequency (PRF) is 400Hz. By the traditional range-Doppler algorithm, we can obtain the TSAR images shown in FigA (a), (c), and (e). Observe that the interference induced by incomplete pulses strongly influences the quality of TSAR images. Using the proposed algorithm, the interference can be avoided as it is shown using the results in FigA (b), (d) and (f).
1878
[2] [3]
[4]
Fig. 5. Experimental data results for the proposed algorithm (a)
Range-Doppler
(b)
imaging with 384 observations Proposed algorithm result for the experimental data imaging
algorithm
result
for
the experimental
data
(c)
Range-Doppler
[5]
with 384 observations algorithm
result
for
the experimental
data
imaging with 256 observations (d)
Proposed algorithm result for the experimental data imaging
(e)
Range-Doppler
[6]
with 256 observations algorithm
result
for
the experimental
data
imaging with 128 observations (f)
Proposed algorithm result for the experimental data imaging with 128 observations
5.
ACKNOWLEDGMENT
[9]
[10]
[II] [12] [13]
This work was supported in part by the Fundamental Research Funds for the Central Universities under grant HTT.BRETTTT.201207, the Program for New Century Excellent Talents in University under grant NCET-12-0149, the National Science Foundation for Post-doctoral Scientists of China under grant 2013M540292, the postdoctoral science-research developmental foundation of Heilongjiang province under grant LBH-Qll092 and the Heilongjiang Postdoctoral Specialized Research Fund.
REFERENCES [1]
[8]
CONCLUSION
Tn real scenarios, the random missing observations cannot be avoided during the process of ISAR imaging because of the unstable radar system but also due to many other influences. The effect of missing observations causes the strong interferences over the cross-range, which decreases the quality of ISAR image. Based on the signal recovery algorithm [16], this paper proposes a new TSAR imaging algorithm for random missing observations, which has been shown effectively to recover high-quality TSAR image. As a future work, this approach can be extended to the scenario of more complex maneuvering target. 6.
[7]
[14] [15] [16]
[17]
E. giusti, M. Martorella, "Range Doppler and Image Autofocusing for FMCW Tnverse Synthetic Aperture Radar," IEEE Trans. Aerosp. Electron. Syst., vol. 47, no. 4, pp. 2807-2823, Oct. 2011.
[18]
1879
M. Martorella et aI, "Target Recognition by Means of Polarimetric ISAR Images," IEEE Trans. Aerosp. Electron. Syst., vol. 47, no. 1, pp. 225-239, Jan. 2011. F. Berizzi, E. D. Mese, M. Diani, and M. Martorella, "High-resolution ISAR imaging of maneuvering targets by means of the range instantaneous Doppler technique: Modeling and performance analysis," IEEE Trans. Image Process., vol. TO, no. 12, pp. 1880-1890, Dec. 2001. Pacifici. F, Del Frate. F, Emery. W.J, Gamba. P, Chanussot. J, "Urban Mapping Usmg Coarse SAR and Optical Data: Outcome of the 2007 GRSS Data Fusion Contest," IEEE Geosci. Remote Sens. Lett., vol. 5, no. E. Candes, J. Romberg, and T. Tao, "Robust uncertainty principles: Exact signal reconstruction from highly Illcomplete frequency information," IEEE Trans. Irif. Theory,vol. 52,no. 2,pp. 489-509,Feb. 2006. E. Candes, J. Romberg, and T. Tao, "Near-optimal signal recovery from random projections: Universal encoding strategies?" IEEE Trans. Iif. Theory, vol. 52, no. 2, pp. 489-509, Feb. 2006. D. Donoho, "Compressed sensing," IEEE Trans. In! Theory, vol. 52, no. 4, pp. 5406-5425, Apr. 2006. XX. Zhu, and R. Bamler, "Tomographic SAR Inversion by L I Norm Regularization - The Compressive Sensing Approach," IEEE Trans. Geosci. Remote Sens., vol. 48, no. 10, pp. 3839-3846, Oct. 2010. L. Zhang et ai, "Achieving Higher Resolution TSAR Imaging With Limited Pulses via Compressed Sampling," IEEE Geosci. Remote Sens. Lett., vol 6, no. 3, pp. 567571, Jul. 2009. E. G. Larsson, G. Q. Liu, P. Stoica, and J. Li, "High resolution SAR imaging with angular diversity," IEEE Trans. Aerosp. Electron. Syst., vol. 37, no. 4, pp. 13591372, Oct. 2001. J. Salzman, D. Akamine, R. Lefevre, and J. C. Kirk, Jr., "Interrupted synthetic aperture radar (SAR),"IEEE Aerosp. Electron. Syst. Mag., vol. 17, no. 5, pp. 33-39, May 2002. L. Zhang et ai, "High-Resolution TSAR Tmaging by Exploiting Sparse Apertures," IEEE Trans Antennas. Propag., vol. 60, no. 2, pp. 997-1008, Feb. 2012. Djurovic, 1., Stankovic, L., Boehme, J.F. "Robust L estimation based forms of signal transforms and time frequency representations", IEEE Trans. Signal Processing, vof. 51, no. 7, pp.1753-1761, Jul. 2003. Huber, P.J. "Robust Statistics" (John Wiley&Sons Tnc., 1981) Stankovic, L., Dakovic, M., Thayaparan, T. "Time frequency Signal Analysis with Applications" (Artech House, Boston, 2012) Srdjan Stankovic, Ljubisa Stankovic, and Irena Orovic, " Relationship between the robust statistics theo� and sparse compressive sensed signals reconstruction' , lET Signal Processing, Special issue in Compressive Sensing and Robust Transform, April 2014 v. C. Chen and S. Qian, "Joint time-frequency transform for radar range-Doppler imaging," IEEE Trans. Aerosp. Electron. Syst., vol. 34, no. 2, pp. 486-499, Apr. 1998. Y. Wang and Y. Jiang, "A novel algorithm for estimating the rotation angle III TSAR imaging," IEEE Geosci. Remote Sens. Lett., vol. 5, no. 4, pp. 608-609, Oct. 2008.
