. RESEARCH PAPER . Special Issue SCIENCE CHINA Information Sciences August 2012 Vol. 55 No. 8: 1830–1837 doi: 10.1007/s11432-012-4614-7 c Science China Press and Springer-Verlag Berlin Heidelberg 2012 info.scichina.com www.springerlink.com Suppressing azimuth ambiguity in spaceborne SAR images based on compressed sensing YU Ze ∗ & LIU Min School of Electronics and Information Engineering, Beihang University, Beijing 100191, China Received November 29, 2011; accepted May 13, 2012; published online June 22, 2012 Abstract In spaceborne synthetic aperture radar, undersampling at the rate of the pulse repetition frequency causes azimuth ambiguity, which induces ghost into the images. This paper introduces compressed sensing for azimuth ambiguity suppression and presents two novel methods from the perspectives of system design and image formation, known as azimuth random sampling and ambiguity separation, respectively. The first method makes the imaging results for the ambiguity zones as disperse as possible while ensuring that the imaging results for the main scene are affected as little as possible. The second method separates the ambiguity signals from the echoes and achieves imaging results without the ambiguity effect. Simulation results show that the two methods can reduce the ambiguity levels by about 16 dB and 99.37%, respectively. Keywords synthetic aperture radar, compressed sensing, azimuth ambiguity, orthogonal matching pursuit Citation Yu Z, Liu M. Suppressing azimuth ambiguity in spaceborne SAR images based on compressed sensing. Sci China Inf Sci, 2012, 55: 1830–1837, doi: 10.1007/s11432-012-4614-7 1 Introduction In spaceborne synthetic aperture radar (SAR), the upper limit of the pulse repetition frequency (PRF) is determined by the swath width and the range ambiguity, which confines the PRF to be far lower than the Doppler spectrum width. Thus, discrete sampling at the rate of the PRF causes aliasing of the azimuth spectrum, i.e., azimuth ambiguity. When strong scatterers exist in the ambiguity zones, with Doppler frequencies that are higher than the PRF, obvious ghosts appear in the SAR images [1]. Azimuth ambiguity decreases the image quality, and affects the interpretation accuracy. Ref. [2] defined the azimuth ambiguity signal ratio (AASR) as a numerical description of the azimuth ambiguity, which is now an important index for evaluation of SAR system performance. Based on an analysis of AASR, two methods have been developed to suppress azimuth ambiguity. The first approach adjusts the SAR system parameters, such as the antenna pattern and the PRF, to let a notch correspond to the ambiguity zones or to increase the ratio of the PRF to the azimuth processing band, which is difficult to perform because of the complexity of the antenna structure and other system performance constraints. The second method improves the estimation accuracy of the Doppler centroid in combination with improved imaging algorithms. However, the precision depends on the complexity of the observed scene. ∗ Corresponding author (email: [email protected])
Yu Z, et al. Sci China Inf Sci August 2012 Vol. 55 No. 8 1831
A
Δ
R0
R0PA
RN
PM
Flight direction
DR
DAPN
DN
θ
θ
θ
Figure 1 Geometry of unambiguous and ambiguous points. For spaceborne SAR, the relative movement between the
satellite and the ground target approximates to a straight line in the aperture time.
Refs. [3,4] suppress the azimuth ambiguity by designing ideal filters and cancelling the ambiguity
through multiple imaging, but the method is only suitable for the case of point scatterers. Ref. [5] uses
post-processing techniques to obtain ambiguity-free results. It requires the azimuth ambiguity zones to
correspond to nulls in the azimuth antenna pattern. The energy in the nulls is considered to belong
only to the main scene. A Wiener adaptive filter is then designed to reduce the ambiguity. Because
the parameters of the filter, including the main scene and the ambiguity intensity, are estimated after
imaging, the ambiguity removal effect relies on the properties of the scene, the precision of the imaging
algorithms, and the antenna pattern measurement accuracy.
