 Open Access
 Total Downloads : 260
 Authors : Ahmed Mohamed Awadallah, Guanghui Zhao, Guangming Shi, Zhaozhao Gao
 Paper ID : IJERTV3IS060557
 Volume & Issue : Volume 03, Issue 06 (June 2014)
 Published (First Online): 11062014
 ISSN (Online) : 22780181
 Publisher Name : IJERT
 License: This work is licensed under a Creative Commons Attribution 4.0 International License
TwoDimensional Compressive SAR Imaging with Nonconvex Model Constraint
AhmedM.Awadallah,Guanghui Zhao,Guangming Shi
School of Electronic Engineering Xidian University Xian,P.R. China
ZhaozhaoGao
Science and technology on Electronic Information Control Laboratory
Chengdu, P.R. China
AbstractCompressed sensing (CS) based SAR imaging approacheshave shown its superior capability in providing high quality SAR images and reducing the storage pressure. However, such performance will degraded when the number of measurements is not sufficient. To significantly reducing the sampling number, we adopt a nonconvex model to deeply exploit the twodimensional (2D) sparsity in both range and azimuth dimensions, and a twodimensional gradient projection (TDGP) scheme is employed for fast reconstruction, finally, a new compressive SAR imaging is proposed. The simulation results demonstrate the superior reconstruction performance of the proposed algorithm.
IndexTermsSAR Imaging; Compressed Sensing (CS); Nonconvex Model; Gradient Projection.

INTRODUCTION
Synthetic aperture radar (SAR) is a radar imaging technology that is capable of producing high resolution images of the stationary surface targets. The main advantages of SAR are that it can reduce the effects of clouds and fog and allow them to be independent of external sources for imaging, having day and night and allweather imaging capability. Traditional compressions of SAR data utilize the redundancy inherent in sampled data under the Nyquist theorem to achieve compressed representation and profitable transmission. This theory claims that one must sample at least two times faster than the signal bandwidth while capturing it without losing information. Thereby there are large amounts of onboard data that have to be stored and it inevitably results in complex computation and expensive hardware.
CandÃ¨s, Tao and Romberg [1] and Donoho [2] have proposedan approach, known as compressed sensing (CS), in which arandom linear projection is used to acquire efficient representations of compressible signals directly. The theory of CS states that it is possible to recover sparse images from a small number of random measurements, provided that the undersampling results in noise like artifacts in the transform domain and an appropriate nonlinear recovery scheme is used [3]. Because of its compressed sampling ability, compressed sensing has found many applications in radar and remote sensing, and other fields.
R.Baraniuk et al. proposed the radar imaging system based on CS for the first time [4]. The papers [5, 6] use CS
in alongtrack interferometric SAR imaging and moving target velocity estimation. Some major open questions related with the application of CS to SAR and ISAR are listed in [7].
In [8], Li introduces a novel twodimensional (2D) SAR imaging algorithm based on CS theory to reconstruct 2D targets in the range and azimuth dimension, respectively. This algorithm provides the approach of receiving the echo data via 2D random sparse sampling beyond the Nyquist theorem. This radar system randomly transmits fewer pulses in azimuth direction and samples fewer data than traditional systems at random intervals in range direction.
In this paper, we focus on SAR imaging from under sampled target return in both range and azimuth dimensions, and finally, propose a nonconvex model for robust image reconstruction.
The reminder of this paper is organized as follows.Section II reviews the CS theory. Section III presents the SAR imaging with 2Ddownsampling. Section IV presents the proposed twodimensional nonconvex gradient projection (TDNGP) algorithm. The experimental results are shown in section V. Finally, Conclusions are given in section VI.

