Post on 13-Mar-2021
transcript
SPARSE MODELING FOR POLARIMETRIC RADAR
M. Hurtado, N. von Ellenrieder, C. Muravchik ∗
Dept. of Electrical Engineering
National University of La Plata
Argentina
A. Nehorai †
Dept. of Electrical and Systems Engineering
Washington University in St. Louis
USA
ABSTRACT
In this paper, we develop a sparse model to represent the
data recorded by a polarimetric, coherent radar. We produce
this model defining an over-complete library of possible tar-
get responses in the range-polarization space. Then, we em-
ploy compressive sensing methods to infer the position and
the scattering matrix of the target. Using real radar data, we
show that this new approach offers better interference rejec-
tion over other methods.
1. INTRODUCTION
A relatively new research topic, known as sparse represen-
tation and compressive sensing, has found many practical
applications, including image processing, sensor array, and
communications. Radar has become another flourishing area
where this technique promises interesting results. The idea of
using compressive sensing to reduce the number of required
measurements and eliminate the matched filter for processing
radar images was introduced in [1]. Then, it was applied
to improve target resolution of mono-static radars [2] and
MIMO radars [3]. Recently, it was used to track multiple
targets [4]. Herein, we propose sparse modeling to address
the problem of polarimetric radar systems.
It is well known that exploiting polarimetric informa-
tion can enhance the radar capabilities (see [5] and references
therein), particularly when the target echoes are contaminated
by strong reflections from the environment (clutter). In this
paper, we present a new approach based on sparse representa-
tions which exploits polarization diversity for discriminating
the target from the clutter. This model considers that the
presence of targets is not dense and that each target can be
represented by a few canonical shapes. When solving the
inverse problem, we retrieve the target position and scattering
coefficients, mitigating the clutter effects.
∗This work was supported by ANPCyT PICT 2007-11-00535 and UNLP.
NvE and MH are supported by UNLP and CONICET; CM is supported by
CICPBA and UNLP. E-mail: martin.hurtado@ing.unlp.edu.ar†The work of A. Nehorai was supported by the US Dept. of De-
fense under the AFOSR MURI Grant FA9550-05-1-0443 and ONR Grant
N000140810849.
The paper is organized as follows. We first state some
assumptions and formulate the problem in Section 2. In Sec-
tion 3, we develop the polarimetric sparse model. We briefly
discuss the algorithm for solving the sparse problem in Sec-
tions 4. We show results using real radar data in Sections 5.
Finally, we provide concluding remarks in Section 6.
2. PROBLEM FORMULATION
We consider a mono-static, coherent radar provided with po-
larization diversity. The data recorded by the radar can be rep-
resented as the contribution from the target echoes, the clutter,
and the noise. This kind of problem can be properly described
by a linear mixed model [6]:
y = Xβ +
K∑
k=1
Zkuk + e = Xβ + Zu+ e, (1)
where y is the (N × 1) data vector, β is the (M × 1) vector
of fixed effects (target), uk is a (Q× 1) vector that represents
the kth component of the random effects (clutter), X and Zk
are the matrices of regressors, and e is the (N × 1) residual
vector (noise). The vector u of size (KQ × 1) is formed
by concatenating the vectors uk, u′ = [u′
1, . . . ,u′
K ], with
corresponding matrix Z = [Z1, . . . , ZK ].In our problem at hand, we propose to represent the data
as a linear combination of a set of predetermined target re-
sponses that forms an over-complete dictionary stored in the
matrix X . Then, the vector β is an unknown sparse vector
representing the weighting coefficients of the possible signals.
Our goal is to identify and estimate these coefficients. The
main challenge consists of solving the under-determined in-
verse problem given that M > N . Furthermore, the distribu-
tion parameters of the random vectors u and e are unknown.
Although these parameters are not the primary interest, they
have to be estimated.
3. MODELING
In this section, we develop the sparse linear mixed model
stated in (1) for the data collected by a polarimetric radar.