Recently, a new theory called compressed sensing (CS) was proposed for sparse signal reconstruction
[6,7]. Based on CS, signals can be recovered from samples obtained at a rate which is lower than
the standard Nyquist rate [8,9]. In this paper, we introduce CS into azimuth ambiguity suppression.
Section 2 discusses the impact of traditional imaging on azimuth ambiguity signals. Then, two imaging
methods are presented from the perspectives of system design and image processing. Section 3 reviews
azimuth ambiguity by analysis of the correlation of the measurement matrices and proposes azimuth
random sampling. Another image formation method without ambiguity effects, ambiguity separation, is
presented in Section 4. Finally, a brief conclusion is given.
2 Traditional imaging of azimuth ambiguity signal
The SAR imaging filter is designed only to focus the main scene echoes. Because azimuth ambiguity
signals have different Doppler centroids, Doppler frequency rates, and range migration values when com-
pared with those for the main scene, two effects on the ambiguities emerge: displacement and dispersion.
The basic methodology of SAR imaging is to resolve of the coupling between the range and the
azimuth, and compress echoes in two dimensions [10,11], which can be regarded as deconvolution. Here,
we apply the back-projection (BP) algorithm, which avoids complicated time-frequency transforms, and
is convenient for explanation of the ambiguity.
Figure 1 shows the geometry of the unambiguous and ambiguous points. Because the nearest ambiguity
zones provide more than 85% of the ambiguity energy [4], we will focus on the +1 and −1 ambiguity zones
in the following sections. PA is a single point in the +1 zone. The impulse response of PA will be dispersed,
and will be centered around PM . PN is DN meters away from PM along the direction perpendicular to
the beam. According to the definition of the azimuth ambiguity, the frequency difference between PA
and PM is k · fp. Thus, the range and azimuth displacement for PA can be derived as
DA = k · λ
2V· RN
cos θ· fp, (1)
DR =RN
cos θ· cos θA −RN , (2)
1832 Yu Z, et al. Sci China Inf Sci August 2012 Vol. 55 No. 8
where k equals +1 or −1, λ is the wavelength, V is the equivalent velocity, RN is the nearest range from
the radar to the target, θ is the squint angle, fp denotes the PRF, and
θA = arcsin [(RN · sin θ + k ·DA · cos θ) /RN ] .
The BP algorithm divides the image into grids according to the resolution, and calculates the two-way
propagation delay time for every grid. Then, the corresponding signals are extracted from the range-
compressed data, and are coherently summed. The imaging result at (R,X) is given by
sa (R,X) = sr (r, x)⊗ hR,X (r, x) , (3)
where sr (r, x) is the range compressed data, hR,X (r, x) is the filter designed for the position (R,X), and
r and x denote the range and the azimuth direction, respectively.
According to (3), the response in PN is
sa (PN ) ≈∫ T/2
−T/2
sin c
(π · λ ·Br · fp
ct
)· ej2π·Δfd·tdt
=c
π · λ · Br · fp
{Si
[πT
(Δfd +
λ · Br · fp2c
)]− Si
[πT
(Δfd − λ ·Br · fp
2c
)]}, (4)
where c is the light velocity, Br is the signal bandwidth, Δfd is the Doppler centroid difference between
PN and PM , T is the aperture time, sin c(τ) = sin(τ)/τ , and Si (z) =∫ z
0 sin c(τ)dτ .
The part in the brackets of (4) represents a rectangular signal, for which half of the distance between
the rising edge and the falling edge is defined as the radius WA for the dispersed energy along the azimuth:
WA = RN · Δθ
cos θ · [cos θ −Δθ · sin θ] , (5)
where
Δθ =Br · fp · λ2
4 · V · c · cos θ .The impulse response of PA across the range also has a finite width, which is determined by the range
migration. Half of this width is defined as the range dispersion radius, which is expressed as follows:
WR =RN · fp · λ2
8ρa · V · cos3 θ , (6)
where ρa is the azimuth resolution. By comparing (5) with (6), it is shown that WA has no relationship
with ρa, while WR does.
