Signal Processing - November 2016 - 87

a more promising research area for radar imagery applications. Compared to the deterministic sparsity-inducing
framework, statistical sparsity-based techniques provide
new opportunities to significantly improve the performance
of radar imagery. This is mostly due to the capability of
avoiding regularization parameter tuning, providing desirable statistical information, and allowing flexible modeling. These advantages are due respectively to the inherent
advantage of the statistical framework, the desirable statistical information obtained from the estimation of the full posterior distribution, and the inherent flexibility of statistical
sparsity-based modeling. To benefit from these advantages,
sophisticated design is required. This article is a companion
of recent tutorial articles on sparsity-based radar imagery
[3]-[5], but with particular emphasis on sparsity-based radar
imagery from a statistical perspective, which is missing in
the recent literature.
We show how this design is to be performed and demonstrate how the various radar imagery problems can be formulated within a sparse Bayesian framework. We illustrate
in detail why the statistical formulation greatly enhances the
radar imaging performance in various practical problems.
The introduced framework has much promise for future radar
imaging systems, as it provides substantial improvements as
well as new opportunities. The notations used in this article are
summarized in Table 1.

Statistical sparsity formulation of radar imagery
We begin our treatment by reviewing the fundamentals of statistical sparsity-inducing models in radar imagery. We formulate the statistical sparsity-based framework and highlight
from where the advantages arise along with the limitations of
statistical sparsity-based methods.

Data modeling
In high-resolution radar imagery, the scattering response of a
target of interest is often expressed as a sum of scatterers'
responses. Without loss of generality, assuming that the radar
emits successive pulses with time interval Tr and that there
exist K strong scatterers in an imaging scene, the received
radar signal can be given by
s r (t, t n) =

/ v k ·sc t - 2R kc(t n) m + n (t, t n),
K

(1)

k=1

where v k represents the amplitude of the kth scatterer, c is
the speed of light, and R k (t n) represents the range from the
radar to scatterer k in slow time t n. The fast time and slow
time of pulse n are denoted by 0 # t # Tr and t n = nTr ,
respectively. To achieve high-range resolution, the emitted
pulses s (t, t n) are often chosen as linear frequency modulated (LFM) signals, but other waveforms such as sparse
stepped-frequency signals [22], sparse probing-frequency
signals [23], and adaptively optimized signals [24] can
also be used for the purpose of improved imaging performance. To achieve high-cross-range resolution, a large
aspect angle between the radar and the target is required

Table 1. Notation summary.
Notations
CM # N

The set of a complex M × N matrix

a, a

Scalar and vector

A, A i·, A ·j, A n, m

Matrix, the ith row, the jth column, and the (n, m)th
entry of a matrix

^ $ hT and ^ $ hH

Matrix or vector transpose and conjugate transpose

A -1

Matrix inverse
The absolute value of a scalar

$
$

The , p norm of a vector

The Frobenius norm of a matrix

exp ^ $ h

The exponential function

CN (n, R)

The complex Gaussian distribution with mean n
and covariance matrix R

Beta (a, b)

The beta distribution with parameters a and b

C (a, b)

The gamma distribution with parameters
a and b

during the CPI. Note that sparsity-based methods go
beyond the convention in the sense that high-range resolution can be obtained with less bandwidth, and high-crossrange resolution can be obtained with a reduced CPI.
After preprocessing and arranging the cross-range measurements column-wise, a linear model is obtained [1] as
Y = AXB + N,

(2)

where Y ! C M # N is the preprocessed radar echoes, X ! C M # N
is the unknown scattering coefficient, and A ! C M # M and
B ! C N # N are the measurement matrices constructed from
(1) for cross range and range, respectively. In general, A and
B are Fourier matrices. There exist, however, other ways to
construct the dictionary other than simply employing the
Fourier matrix. These include the frame-based matrix [25]
and the matched filter-based matrix [26]. Note that the model
in (2) does not yet include the case of an undersampled or
incomplete measurement of Y. Modifications to capture
these are straightforward. A more detailed discussion on
this issue is presented in the "Superresolution Radar
Imagery" section.
In (2), N is assumed to be independently circularly symmetric complex Gaussian distributed, so that the received
signal Y follows a complex Gaussian distribution with a likelihood function given by
p (Y ; X, a 0) =

% % CN^Yij ; A i· XB ·j, a 0-1 Ih.
M

(3)

i=1 j=1

The a 0 is the noise precision or the reciprocal of the
variance, which is assumed to be random. For the sake of

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