Signal Processing - November 2016 - 90
p (Y) =
#H p (H) p (Y | H) dH,
and thus the posterior distribution in (9) is not attainable.
Although MAP estimation can be obtained from this model,
the full posterior is more desirable, as it provides a more
accurate description of the estimated parameter.
Because the exact inference is not attainable from this
model, two classical inference techniques known as the Markov chain Monte Carlo (MCMC) method and the variational
Bayesian (VB) method are often used to approximate the posterior from sampling and optimization, respectively. In this
way, the approximated posterior can be obtained at the cost
of increased computational complexity compared with other
sparse signal recovery methods. The MCMC method is accurate when the number of samples becomes large, while the VB
method provides a desirable approximation with a reasonable
computational complexity.
The MCMC method
This method approximates the posterior by sampling. MCMC
is a strategy for generating samples, while the equilibrium
distribution of the Markov chain is the same as the desired
probability distribution [34]. The most widely used MCMC
algorithms are the Metropolis-Hastings and the Gibbs sampling algorithms [27]. Under the assumption that all the conditional distributions are available, Gibbs sampling is easily
applicable. In fact, Gibbs sampling can be considered as a
special case of the Metropolis-Hastings algorithm if the conditional distributions are provided [34].
Since the conditional distribution is available in our graphical model, as shown in Figure 1, we will briefly review Gibbs
sampling. In this approach, sequential sampling of the conditional distribution, expressed as
H i ~p (H i | H k ! i, Y),
(10)
is performed. Therefore, the algorithm iterates until the desirable posterior is obtained. A more detailed description of the
algorithm can be found in [34] and the references therein.
The VB method
The main idea of this method is to approximate the true posterior by a factorizable form
q (H) =
k
% q (H i) .
(11)
i=1
This is known as the mean-field assumption [27]. The
objective is to find a factorizable q (H) that is as close as
the true posterior p (H ; Y). The closeness of the estimated
posterior to the true one in the VB method is measured by
the Kullback-Leibler (KL) divergence, and thus the optimal
approximated posterior is obtained by minimizing the following KL divergence:
q* (H) = argmin
q (H)
90
# q (H) ln pq(H(H| )Y) dH.
(12)
Based on (11) and (12), it can be shown that the approximated posterior for each of the variables can be calculated as
[27], [35]
q* (H i) = exp " ln p (H, Y)
q (H\H i) ,,
(13)
where · q (·) represents expectation with respect to the
probability density function q (·).
In the MCMC-based methods, sampling is required for
each step during iterations. In contrast, the VB-based methods
require matrix inversion in each step during iterations. Notably, sampling and matrix inversion would generally induce
high computational complexity for MCMC and VB, respectively. Due to these reasons, statistical sparsity-based methods generally cost more computations than the nonstatistical
approaches. The MCMC method can achieve better estimation
accuracy than the VB method, but at a higher computational expense. In a practical application, one should choose the
method according to its computational cost tolerance.
In summary, the key advantages of statistical sparsity-based
methods are:
■ They avoid regularization parameter tuning. Parameter tuning is not required in statistical sparsity-based methods,
which will be demonstrated in all the applications reviewed
in this article.
■ They provide full posterior. With this capability, desirable
improvements can be achieved by properly manipulating
the statistical model, particularly as shown in the sections
"Statistical Sparsity-Based Autofocus Techniques
in Radar Imagery" and "Statistical Sparsity-Based
SAR GMTIm."
■ They offer flexible modeling. Since the model is constructed probabilistically, encoding the prior can be carried
out in a rather flexible way, which is demonstrated in the
section "Enhanced Target Imagery by Exploiting Structured Sparsity."
Despite their remarkable advantages, the key limitations of
the statistical sparsity-based methods lie primarily in the following facts.
■ They have high computational complexity. The generally
required computational cost of statistical sparsity-based
methods is higher than that required by greedy or regularized methods.
■ They require sparsity assumption. The success of almost
all sparsity-based methods depends on the existence of
sparsity or compressibility. If the radar target scene does
not exhibit sparsity, modifications should be made to allow
a sparse representation [3], [5].
Superresolution radar imagery
In conventional Fourier-based radar imagery, the resolutions
in cross range and slant range are bounded by the Rayleigh
limit, which can be overcome by superresolution techniques.
In general, superresolution radar imaging can be well formulated as an inverse problem, where the scattering field is
required to be inversely estimated from the received radar
IEEE SIgnal ProcESSIng MagazInE
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November 2016
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Table of Contents for the Digital Edition of Signal Processing - November 2016
Signal Processing - November 2016 - Cover1
Signal Processing - November 2016 - Cover2
Signal Processing - November 2016 - 1
Signal Processing - November 2016 - 2
Signal Processing - November 2016 - 3
Signal Processing - November 2016 - 4
Signal Processing - November 2016 - 5
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Signal Processing - November 2016 - 7
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Signal Processing - November 2016 - 131
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Signal Processing - November 2016 - 134
Signal Processing - November 2016 - 135
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Signal Processing - November 2016 - 138
Signal Processing - November 2016 - 139
Signal Processing - November 2016 - 140
Signal Processing - November 2016 - 141
Signal Processing - November 2016 - 142
Signal Processing - November 2016 - 143
Signal Processing - November 2016 - 144
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Signal Processing - November 2016 - 146
Signal Processing - November 2016 - 147
Signal Processing - November 2016 - 148
Signal Processing - November 2016 - Cover3
Signal Processing - November 2016 - Cover4
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