The above analysis shows that the azimuth ambiguity is much more obvious as the squint angle
decreases and the wavelength becomes shorter. However, increasing numbers of SAR satellites apply
attitude steering technology [12], and the X band is also the most frequently used frequency band at
present. These factors make ambiguity suppression imperative.
3 Azimuth random sampling method
3.1 Application of CS theory in SAR
Candes et al. [6] and Donoho [7] systematically presented CS theory, and showed that the original signal
can be recovered when the sampling rate is lower than the Nyquist rate. The core of CS theory consists
of three parts:
1) In principle, it is possible to recover σ ∈ RN by achieving K observations if σ is a K-sparse signal,
which means σ can be approximately represented by only K nonzero elements. Analysis of the signal
Yu Z, et al. Sci China Inf Sci August 2012 Vol. 55 No. 8 1833
characteristics to select the base which minimizes the number of nonzero decomposition coefficients in
the transformation domain is the basic idea of CS theory.
2) The linear measurement s can be obtained by applying the sensing matrix Φ ∈ RM×N , i.e., s = Φσ·Φmust satisfy the restricted isometry property (RIP) to guarantee recovery precision [13]. RIP can be
described as
(1− ξ) ‖σ‖2 � ‖Φσ‖2 � (1 + ξ) ‖σ‖2 ,where ξ is a small number. The smaller ξ is, then the more accurate the reconstruction becomes.
3) The original signal can be reconstructed perfectly by applying convex optimization to solve the
norm problem with restrictions. Basis pursuit and Bregman iteration are appropriate when the signal
to noise ratio (SNR) is high. Basis pursuit de-noising, second-order cone programming, and the Dantzig
selector can be applied under low SNR conditions.
Selection of the base, design of the measurement matrix and development of the recovery algorithms
are three basic problems in CS theory. In particular, the sensing matrix determines the reconstruction
accuracy, which means that the RIP must be satisfied as far as possible. However, it is difficult to prove
whether Φ follows the RIP. In reality, this condition can be simplified in that Φ is an uncorrelated matrix.
The SAR echo can be regarded as a convolution of the chirp signal and the scene. If we review this
process from the perspective of CS theory, the sensing matrix in the range reconstruction becomes:
Φr =
⎡⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎢⎣
sT (1) 0 · · · 0 · · · 0 0
sT (2) sT (1) · · · 0 · · · 0 0... sT (2) · · · 0 · · · 0 0
sT (Q)...
. . . sT (1). . .
... 0
0 sT (Q) · · · sT (2) · · · 0...
......
. . ....
. . . sT (1) 0
0 0 · · · sT (Q) · · · sT (2) sT (1)
⎤⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎥⎦Mr×Nr
, (7)
where sT (i) is the ith sampling point of the emitted chirp signal, and Q denotes the total number of
nonzero sampling points along the range.
In SAR modeling, it is usually assumed that the targets are only illuminated by the antenna pattern
within the 3 dB beam width, whereas, in reality, this is not true. The azimuth echo length corresponding
to one isolated target is thus infinite, which makes the azimuth measurement matrix different, as follows:
Φa =
⎡⎢⎢⎢⎢⎢⎢⎢⎣
sA (0) sA (−1) · · · sA (1−Na)
sA (1) sA (0) · · · sA (2−Na)
sA (2) sA (1) · · · sA (3−Na)...
.... . .
...
sA (Ma − 1) sA (Ma − 2) · · · sA (Ma −Na)
⎤⎥⎥⎥⎥⎥⎥⎥⎦Ma×Na
, (8)
where sA(i) is the ith sampling point of the azimuth chirp signal.
