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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