Aerospace and Electronic Systems Magazine July 2017 - 42
Automatic Target Recognition in Missing Data Case
(
)
where τ = xn p , yn p , zn p is the 3-D location of the radar platform,
and vn p ( f m ) is the additive noise for the npth pulse. α pr ,l and c are
the reflection coefficients corresponding to the scatterer pr,l and the
speed of light, respectively [9]. The expression in (2) can be represented in a matrix form as follows:
2
Q r ,l = R − α pr ,l d r ,l d rH,l ,
and
R
L
2
R = α pr ,l d r ,l d rH,l .
r =1 l =1
(3)
y = Dα + v,
αˆ p =
r ,l
T
y = y1 ( f1 ) y1 ( f M ) y2 ( f1 ) y N p ( f M ) is the measured data
vector, D = d1,1 d1, L d R , L is the known dictionary that relates the received data to the various scatterer locations, and
T
v = v1 ( f1 ) v1 ( f M ) v2 ( f1 ) vN p ( f M ) is the additive noise vector. The vector dr,l has the n p − 1 × M + m th component of the
4π
−j
f m ( τ − p r ,l )
c
e
term
for np = 1,...,Np and m = 1,...,M in (2). To obtain estimates of the vector α in (3), we minimize the following
weighted least squares problem [12]:
(
min y − α pr ,l d r , l
α pr ,l
where u
2
Q −r 1,l
2
Q −r 1,l
,
u H Q r−,1l u,
)
(4)
(6)
Minimizing the cost function in (4) gives the following solutions:
T
where α = α p1,1 α p1,L α pR ,L is a sparse vector of
the reflection coefficients of the image to be obtained,
(5)
d rH,l Q −r ,1l y
d rH,l Q r−,1l d r ,l
,
(7)
which can be rewritten as
αˆ p =
r ,l
d rH, l R −1y
d rH,l R −1d r ,l
,
(8)
by using the Woodbury matrix inversion lemma [20] and (5). Instead of computing Q −r ,1l for each pixel in (7), (8) only computes
R−1 once and thus saves computations significantly. Note that the
estimate of α pr ,l requires the knowledge of R and vice versa.
The BP image formation algorithm, while computationally attractive, offers poor resolution and high sidelobe levels in comparison with many data-adaptive imaging algorithms. For example, consider Figure 5, which shows the resulting image when the
Figure 5.
Hybrid high-resolution SAR images: full data {(a),(d),(g)}, 30% missing data {(b),(e),(h)}, and 50% missing data {(c),(f),(i)}. Each image is in 5m by
5m and 0.05m pixel size.
42
IEEE A&E SYSTEMS MAGAZINE
JULY 2017
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