We define the correlation coefficient ρi,j of the ith column and the jth column as:
ρi,j =|〈Ci,Cj〉|√|〈Ci,Ci〉| · |〈Cj ,Cj〉|
(9)
where Ci is the ith column, and 〈a, b〉 denotes the inner product of a and b. The simulation parameters
for the correlation analysis of Φr and Φa are presented in Table 1. Figure 2 demonstrates that the different
columns of Φr are nearly uncorrelated. Figure 3 shows that there is a large correlation coefficient between
one column of Φa and another column in the same matrix, which shows that the energy of PA and PM
is confused.
1834 Yu Z, et al. Sci China Inf Sci August 2012 Vol. 55 No. 8
Table 1 Simulation parameters
Parameters Quantity
Orbital height (km) 500
Equivalent velocity (m/s) 7600
Signal bandwidth (MHz) 90
Pulse width (µs) 1
Sampling rate (MHz) 99
PRF (Hz) 3000
Resolution (m) 3
Looking angle (Degree) 30
Wavelength (m) 0.03
1.0
0.8
0.6
0.4
0.2
0
20003000
10000 0
10002000
3000
Cor
rela
tion
coef
fici
ent 1.0
0.8
0.6
0.4
0.2
0
20003000
10000 0
10002000
3000
Cor
rela
tion
coef
fici
ent
Figure 2 Correlation coefficients of different columns
in Φr .
Figure 3 Correlation coefficients of different
columns in Φa.
3.2 Azimuth random sampling
The above analysis shows that the azimuth reconstruction matrix determines the azimuth ambiguity.
If the properties of the matrix can be changed, ambiguity suppression may be possible. However, the
azimuth properties of the SAR signal are decided by the relative movement between the platform and
the targets, which cannot be changed arbitrarily like the transmitted signal. The only element that can
be altered is the position at which the signal is transmitted and received.
Azimuth sampling is usually performed at the rate of the PRF. Here, a novel method named azimuth
random sampling is presented. In this method, the sampling interval along the azimuth is not uniform.
Each new sampling time is generated randomly according to the following expression
t′i = ti +lifp
, (10)
where li is a random variable which obeys the uniform distribution in the interval [0, 1], and ti and t′iare the original and new sampling times, respectively.
Figure 4 gives the correlation coefficients of the different columns of Φa based on the new scheme.
Compared to Figure 3, the large correlation values corresponding to the azimuth ambiguities have disap-
peared.
Using the same parameters, the original imaging results of strong scattering targets in the +1 az-
imuth ambiguity zone and the results based on the proposed random sampling scheme are shown in
Figures 5 (a) and (b), respectively. It is obvious that, by using the proposed method, the ambiguity
energy has spread over a much larger area. The result in Figure 5(a) can be interpreted as a false target.
However, that is impossible for the result shown in Figure 5(b). Our simulation shows that a suppression
level of approximately 16 dB has been achieved.
Imaging results for the main scene are presented in Figure 6, where no significant differences are
observed between them, as required for a good azimuth ambiguity suppression scheme.
Yu Z, et al. Sci China Inf Sci August 2012 Vol. 55 No. 8 1835
1.0
0.8
0.6
0.4
0.2
0
20003000
10000 0
10002000
3000
Cor
rela
tion
coef
fici
ent
Figure 4 Correlation coefficients of different columns in Φa. The sampling time is selected according to (10).
20
(a)
40
60
80
100
50 100 150 200 250 0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
×104
20
40
60
80
100
50 100 150 200 250 0
0.5
1.0
1.5
2.0
2.5
3.0
3.5
×104
(b)
Figure 5 Comparison of the imaging results for strong scattering targets in the +1 azimuth ambiguity zone based on the
original and proposed sampling schemes. (a) The imaging results based on the original uniform sampling scheme; (b) the
imaging results based on the proposed random sampling scheme.
20
40
60
80
100
50 100 150(a) (b)
200 2500
0.5
1.0
1.5
2.0
2.5
3.0
×104
0
0.5
1.0
1.5
2.0
2.5
3.0
×104
20
40
60
80
100
50 100 150 200 250
Figure 6 Comparison of the imaging results of targets in the main scene based on the original and proposed sampling
schemes. (a) The imaging results based on the original uniform sampling scheme; (b) the imaging results based on the
proposed random sampling scheme.
