IEEE Geoscience and Remote Sensing Magazine - December 2022 - 43

concept is given in the following sections. In particular, we
also provide necessary discussions of the challenges encountered
for current sparse SAR imaging together with potential
compelling solutions.
MEASUREMENT MATRIX CONSTRUCTION
The construction of measurement matrix U in (2) is a very
important issue in sparse SAR imaging. There are two major
terms. First, U is related to the undersampling matrix U
from compressive sampling [39]. Second, the construction of
the SAR observation matrix H is necessary during the solution
of sparse SAR imaging [20], [30], [157], [158], which
directly determines the accuracy and efficiency of sparse imaging
implementation.
In CS, the compressive sampling is related to the RIP of
U , which is the basic precondition for the potential success
of sparse SAR imaging [39]. Usually, the compressive
sampling in the azimuth dimension is implemented in the
slow time domain, indicating the variation of pulse repetition
frequency. Meanwhile, the compressive sampling in
the range dimension is realized by transmitting the sparse
frequency waveform or undersampling in the range time
domain [41], [159]. From the theory of CS, the type of
random sampling is one of the most ideal choices due to
the perfect RIP of the random matrix [38], [39], [40]. Typically,
the random or pseudorandom sampling [26], [83]
has been suggested as an optimal option in theory. However,
the truly random sampling in the 2D time domain
may not be practical for existing hardware, such as analogto-digital
converters.
In contrast, random undersampling [26], jittering/
staggering [83], and others, which are derived and modified
based on conventional uniform sampling, are suggested
due to the simple implementation. Taking the azimuth
dimension as an example, the undersampling tends
to introduce the sparse aperture, where the compressed
gap sampling (CGS) and compressed random sampling
(CRS) are commonly used [27], [114]. The details on the
introduction of CGS and CRS can be found in [27], [31],
and [114]. In the waveform design, the sparse bandwidth
waveforms [24], [160], [161], [162] have been investigated,
including sparse frequency waveforms [160] (such as
sparse step -frequency [24] and sparse probing-frequency
[159]); coded incoherent/orthogonal signals [83], [163]
(such as random noise [159], [163] and frequency division
multiplexing [201]); and other optimization-derived
waveforms [159].
Specifically, the undersampling can be applied in the
range time domain by transmitting the LFM signal, which
only needs a minor change to the uniform sampling schedule
of the radar system. From the aspect of CS, the scheme
of waveform design or sparse sampling in the range time
domain does contribution to compressive sampling in SAR,
which partly determines the RIP of
U . Meanwhile, the compressive
sampling in the slow time dimension is another
important issue of the sampling scheme in SAR, which is
DECEMBER 2022 IEEE GEOSCIENCE AND REMOTE SENSING MAGAZINE
equivalent to the design of the sparse synthetic aperture array.
Generally, 2D sparse sampling has better performance
than the 1D case [138]. It is believed that, in the future, the
sparse sampling strategy in SAR may lead to great progress in
the development of radar systems.
Usually, the solution of sparse SAR imaging needs several
iterations to find an optimal result. In this case, the usage
of the exact form of H in (2) is very time-consuming,
hindering practical real-time application. The reason lies
in the fact that the calculation of H and its inversion HH
corresponds to forward projection (FP) and back projection
(BP) during sparse SAR imaging [18], which usually
require a high computational burden. Taking into account
that the MF is efficient by using the fast Fourier transform
(FFT), a well-defined approximation of H can be deduced
from the inverse of the SAR focusing procedure, which is
named fast CS SAR imaging [157]. In this manner, FFT can
be applied during the solution of sparse SAR imaging with
high efficiency. Accordingly, the approximation of H can
be derived from the range Doppler algorithm [157], chirp
scaling algorithm [164], wavenumber-domain algorithm
[158], etc. Essentially, the efficiency improvement comes
from the ability of the FFT to reduce the computational
complexity. However, if the compressive sampling of H
invalidates the application of FFT by transforming s into
the 2D frequency domain, the meaning of " fast " may be
lost. In this case, it is necessary to find alternative solutions,
such as the fast nonuniform Fourier transform, fast
BP, and fast factorized BP [165], and further study of these
is very important.
A more detailed discussion of this issue with a review of
the existing work is presented in the " Superresolution Imaging "
section.
SPARSE REPRESENTATION
One important perspective of sparse SAR imaging is that the
image needs to be represented in a sparse way. For a SAR image,
it is hard to evaluate the sparsity because it all depends
on the observed scenes. In special cases, the SAR image is
assumed to be sparse only when the observed scenes are
sparsely distributed in space, usually referred to as ISAR images,
clutter-reduced SAR GMTI images, and 3D images. However,
the condition of sparsity from scenes cannot always be
satisfied because a SAR image is usually not sparse when the
dense scene captures almost the entire space. Therefore, it is
very important to find a useful transform dictionary for the
sparse representation of SAR images.
In the literature of CS, sparse representation using a predesigned
dictionary, including the orthogonal basis and
redundant dictionary, has been widely applied in optical
image processing. Unfortunately, the conventional dictionaries
cannot be directly applied due to the systemic characteristic
of SAR images. Built upon coherent processing, SAR
images are complex valued with a random phase [43]. The
unique property of the random phase dramatically reduces
the correlation between neighboring pixels. Meanwhile, SAR
43
IEEE Geoscience and Remote Sensing Magazine - December 2022

Table of Contents for the Digital Edition of IEEE Geoscience and Remote Sensing Magazine - December 2022