2011 IEEE Statistical Signal Processing Workshop (SSP)
978-1-4577-0570-0/11/$26.00 ©2011 IEEE 17
3.1. Polarimetric Data
When a transmitted electromagnetic wave of polarization f
is reflected back, the normalized complex amplitude of the
voltage at the terminals of the receiver antenna is given by [7]
v = gTSf , (2)
where g is the polarization of the receiver antenna and S is
the scattering (Sinclair) matrix of the reflecting matter
S =
[
S11 S12
S21 S22
]
. (3)
For a specific polarization basis (u1,u2), the variables S11
and S22 are co-polar scattering coefficients and S12 and S21
are cross-polar coefficients. One property of the mono-static
configuration is that S12 = S21. Frequently, the polarization
basis are the horizontal and vertical linearly-polarized compo-
nents; however, other polarization basis less commonly used
are left and right circular polarization, and left and right slant
polarization.
Assuming that the radar transmits J diversely polarized
pulses and measures the incoming reflections using R anten-
nas with different polarization, the noise-free measurements
in a time window corresponding to one of the K range-cells
are
ykj = GTSkf j , (4)
where k = 1, . . . ,K and j = 1, . . . , J . The matrix G =[g1, . . . , gR] of size (2×R) represents the polarizations of the
receiving antenna array, Sk is the scattering matrix of the kth
range-cell, and f j is the polarization of the jth pulse transmit-
ted by the radar. The measurements of different range-cells
can be stacked in a vector of size (KR× 1)
¯yj =(
IK ⊗GT)
Sf j , (5)
where⊗ is the Kronecker product, IK is the identity matrix of
size K , and the matrix S =[
ST1 , . . . , S
TK
]Tof size (2K × 2)
concatenates the scattering matrices of the range-cells illumi-
nated by the radar. The data which correspond to all the trans-
mitted pulses can be arranged in a matrix of size (KR× J)
¯Y =(
IK ⊗GT)
SF, (6)
where the matrix F = [f1, . . . ,fJ ] of size (2 × J) repre-
sents the polarizations of the transmitted pulses. Applying
the properties of the Kronecker product, the data is piled in a
vector of length N = JKR
¯y = vec(
¯Y)
=(
FT ⊗ IK ⊗GT)
vec (S) , (7)
where the function vec stacks the columns of a matrix in a
vector. In addition, we can rewrite vec(S) in a form more
suitable for our model by vectorizing the scattering matrix of
each range-cell as follows
uk =[
Sk11 Sk
22 Sk12
]T, (8)
where uk is a vector of length Q = 3. Then, we write
vec(S) =
K∑
k=1
Hkuk (9)
where Hk are (4K ×Q) matrices with zero entries except for
the elements
[Hr]2k−1,1 = 1 [Hr]2k,3 = 1
[Hr]2(k+K)−1,2 = 1 [Hr]2(k+K)−1,3 = 1. (10)
The aim of these matrices is to map the elements of uk in the
proper position to form the vector vec(S).As we have discussed previously, the back-scattered sig-
nal from a certain range-cell corresponds to the superposition
of the electromagnetic field reflected by the target and its en-
vironment. Hence, the scattering matrix in (4) can be decom-
posed in two terms: Sk = Stk + Sc
k; similarly, its vector form
is uk = utk + uc
k, where the superscripts t and c refer to the
target and the clutter. Then, the vector of signal plus clutter is
¯y =(
FT ⊗ IK ⊗GT)
K∑
k=1
Hk(utk + uc
k). (11)
3.2. Sparse Linear Mixed Model
In order to generate an over-complete library of target re-
sponses, we apply Krogager’s decomposition of the scattering
matrix [8]. This decomposition is based on the claim that the
scattering matrix of an object can be represented by the com-
bination of three canonical shapes: a sphere, a diplane, and a
helix. Then, the vector form of the target scattering matrix at
the kth range-cell is
utk =
L∑
l=1
βklukl (12)
where βkl are the weighting coefficients of each canonical tar-
get and L is total number of components including the sphere,
the left and right helix, and the diplanes at different orienta-
tion angles ϕ. The scattering vectors of each shape are:
usphere =[
1 1 0]T
/√2
uhelix =[
1 −1 ±j]T
/2
udiplane =[
cosϕ − cosϕ sinϕ]T
/√2
. (13)
The contribution of the target reflections to the measured data
is the fixed effect of the model
(
FT ⊗ IK ⊗GT)
vec(
St)
=
=
K∑
k=1
L∑
l=1
(
FT ⊗ IK ⊗GT)
Hkutklβkl
=
M∑
m=1
Xmβm = Xβ, (14)
18
where M = KL and Xm are the columns of the matrix X .