REVIEW OF CS THEORY
The CS theory points out that, any sparse or compressible signal vector has a sparse representation in the basis dictionary, i.e., = , lowdimensional measurements vector of the signal is calculated through an irrelevant observation matrix Ã— ( ). Then the information of the signal can be obtained by resolving the following optimization problem.
0 0
min , s.t. = (1)
where . 0represents 0norm and = .
Since resolving program in Eq. 1 is an NPhard problem, Donoho [2], Cands [9], Romberg [10], and Tao [1] proposed the following relax convex optimal model:
1 1 , s.t. = (2)
They also pointed out that Eq. 2 has the same solution with the Eq. 1 when the number of measurements satises the
inequality constraint log (b is a constant, K represents the sparsity degree of the signal, N is the length of the signal). Many algorithms have been developed to solve this model, such as LASSO [11], BP [12], and GPSR
[13] which is often being significantly faster (in terms of computation time) especially in largescale settings.The research of Cands [9] and Chartrand [10] discovered that, when the constraint condition logcannot be satisfied, the minimization of (1) has different solutions from that of (0) norm, which means program (1) cannot fully exploit sparsity of the signal. So Cands introduced a sparsity promoting weighted 1 norm to better approximate (0) norm. Since 0 = lim0 , Chartrand proposed an iterative weighted least squares (FOCUSS/IRLS) to solve the minimization of 01norm, and the corresponding nonconvex model is as follows:
min () , s.t. = (3)
e
Azim
tm
X max
uth
Ri (tm )
i
Rang
0
Rmin
X min
Rmax
M N
Figure 2, 2D scene division map
If the slow time ( ) sequence length is , fast time () sequence length is , then the echo signal is Ã— dimensional matrix. The scene according to the azimuth and range dimensions respectively wasdivided equally into Ã— small squares = ( )/,
= ( )/, as shown in Fig. 2.
where = = . However, for a large
=1
sized image, FOCUSS algorithm process (the process of calculating leastsquares solution) requires huge calculations which will seriously influence the reconstructed speed and decrease the recovered image quality and consequently prevent this algorithm from widely used in the field of image.
Consider each small square can only have one scattering point (with the same resolution), if there is a target then scattering coefficient is not zero, and vice versa. The scene grid is divided into small squares with a unified numbers of(1,2,3, , ), where = Ã— .
Assuming LFM signal is transmitted as
S (t)
exp( j 2 f t j t2 )
(4)
Inspired by the fast performance of the GPSR (convex CS) and the ability FOCUSS (nonconvex CS) to overcome the insufficient measurements by solving the 01 norm in which the nonconvex problem is converted into convex one, we propose a novel fast algorithm named TwoDimensional Nonconvex Gradient Projection (TDNGP) based on mini
mization of 01 norm, which can succeed in the process
t r c
where is the transmission fast time, is the range window function, is the carrier frequency, is the modulation frequency.
Assume = 0, then the echo baseband signal is
n j 4 Ri t m 2Ri t m
of largescale image, and adopt it in SAR imaging.
S r (t ,t m ) i exp
r t c
i 1

SAR IMAGING WITH 2D DOWNSAMPLING
This section will describe the 2D downsampling scheme
t
2
2R t
exp j t i m
(5)
a m
c
for the stripmap SARimaging,the imaging geometry can be referred to Fig.1.According to SAR imaging process to construct a twodimensional joint basis matrix, the coming echo will be transformed into a product form of the basis matrix and the scene scattering coefficient vector while two dimensional reconstruction.
where is the radar time (slow time), and ( )are the ithscattering coefficient and point target rangerespectively, is the signal wavelength, is the light speed, is the azimuth window function.
The echo baseband signal is analyzed line by line, where the kth line signal is:
n j 4 Ri t k 2Ri t k
Flight Path
S r (t ,t k ) i exp
r t c
i 1
2
Radar Beam
t exp j t 2Ri t k (6)
a k c
Ground Path
a k
with the a assumption that:
Swath
Range
hi (t k ) r t
2Ri t k t exp j
4 Ri t k
Azimuth
c
Figure 1, Stripmap SAR data acquisition mode
2R t 2
exp j t
i k
(7)
c
thenEq. 6 can be written as:
T (8)
As been shown in Fig. 1, in stripmap SAR mode the echo can be written as:
S r (t ,t k )N 1 H (t k )N n n 1
S t,t
T
(10)
where = , , , , ,
,
R m M N
M M M N N N
1
2
Ã—
Ã—1
= 1, 2, , . Similarly, the baseband echo signal can be expressed as:
whereis the scene scattering coefficient. and are the basis matrices for azimuth and range dimensions
T
S
r (t ,t m ) H (t m )(N M )n n1
(9)
respectively, so we can get the following imaging solution
model:
where
= , , ,
, namely the
min u p , s.t Echo u
0 p 1
(11)
1 2 Ã— Ã— p A R
twodimensional joint basis matrix.
Analysis of the above formula, we find that the left side of the equation is the echo signal pulled into one column, which contains all the echo information and related information. Thus as the observed signal will contain richer
where is a vectorized form for , and .are the azimuth and range dimension basis matrices respectively
In gradient projection framework, the first step is to compute the gradient of
information than a single line signal contains, so the
d
1 u
p u u
(12)
reconstruction results will be improved. Moreover,
k u p
k 1 p
k 1
k 1
traditional algorithm should first perform range migration correction as the coupling between the echo range and azimuth dimensions does not exist, so the twodimensional separate reconstruction needs more processing. In this formula the echo signal will be written in vector multiplication form of a large matrix and the objectives
where 1 = 1 2 + 2, and 2 < 0, the parameter is introduced into the weight to avoidthe appearance of singular matrix at each iteration.
To make sure the solution converges stably, the stepsize
should satisfy
scattering coefficients, without range migration correction, therefore no distinction between broadside and squint modes
uk min u
k
k 1
p

