IEEE Geoscience and Remote Sensing Magazine - December 2022 - 36

where the first term is used to preserve the data fidelity during
the solution; the second term of the Lp-norm is used to enforce
the sparse constraint (known as the regularizer); and p
c
is defined as the sparsity coefficient to balance between the
two terms, directly determining the performance of sparse reconstruction.
Empirically, the choice of p
c can be obtained
from the statistical aspect. Essentially, (6) can also be derived
from a maximum a posteriori (MAP) estimator [81]. It is necessary
to mention that the group of reweighted least-squares
methods has been advocated by employing a sequence of
weighted L2-norm minimization [104] and weighted L1norm
minimization [105]. The reweighted L2-norm has the
advantage of a closed-form solution, while the reweighted
L1-norm has a sparser solution with better convergence.
Furthermore, the data-driven and learning-based models
are investigated, such as dictionary learning algorithms [106].
To overcome the basis mismatch of the sparse dictionary ,W
the dictionary learning approach is adopted in CS. Typically,
sparse coding is one famous technique, which can be derived
from (6) in a refined formulation as
min/_ rr r
air W i
UW
i 2
2
c
,
where si
r
and ai
r
are the ith patch samples of s and a, used for
training the dictionary W . In this way, the optimal sparse dictionary
can be learned from the data in an adaptive manner
to effectively improve the sparsity enforcement on a, which is
very useful in sparse SAR imaging.
The convex algorithms also have a long history in radar
imaging, such as Tikhonov regularization, which can be
traced back to the image processing community [19]. For
sparse scenes, the aforementioned algorithms are widely
used in SAR imaging in a straightforward manner and usually
outperform the greedy algorithms. To sparsely represent
SAR images, many different algorithms are proposed and can
be mainly classified into three groups.
The first group employs some predesign dictionaries, such as
wavelet [107] and discrete cosine transform [108], directly on
the complex SAR images. For the class of wavelet, the multiresolution
property has some advantages in image compression.
Due to the complex property of SAR images with random
phase nature, this group of methods usually has very
limited performance.
The second group only sparsely represents the magnitude
of the SAR image by separating from the phase term
[18], [20], [21], [33], [34], [43], [109]. To overcome the
random phase issue, the sparse SAR imaging algorithms
are proposed using Lp-norm ()p011 # regularization on
the magnitude of the SAR image with a nonlinear operator.
In this manner, the conventional dictionaries, including
wavelet [109], total variation (TV) [110], and so forth, are
utilized for feature enhancement of a SAR image, such as
region smoothing.
The last group refers to the structure-based [111], [112],
[113], [114] and patch-based [21], [115] processing algorithms
in a data-driven manner. The typical one is the sparse
36
sa a-+ iii1
(7)
coding [115] and similar dictionary learning method, which
has been widely applied to SAR and InSAR image denoising.
The other one is to model the cluster or structure sparsity
for improved imaging performance, which has already been
applied to ISAR [114] and TomoSAR [116]. In addition, the
patch-based processing methods are very popular in SAR
imaging for effective multidimensional latent representation,
including block-matching 3D [117], nonlocal sparse
[118], and other methods. The last group can provide perfect
feature enhancement of a SAR image due to its data-driven
characteristics and has found a wide application in speckle
reduction and interferometric phase denoising [58].
In addition, various types of jointly sparse regularization
algorithms are proposed in MMV applications, considering
the coherence between multi-images, such as wide-angle SAR
[119]; multichannel SAR [43], [120], [121]; and multitemporal
SAR [122]. One major advantage of the convex regularization
is its tolerance for model errors, which has also
been explored in SAR imaging. Incorporating the turbulence
during the measurement matrix ,U such as nonideal radar
platform motion and noncooperative moving target motion,
sparse autofocusing [81], [123] and parametric sparse methods
[114], [124], [125], [126] have already been studied in
SAR imaging and can effectively improve the focusing quality
by overcoming the turbulence factors.
BAYESIAN ALGORITHMS
The last class includes sparse Bayesian algorithms from the
statistical framework by employing the uncertain information.
The most popular choices are Bayesian CS (BCS) [127]
or sparse Bayesian learning (SBL) [128], which are the basic
foundations of this group method. In this paradigm, the
Bayesian rule is used to construct the posterior density of a
from (2) and (3), which can be expressed as
p_as;Ti =
where pp() pdsa
() ;;
saa$TT;
= #
pp^a;
p^ h
;
i
_sa $TTh
s
(8)
^h is the marginal likelihood,
and D is the set of statistical parameters. From the
Bayesian perspective, the hierarchically Gaussian models
are usually employed for the likelihood, and a sparse prior
is used for conjugating with each other [129]. Various hierarchical
models have been widely studied [82], including
Laplace distribution, Student's distribution, exponential
distribution, the mixture form, and so on. In this way, the
associated Bayesian inference is applied for a closed-form
solution. For clarity, Figure 4 shows a graphical representation
of one typical hierarchical model (the scaled Gaussian
mixture). To distinguish from the MAP estimator (i.e.,
Lp-norm regularization), the full posterior is exploited by
the Bayesian algorithms, which offers great benefits for accurate
parameter estimation and signal recovery. The major
advantage of the Bayesian method is its simple extension
and self-consistency, which have been studied in structured
sparsity [113], dictionary learning [114], and other
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