4586 IEEE TRANSACTIONS ON GEOSCIENCE AND REMOTE SENSING, VOL. 50, NO. 11, NOVEMBER 2012
Fig. 2. Mapping 2-D/3-D array and image into vectors.
The first term in (4) causes the resolving in the x and y direc-
tions, whose angular resolutions ρ
x
and ρ
y
can be calculated as
[14], [33]:
ρ
x
=
υ
L
x
(5.1)
ρ
y
=
υ
L
y
(5.2)
where L
x
and L
y
denote the effective aperture lengths in the x
and y-directions.
The second term causes the resolving in the range direction.
The ambiguity function χ
R
(t) could be regarded as a rectangu-
lar impulse function for pulsed radar or sinc function for pulse-
compression radar, whose resolution ρ
r
can be calculated,
respectively as:
ρ
r
=
cT
pu
2
or ρ
r
=
c
(2B)
(6)
where c denotes the speed of light, T
pu
denotes the pulse
duration of pulsed radar, B denotes the bandwidth of received
signal.
For multiple-scattering target, the echo is the sum of all
satterers, i.e.,
D(t, n)=
M
m=1
σ
m
d(t, n; m) (7)
where M denotes the total number of scatterers (including those
scatterers with zero scattering coefficients), σ
m
denotes the
scattering coefficient of the mth scatterer.
In the term of sparse recovery, the signal processing is
always treaded in the matrix form. For microwave 3-D imaging,
however, the observed data and image space are both multidi-
mensional array, and we need to rearrange the acquired data
and imaging space in vector form (shown in Fig. 2) before
expressing the microwave 3-D imaging problem in matrix form:
Y =[y
i
],y
(n−1)N+t
= D(t, n) n =1, 2,...,N,
t =1, 2,...,P (8.1)
X =[x
m
] m =1, 2,...,M (8.2)
where P denotes the sampling number in the range domain.
Thus, the measurement matrix A can correspondingly be
written as:
A =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎢
⎣
d(1, 1; 1) d(1, 1; 2) ... d(1, 1; M)
d(2, 1; 1) d(2, 1; 2) ... d(2, 1; M)
... ... ... ...
d(P, 1; 1) d(P, 1; 2) ... d(P, 1; M )
d(1, 2; 1) d(1, 2; 2) ... d(1, 2; M)
... ... ... ...
d(P, N;1) d(P, N;2) ... d(P, N; M)
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎥
⎦
PN×M
(9)
where Y ∈ C
PN
, X ∈ C
M
, A ∈ C
PN×M
.
And the matrix expression of microwave 3-D array
imaging is:
Y = AX (10)
where Y denotes the observation vector, A denotes the mea-
surement matrix, X denotes the scattering coefficient vector.
Mathematically, given the observation vector Y and mea-
surement matrix A, one can directly obtain the scattering
coefficient vector X by solving a linear equation group. In
practice, however, the sizes of the observation vector, scatter-
ing coefficient vector, and measurement matrix are extremely
huge (might be hundreds of thousands or more), and it is ill-
conditioned and time-consuming to solve the linear equation
group directly.
To reduce the computational cost, one can divide the imaging
space into a series of equiv-range slices (under far-field assump-
tion), which is shown in Fig. 1 (right). Thus, the measurement
matrix A(t
0
) in one equiv-range slice can simply be rewritten
as:
A(t
0
)=
⎡
⎢
⎣
d(t
0
, 1; 1) d(t
0
, 1; 2) ... d(t
0
, 1; M
0
)
d(t
0
, 2; 1) d(t
0
, 2; 2) ... d(t
0
, 2; M
0
)
... ... ... ...
d(t
0
,N;1) d(t
0
,N;2) ... d(t
0
,N; M
0
)
⎤
⎥
⎦
N×M
0
(11)
where t
0
denotes the distance between the receiver array center
to the equiv-range slice, M
0
denotes the scatterer number in one
equiv-range slice.
In the remaining part of this paper, we just consider one slice,
and one can obtain 3-D image one slice by one slice. A(t
0
) and
M
0
will be replaced by A and M, respectively, without loss of
generality.
B. Sparsity of 3-D Imaging
Imagining a 3-D scene illuminated by EM wave, for both
2-D and 3-D imaging systems, their echoes at every receiver
with same position are identical, and contain the same scatterers
exactly. The difference is that for 2-D imaging system, the
scatterers of the whole scene are projected onto the image plane
(no matter which image plane is selected.). As a result, the
projection plane is logically filled with scatterers, though some
of them are dim and can be ignored in practice.
For 3-D imaging system, however, the signal characteristic
guarantees that the scatterers in different slices (such as scat-
terers with different distances to the radar.) hardly influence to
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