d
k k p
(13)
which makes the algorithm more pervasive.


PROPOSED IMAGING ALGORITHM (TDNGP) Traditional SAR imaging algorithms are performed based
on matched filtering and Nyquist theory. Our proposed algorithm doesnt use matched filtering and can reconstruct a high quality image from echo data beyond Nyquist theorem using CS theory and even without the sparsity constraint satisfied by using the nonconvex approach. Figure 3 shows
simple diagrams for the traditional range Doppler (RD)
Considering the nonconvex property of the 01 norm, the solution of cant be achieved precisely according to Eq.13. In the methodology of gradient projection (GP), many methods can be utilized to get proper estimation of , such as linear search. However, it has been testified that such option is timeconsuming [14], especially for nonconvex model. In the proposed method, inspired by FOCUSS/IRLS, we make use of the methodology of the reweighted 2 norm, and the stepsize should satisfy
algorithm and the proposed twodimensional nonconvex
min E u
T
k 1
k d k
min u
k 1
k d k
gradient projection (TDNGP) algorithm.
k k
u
k 1
u
k 1
k dk
(14)
Random sampling of echo data
in range and azimuth dimensions
(below Nyquist rate)
Full Echo data sampling (according to Nyquist theorem )
where = 1 , then making a differential with
respect to deduces:
E u d
d T d d T u d
(15)
k 1 k k
k k k
k k 1 k
Let Eq. 15 equals zero, the stepsize is:
Two dimensional Nonconvex CS
reconstruction with Gradient Projection (TDNGP)
Range compression (matched filtering)
+ range cellmigration correction
Twodimensional combined basis matrix
k
d k ,d k
d k , uk 1
d k
(16)
where . denotes the inner product process. To make sure the equality constraint = , the descending
direction can be obtained via projecting the gradient
Azimuth compression (matched filtering)
into the null space of as:
d d
d
(17)
High quality
k k A A k R R
where( . ) denotes the generalized inverse procedure. Then the new estimation of in the (k+1)th iteration can be updated as:
SAR iamge
SAR image
uk uk 1 k d k
(18)

(b)
Figure 3, (a) Traditional RD(b) The TDNGP
The process ends when reaching the maximum number of iteration or threshold condition is satisfied.
The TDNGP algorithm can be concluded as: (12.66%, 3.42%), the random sampling is in both range and
0 A R 0

Initialize u Echo k : 1

Calculate gradient of 1 1
azimuth dimensions. Comparing images quality, it can be seen that the GPSR is the lowest resolution imaging algorithm, TDNGP algorithm and CVX algorithm imaging quality are equivalent. Table II shows measured parameters
d
1 u
p u u
for the bottom target in the reconstructed images using
k u p k 1 p
k 1 k 1
3.42% sampling rate. Comparing running time, TDNGP
k 1
uk 1 diag
u i 2
p 2 ;
k 1
requires only 4~5 times of the GPSR required time, and CVX costs 150~300 times of the TDNGP required time.
;
Figure 6 shows the time comparison at different sampling

Step calculation: k
d k ,d k
d k , uk 1
d k
rates. The previous experiments show that even with a significant reduction in the SAR echo, the proposed algorithm can make fast reconstruction for a high quality

The null space projection
images.
d k d
d ;
80
k A A k R R
60
update uk uk 1 k d k ;
40
20

Setting a threshold value If
uk uk 1
0
k : k 1 go to step (2), else stop.


