Signal Processing - November 2016 - 94

Neighbors of Site i

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Range Cell

Site i
Types of Cliques

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80
100

Singleton Horizontal Vertical Diagonal Diagonal
3π/4
π/4

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10

FIGURE 5. An example of a second-order MRF model, where eight neigh-

(a)

bors are considered.
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Range Cell

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60
80
100
120
10

30
40
Doppler Cell
(b)

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Range Cell

compared with the simple first-order approach. In Figure 5,
the construction of the MRF model is presented, where a secondorder neighborhood system is employed. This model allows
continuity from four directions, i.e., horizontally, vertically, 3r/4 , and 1r/4 diagonally. The authors argued that
adopting second-order MRF enables the capture of correlations of the investigated scatterer with its nearest eight
neighbors, as shown in Figure 5.
To impose continuity of the target scene, the authors in
[46] considered a more general structured sparse prior as
compared with that in [39] and [45]. A more complicated
continuity prior has been proposed by modifying the spikeand-slab modeling to better preserve the weak scatterers.
Moreover, the hyperparameter selection in [39] and [45] is
avoided by adopting an MRF prior, since all the parameters can be automatically inferred. This is a very desirable
feature for statistical inference. Based on this model, the
authors employed a VB expectation maximization method
for inference, where an improved rate of convergence can
be obtained, as compared with the method in [39] and [45].
As commented earlier, the VB-based method generally
requires less computational complexity than the MCMCbased one.
In Figure 6, we can observe that both the first-order continuity method in [39] and [45] and the second-order continuity
method in [46] produce much-enhanced radar images in the
sense of less noise and a better-concentrated target region,
compared with the , 1-regularized method. As shown in Figure 6(b) and (c), the second-order continuity-based method
performs much better than the first-order one in terms of
removing the undesirable isolated artifacts and preserving weak scatterers outside and within the target region,
respectively. More important, the computational time of the
second-order continuity method is much less than that of the
first-order one [46].
The above discussion demonstrates how, by incorporating
structural priors in addition to sparsity, a statistical framework
provides superior performance compared to a merely sparsitybased framework. The major advantage of these approaches is

30
40
Doppler Cell

60
80
100
120
10

30
40
Doppler Cell
(c)

FIGURE 6. A comparison of radar images with a quarter of the full measurements and SNR = 5 dB obtained by (a) an , 1-regularized method,
(b) a method in [39], and (c) a method in [46].

that they can statistically impose the structured sparsity on the
signal in a rather flexible way, which allows the algorithm to
adapt the structured sparse estimation in a data-driven manner. More specifically, in all the introduced models, the structural information is not directly imposed on the sparse signal
itself but on the probability distribution that determines the
sparsity profile.

IEEE SIgnal ProcESSIng MagazInE

November 2016

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