Signal Processing - November 2016 - 92

Range Cell

Cross-Range Cell

(m, n)

Pattern

FIGURE 3. The first-order continuity patterns.

In summary, there are two important points to be noted
here. First, the success of statistical sparsity-based methods
depends on the proper selection of sampling patterns, which
is also the case for conventional sparsity-based methods. In
fact, statistical sparsity-based methods are less sensitive to
highly correlated measurement matrices induced by some
sampling patterns than sparsity regularized methods. A
detailed analysis can be found in [43] and the references
therein. Second, a tradeoff between design convenience and
performance is to be properly balanced, depending on the
performance requirements for superresolution in the problem at hand.

Enhanced target imagery by exploiting
structured sparsity
In the "Superresolution Radar Imagery" section, we demonstrate that statistical sparsity could lead to improvements
over deterministic sparsity in superresolution radar imagery.
To carry out sparse estimation, the scaled Gaussian mixture
is imposed on the scattering coefficients, which are
assumed to be independently distributed. However, in practice, targets in radar images always exhibit strong spatial
correlation due to the fact that a real target is physically
continuous [39], [44]-[46]. For example, the radar returns
from a tank or an airplane will often exhibit strong spatial
correlation, i.e., nonzero-valued scatterers in the target
region continuously residing in the range and/or cross-range
dimensions. This phenomenon motivated the research in
[39], [45], and [46], which modifies the statistical sparse
model accordingly. In these works, continuity in the target
scene is exploited by incorporating a correlated prior in a
probabilistic framework. In what follows, we review methods that impose first-order and higher-order correlations on
the sparse scattering coefficients.

First-order correlation
In [39], [44], and [45], a modification in spike-and-slab
modeling was made so as to impose first-order spatial
correlation of the coefficients. The reason for choosing
the spike-and-slab prior rather than the scaled Gaussian
mixture is because imposing correlation on the support of
the sparse signal is more accurate and justifiable than
imposing it on the amplitude of the sparse signal. As
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discussed in the "Statistical Sparsity Formulation of
Radar Imagery" section, the sparsity pattern of the signal
is determined by W in the spike-and-slab model in (7),
where the parameter q controls the probability of W
being nonzero. Therefore, a straightforward way to
impose a continuity prior on the signal can be carried out
directly on W. However, this treatment deviates from the
original intention to perform a flexible statistical modeling step. For this particular reason, it is suggested in [39]
and [45] to encode the first-order structural information
on q in an intermediate way rather than straightforwardly
on W. The key modification is to replace the single beta
prior for parameter q by a set of beta priors that consist of
three different sets of parameters, " e k, fk ,k = 0, 1, 2, so as to
capture strongly independent, strongly continuous, and
noninformative priors, respectively.
The proposed sparsity patterns in [39] and [45] that both
encourage continuity and preserve sparsity are summarized
as follows.
■ Strong rejection: If the first-order neighborhoods of X mn
are all zero, it would be very likely that X mn is also zero,
due to the continuity of the target scene. The prior
Beta(e 0, f0), with e 0 1 f0, is utilized to make the probability qmn of Wmn = 0 being large. This means that the absence
of a first-order neighborhood implies the investigated scatterer being zero with a high probability. This rejection
pattern can eliminate the undesired isolated speckles or
artifacts in the radar image.
■ Strong acceptance: If any of the continuity patterns for
X mn in Figure 3 is observed, the prior that a nonzero-valued X mn arises with a high possibility should be imposed.
This step imposes continuity of the target image. In this
case, the prior Beta (e 1, f1), with e 1 2 f1, enforces the
probability qmn of Wmn = 1 to be large, and thus the scatterer under test can be accepted. This implies that the occurrence of any pattern in Figure 3 leads to one that is nonzero
with a high probability. This pattern enforces first-order
correlation of the scattering coefficient and therefore continuity of the target.
■ Weak rejection: Apart from the scenario of strong rejection
and strong acceptance patterns, a noninformative prior is
imposed on any other neighborhood patterns for Xmn. The
prior Beta (e 2, f2), with e2 = f2, is used to impose a noninformative prior on qmn. This appropriately allows the model
to be effective in imposing the prior whenever necessary
and to remain noninformative whenever no strong rejection
or acceptance patterns appear.
By adaptively selecting from different beta hyperpriors, the statistical model can either encourage continuity or
independence, apart from mere sparsity. In this manner, the
structured information can be flexibly incorporated to obtain
concentrated imagery results. A key component in incorporating the prior is that it is imposed on the parameter q rather
than directly on W. The underlying motivation for this formulation is that it is more flexible to impose a probabilistic belief
than a rigid support W.

IEEE SIgnal ProcESSIng MagazInE

November 2016

Table of Contents for the Digital Edition of Signal Processing - November 2016