1836 Yu Z, et al. Sci China Inf Sci August 2012 Vol. 55 No. 8
−200
20
−20−1001020
×106
Azimuth (m)
Range (m)
Am
plitu
de
8
6
4
2
0−20
020
−20−10010200
2
4
6
8
10
12
×104
Azimuth (m)
Range (m)
Am
plitu
de
Figure 7 The imaging results from application of the
BP algorithm.
Figure 8 The imaging results obtained using the
proposed method.
4 A novel image formation method without ambiguity effect
To accomplish the azimuth random sampling method described in the previous section, the radar sys-
tem parameters must be changed. This section presents another image formation method, ambiguity
separation, designed to suppress the ambiguity by signal processing based on CS.
The important part of the method presented here is the construction of a measurement matrix for the
ambiguous zones, while the currently applied Φ for SAR imaging only contains the information for the
main scene [14]. The detailed steps are illustrated as follows.
1) First, determine the projection area of the main scene on the imaging plane and calculate the
positions of the first azimuth ambiguity zones according to Eqs. (1)–(6). Then, discretize these areas
into grids.
2) If Ψik represents the echo from the ith grid at the kth moment, where
Ψ=ik [. . . , 0, sT (1) , sT (2) , . . . , sT (Q) , 0, . . .]
T ∈ CM×1,
then the measurement vector for the ith grid during the total observation time is
Φi =[ΨT
i1,ΨTi2, . . . ,Ψ
TiN
]T. (11)
Here, ()T represents the transpose of a vector, and M and N denote the sampling numbers along the
range and the azimuth, respectively.
3) The ambiguous measurement matrix can be expressed as
ΦA = [Φ1,Φ2, . . . ,ΦP ] , (12)
where P is the total number of grids in the ambiguity zones.
4) Let s represent the echoes from the entire scene, including the main zone and the ambiguity zones,
and apply the OMP algorithm to achieve the best solution σA from s = ΦAσ [15]. It is obvious that s
must be a column vector here.
5) Construct the main measurement matrix ΦR as in (12), and then solve δ = ΦRσ, where δ =
s−ΦAσA. Then, arrange the solution σR in two dimensions according to the order of the measurement
vectors to form the final image without ambiguity effects.
We present the following simulation results with the parameters shown in Table 1. Echoes of the main
scene and the +1 ambiguity zone were simulated and processed. The main scene contains 3×3 points
at intervals of 12 m. 25×25 targets covering 6×6 square meters are placed in the +1 ambiguity zone.
The intensity of each ambiguous point is 35 dB higher than that of the main target. Figure 7 shows the
imaging results derived by applying the BP algorithm, where the ambiguous energy submerges the main
targets. The fluctuation in Figure 7 was caused by the antenna pattern weighting. The results obtained
using the proposed method are presented in Figure 8, where the ambiguous signals have been filtered out
Yu Z, et al. Sci China Inf Sci August 2012 Vol. 55 No. 8 1837
and 9 main points appear clearly. The results demonstrate that a suppression level of about 99.37% of
the ambiguous energy was reached.
5 Conclusions
This paper proposes two new methods for azimuth ambiguity suppression. Azimuth random sampling
can be implemented in SAR systems easily by changing the PRF and can achieve a suppression level
of about 16 dB. Another image formation method, ambiguity separation, was also proposed and has
the ability to reconstruct the main scene without azimuth ambiguity. The method can be applied to
any operation mode, including stripmap, spotlight or hybrid modes. Also, good performance is achieved
without any assumptions about the observed scene. Future studies will focus on the effects of three
factors: the signal-to-noise ratio, the ambiguity-to-signal ratio and even lower PRF levels.
Acknowledgements
This work was supported by National Key Basic Research Program Project (973 Program) of China (Grant No.
2010CB731902).
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