Signal Processing - November 2016 - 95

Statistical sparsity-based autofocus
techniques in radar imagery

in the scaled Gaussian mixture model introduced previously.
The most straightforward way to obtain an estimate of the
phase error E is to maximize the expected log-likelihood
function as

The CS-based radar imagery techniques discussed in the
previous sections generally depend on the premise that preprocessing procedures, such as range cell migration
t = arg min - ln p (Y, X, a, m; E)
(RCM) correction and phase adjustment, have been per(16)
E
q (X) q (a) q (m) .
E
fectly conducted. Unfortunately, this is not a valid assumption from a practical viewpoint, since the motion of the
Equation (16) is strictly convex with a close-form solution.
target cannot be precisely compensated in coarse preproBy solving the optimization problem in (16), the updating
cessing stages. If these errors are not properly corrected or
formula can be obtained [49]. This updating rule for phase
compensated for before carrying out any CS-based
error is rather similar to that of the regularized approach in
algorithms, the reconstructed radar image will not be
[14] and [15], because the updating formula could use only
well concentrated.
the first-order moment of X to estimate E, and the obtained
Recently, phase error correction has been considered by
covariance matrix R of X does not appear in this updating
utilizing a sparse recovery technique, where alternating
rule. In other words, this formulation deviates from the original intention of utilizing higher-order statistical information
, 1-regularized approaches [14] were proposed to obtain more
in the first place.
focused images. In these methods, the sparse scattering
To properly utilize the uncertainty information, the
coefficient and the phase errors are iteratively estimated to
work in [47] proposed to incorporate
induce sparsity and obtain a focused radar
By incorporating
the obtained covariance matrix R in
image. Although these methods have demonstrated remarkable improvements over
the
estimation of phase errors to obtain
structural priors in
conventional autofocus techniques, these
enhanced
accuracy. Toward this end,
addition to sparsity, a
regularization-based methods might conthe phase error is deliberately modeled
statistical framework
verge to a shallow local minimum duras a complex parameter a i + jb i rather
provides superior
ing the iterative procedure. The alternate
than explicitly as e j{i. By introducing
performance compared to this complex parameter instead of the
optimization between the sparse scattera merely sparsity-based
ing coefficient and the phase error would
angle parameter { i, we will see that the
inevitably result in error propagation [47].
uncertainty
information can be naturally
framework.
More concretely, the alternate optimizaincorporated in the algorithm to achieve
tion scheme would introduce errors, since the estimation
enhanced estimation accuracy of E in each iteration. In
accuracy of one parameter substantially influences that of
the derived updating formula in [47], it can be seen that R,
another. This issue is particularly severe with undersampled
which contains uncertainty information, can be incorpodata and in low SNR conditions. To appropriately overcome
rated into the estimation of the phase error parameter E.
the aforementioned limitations, high-resolution imagery and
It is demonstrated in [47] that by replacing the true phase
phase error correction have been formulated in a statistical
error parameters with complex-valued error parameters, the
sparsity-based model [47]-[49]. In this formulation, probabiresulting scheme could utilize the estimation uncertainty
listic models are imposed on the signal to encode sparsity in a
information and obtain a performance gain as compared
statistical way. Subsequent parameter estimation is conducted
with regularized sparsity-based autofocus techniques.
within a sparse Bayesian learning framework [19], [47].
In Figure 7, an illustrative example is presented to evaluate the performance of the updating rule without and with
high-order uncertainty information. In this simulation, a
Statistical sparsity-based autofocus
total of 11 scatterers are present in the imaging scene. In
Assuming that the phase error in radar imagery exhibits range
Figure 7(a), the random phase error is shown. Figure 7(c) and
invariance [15], the mathematical model can be stated as
(d) shows that both updating rules lead to a more focused
image compared with the RDA shown in Figure 7(b). In parY = EU 1 AX + N,
(15)
ticular, the updating rule without utilizing the uncertainty
information leads to a less focused image, where undesirwhere E = diag (e j{1, ..., e j{ P) denotes the phase er ror
able sidelobe effects exist for almost all the scatterers on
matrix, which is a diagonal matrix representing crossthe imaging scene. In contrast, the image obtained by the
range variant phase errors. In [47], the authors utilized the
updating rule utilizing the uncertainty information is well
scale Gaussian mixture model to impose sparsity on X.
focused, with substantially suppressed sidelobe effects.
The estimation of X, a, and m is obtained individually,
Quantitative evaluation also demonstrates that the radar image
since they are task-specific parameters, while estimation of
in Figure 7(d) provides a lower NMSE X as well as MSE {
a 0 and E is performed in a global manner due to the taskinvariant property.
than those obtained in Figure 7(c) due to the inherent abilAccording to the graphical model [47], the parameters
ity to utilize the uncertainty information of estimation of
X, a, m, and a 0, can be estimated in a way similar to that
X. This validates the motivation of utilizing the uncertainty
IEEE SIgnal ProcESSIng MagazInE

November 2016

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