The vector β is considered sparse because usually only a few
range-cells are occupied by targets; and these targets are rep-
resented by a few components among the L possible canonical
shapes.
Similarly, the clutter reflections form the random effects
of the model
(
FT ⊗ IK ⊗GT)
vec (Sc) =
=K∑
k=1
(
FT ⊗ IK ⊗GT)
Hkuck =
K∑
k=1
Zkuck. (15)
In order to reach the model as stated in Equation (1), we
replace (14) and (15) in (11). Then, we include the term
which corresponds to the residual. This term represents the
additive noise introduced by the sensing system, as well as
the modeling errors.
3.3. Non-polarimetric Model
Many conventional radar systems operate transmitting and re-
ceiving a single polarization to reduce hardware complexity
and building cost. Nevertheless our model is general and can
account for this particular case. This situation is represented
by taking J = R = 1. Without polarization diversity, the
clutter scattering matrix becomes a scalar, with Q = 1. Nei-
ther it is possible to apply Krogager’s decomposition of the
target scattering matrix; then, L = 1.
3.4. Compressive Sensing
The vector of measurements y is a linear combination of only
a few columns of the matrix X ; that is, only a few elements of
the vector β are non-zero. Then, the discrete signal y is said
to be compressible. A consequence is that a reduced num-
ber of measurements are required to reconstruct the original
information. The set of reduced number of measurements is
collected using a linear projection Φy, where Φ is the com-
pressing matrix of size N ′×N , being N ′ < N . For the com-
pressive sensing system, model (1) remains valid by redefin-
ing the matrices of regressors as ΦX and ΦZ . The analysis
of conditions for the matrix Φ to produce the said reduction
without a significant decrease in performance is beyond the
scope of this paper and will be addressed in a future work.
4. SOLVING ALGORITHM
In [9], we developed an algorithm for solving the inverse
problem of a sparse, under-determined linear mixed model.
The proposed method combines the Expectation Maximiza-
tion (EM) algorithm with a decision test. While the first one
solves numerically the estimation problem, the latter prunes
those components which are statistically small producing a
solution with low ℓ0 norm.
We showed that this new algorithm outperforms convex
relaxation methods, such as the Dantzig selector [10], because
it exploits the information about the clutter structure. On the
contrary, conventional methods address only the sparse linear
regression in white noise.
5. REAL DATA RESULTS
We evaluate the proposed sparse model and method using
radar data collected with the McMaster University IPIX
radar [11]. It is a dual polarized radar which transmits and
receives vertical and horizontal polarization. Specifically, we
processed the dataset stare4 recorded on Nov. 9, 1993. The
data corresponds to a beachball wrapped with aluminum foil
floating on the sea surface, located at 2.65km from the radar.
Fig. 1 shows the four polarimetric channels of this dataset,
where clutter reflections are as strong as those from the target.
