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)

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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.

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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.

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Fig. 4. Image built using maximum likelihood estimation.

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