EXPERIMENTAL DESIGN
To verify the validity of the proposed image formation algorithm, the following raw data simulation and experiments have been done. The raw data for a scene with five point targets were simulated according stripmap SAR mode with the parameters listed in Table I. The resulting image using fullsampling data obtained under the traditional RD algorithm is shown in Fig. 4.
Experiments comparison with the proposed TDNGP, GPSR, and the CVX [15] algorithms for SAR imagingwithdifferent random sampling rates in both range
80
60
40
20
0
20
40
60
80
20
40
60
80
3800 3900 4000 4100 4200
Figure 4, SAR image using traditional RD
80
60
40
20
0
20
40
60
80
and azimuth dimensions are tested. We also compare the
3800 3900 4000 4100 4200
3800 3900 4000 4100 4200
quality and CPU elapsed time for image reconstruction under these algorithms. In the GPSR and CVX algorithm, range
(a) 12.66% (GPSR) (b) 3.42%
and azimuth dimensions reconstruction processing are 80
60
performed separately, while our proposed TDNGP algorithm
40
80
60
performs the two dimensional reconstruction directly. Figure
20
20
5 shows two imaging results for the GPSR, TDNGP, and
0
0
CVXalgorithms at two different random sampling rates
20
20
40
40
40
TABLE I. SIMULATION PARAMETERS
60
80
Parameter
Value
Transmit bandwidth
60 MHz
Pulse duration
5 Âµs
Carrier frequency
10 GHz
Antenna length
4m
Platform velocity
100 m/s
Platform height
3000 m
Point targets coordinates
(0,3900) (0,4000) (0,4100)
(50,4000) (50,4000)
3800 3900 4000 4100 4200
60
80
3800 3900 4000 4100 4200
100
50
0
50
(c) 12.66% (TDNGP) (d) 3.42%
80
60
40
20
0
20
40
60
100
3800 3900 4000 4100 4200
80
3800
3900 4000 4100 4200
(e) 12.66% (CVX) (f) 3.42%
Figure 5, Comparison results of imaging algorithms under different
sampling rates
GPSR
TDNGP
CVX
3dB(A) m
1.7319
0.8061
0.9338
3dB(R) m
2.6479
1.1990
1.2490
PSLR(A) dB
14.9170
18.3098
19.0603
PSLR(R) dB
11.7282
16.4402
14.0146
ISLR(A) dB
10.3454
14.1775
14.7972
ISLR(R) dB
9.6142
12.2230
13.9128
Time (s)
3.1
26.5
1215.1
TABLE II. COMPARISON OF RESOLUTION ANALYSIS AT THE BOTTOM TARGET FOR RECONSTRUCTED IMAGES FROM 3.42% SAMPLING RATE
4
10
CVX
TDNGP
GPSR
CPU Elapsed Time (s)
3
10
2
10
1
10
0
10
25 20 15 10 5
Samling Ratio (%)
Figure 6, Comparison results of the CPU elapsed time for different algorithms under different sampling rates

CONCLUSION
The proposed algorithm (TDNGP) is based on solving the nonconvex optimization problem with gradient projection and combining with the twodimensional CS reconstruction. Since the proposed algorithm only involves some matrixvector products, it is easy to implement fast implicit operation and make it possible to take use of the advantage of 0p1 seminorm based model practically in largescale applications such as SAR image reconstruction, which is a hard task for common procedure for 0p1semi norm optimization such as FOCUSS/IRLS. The simulation of SAR image reconstruction shows the superperformance of the proposed algorithm in the quality improvement and
the significant reduction in the required time which has consequently a great advantage on the SAR imaging problem.
ACKNOWLEDGMENT
This work is supported in part by the Major State Basic Research Development Program of China (973 Program, No. 2013CB329402), in part by the National Science Foundation of China under Grants 61372071, 60902079, 61201289,
61033004, 61072104 and 61070138.
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