We generated the over-complete dictionary by allowing
the presence of a target in each of the 68 range-cells which
form the radar footprint. For each range-cell, we consider
nine components to represent the target scattering matrix: a
sphere, left and right helix, and six diplanes with different ori-
entations. In Fig. 2, we show the radar image reconstructed
using the fully polarimetric sparse model. Several false detec-
tions are produced by the clutter; however, their amplitudes
are much weaker than the target. Additionally, we applied
the non-polarimetric sparse model to the VV channel, shown
in Fig. 3. We note that the lack of polarization makes more
difficult rejecting the clutter. For comparison, we also com-
pute the maximum likelihood estimator (MLE) of the target
scattering matrix, developed in [12]. Fig. 4 shows that the
MLE is not as efficient as the polarimetric sparse model for
de-noising the radar signal. Note that Figures 2-4 show sim-
ilar target response at range 2.65km. However, they differ in
the amount of clutter that it is filtered out; being our proposed
method the one with higher interference rejection.
Fig. 1. Radar raw data of the four polarimetric channels.
19
Fig. 2. Image built using the polarimetric sparse model.
Fig. 3. Image built using the non-polarimetric sparse model.
6. CONCLUSIONS
We addressed the problem of parameter estimation of tar-
gets in heavy clutter by exploiting polarization diversity. We
proposed a new approach based on sparse representation of
the target response. We applied Krogager’s decomposition of
the scattering matrix to generate an over-complete library of
target responses which gives rise to the polarimetric sparse
model. We tested our model with real radar data, and it
showed significant improvement with respect to other meth-
ods. In future work, we will study the design of polarimetric
compressive radars to reduce the number of required data.
7. REFERENCES
[1] R. Baraniuk and P. Steeghs, “Compressive radar imag-
ing,” in Proc. Radar Conf., Apr. 2007.
[2] M. A. Herman and T. Strohmer, “High-resolution radar
via compressed sensing,” IEEE Trans. Signal Process.,
vol. 57, no. 6, pp. 2275–2284, June 2009.
Fig. 4. Image built using maximum likelihood estimation.
[3] Y. Yu, A. P. Petropulu, and H. V. Poor, “MIMO radar
using compressive sampling,” IEEE J. Sel. Topics Signal
Process., vol. 4, no. 1, pp. 146–163, Feb. 2010.
[4] S. Sen and A. Nehorai, “Sparsity-based multi-target
tracking using OFDM radar,” IEEE Trans. Signal Pro-
cess., vol. 59, no. 4, pp. 1902–1906, Apr. 2011.
[5] M. Hurtado, J.-J. Xiao, and A. Nehorai, “Adaptive po-
larimetric design for target estimation, detection, and
tracking,” IEEE Signal Process. Mag., vol. 26, no. 1,
pp. 42–52, Jan. 2009.
[6] G. Molenberghs and G. Verbeke, Models for Discrete
Longitudinal Data, Springer, New York, 2005.
[7] D. Giuli, “Polarization diversity in radars,” Proc. IEEE,
vol. 74, no. 2, pp. 245–269, Feb. 1986.
[8] E. Krogager, “New decomposition of the radar target
scattering matrix,” Electronics letters, vol. 26, no. 18,
pp. 1525–1527, Aug. 1990.
[9] M. Hurtado, N. von Ellenrieder, C. Muravchik, and
A. Nehorai, “Sparse component analysis for linear
mixed models,” in Sensor Array and Multichannel Sig-
nal Processing Workshop, Oct. 2010.
[10] E. Candes and T. Tao, “The Dantzig selector: Statistical
estimation when p is much larger than n,” Annals of
Statistics, vol. 35, no. 6, pp. 2313–2351, Dec. 2007.
[11] S. Haykin, C. Krasnor, T. J. Nohara, B. W. Currie, and
D. Hamburger, “A coherent dual-polarized radar for
studying the ocean environment,” IEEE Trans. Geosci.
Remote Sens., vol. 29, no. 1, pp. 189–191, Jan. 1991.
[12] M. Hurtado and A. Nehorai, “Polarimetric detection of
targets in heavy inhomogeneous clutter,” IEEE Trans.
Signal Process., vol. 56, no. 4, pp. 1349–1361, Apr.
2008.
20