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Sparsity-Driven ISAR Imaging Based on
Two-Dimensional ADMM
Hamid Reza Hashempour
Abstract—Compressed sensing (CS) can achieve high resolu-
tion inverse synthetic aperture radar (ISAR) imaging of mo-
ving targets with limited measurements. Recently, alternating
direction method of multipliers (ADMM) has been introduced
to solve the optimization problem for one dimensional (1D)
sparse signal recovery. The main drawback of 1D sparsity-driven
algorithms are the high memory usage and the computational
complexity. Thus, in this paper a novel two dimensional (2D)
ADMM approach is presented which can be directly applied
to the ISAR model in matrix form, and needs lower memory
and computations compared to the 1D algorithm. Moreover, the
performance of the 2D-ADMM method is better than the 2D
smoothed L0 (2D-SL0) and 2D gradient projection sequential
order one negative exponential (2D-GP-SOONE) algorithms in
different signal-to-noise ratio (SNR) conditions and sampling
rates. Joint simulations and measured data results based on real
data of Yak-42 airplane, validate the superiority of the proposed
approach.
Index Terms—Inverse synthetic aperture radar (ISAR), alter-
nating direction method of multipliers (ADMM), sparse matrix
recovery, two dimensional compressed sensing (2D-CS)
I. INT ROD UC TI ON
Inverse synthetic aperture radar (ISAR) is an effective signal
processing tool which can obtain high resolution images of
moving targets from all-day and all-weather environment.
ISAR widely applied in various military and civilian appli-
cations, e.g., target identification and classification, air/space
surveillance, etc. [1], [2]. In order to attain high resolu-
tion ISAR images in range and cross-range directions, wide
bandwidth signal and long coherent processing interval (CPI)
are required, respectively. However, by increasing the radar
bandwidth and CPI, the amount of data increased significantly,
which causes a big problem in data acquisition and storage
system. On the other hand, in a long CPI, the assumption of
constant rotation rate of the target is not valid, which degrades
the ISAR image obtained from the conventional range-Doppler
Algorithm (RDA).
Compressed sensing (CS), as an emerging technique in
signal processing, states that a high-dimensional unknown
sparse or compressible signal can be reconstructed exactly
from limited measurement with overwhelming probability by
solving a sparsity-driven optimization problem [3]–[6]. CS has
been widely exploited and developed in radar imaging because
of intrinsic sparsity of ISAR images, and can achieve high
resolution images with limited measurements [7]–[13]. In [7],
a CS-based approach is presented to achieve high resolution
with limited pulses. The problem of ISAR imaging based
The author is with the School of Electrical and Computer Engineering, Shi-
raz University, Shiraz 7134851154, Iran e-mail: hrhashempour@shirazu.ac.ir.
on CS from incomplete data is also considered in [9]. The
problem of sparse vector recovery can be solved by several
methods, e.g., basis pursuit [14], matching pursuit [15], and
smoothed L0 (SL0) [16]. However, the aforementioned met-
hods cannot be directly used to solve a sparse matrix. In [17] a
2D-SL0 algorithm has been proposed, which could be applied
directly to 2D signals. Based on 2D-SL0, a 2D compressive
sensing (2D-CS) algorithm for high resolution ISAR imaging
is presented in [10]. A gradient projection sequential order one
negative exponential (GP-SOONE) method is introduced in
[11] to solve a constrained nonconvex problem and reconstruct
the sparse signal. This method has been further developed for
matrix recovery and the 2D-GP-SOONE algorithm has been
exploited for high resolution ISAR imaging of maneuvering
targets [12], [13].
Alternating direction method of multipliers (ADMM), is
a proximal splitting algorithm which can solve non-smooth
and large-scale optimization problems [18]–[20]. ADMM uses
auxiliary variables and divides the problem to individual
convex sub-problems which are solved easily based on prox-
imal operators. In this paper, based on this method, first a
one dimensional ADMM (1D-ADMM) approach for ISAR
imaging is presented, then a generalized 2D-ADMM algorithm
which can directly reconstruct sparse matrices is proposed. The
presented 2D approach has a lower computational complexity
and requires less memory storage for computation compared to
the 1D-ADMM method. Furthermore, 2D-ADMM is more ro-
bust to noise than the 2D-SL0 and 2D-GP-SOONE algorithms.
We also develop our algorithm for 2D-CS to reconstruct ISAR
images from undersampled data in both range and cross-range
directions. Several simulations and measured data of Yak-
42 aircraft, are exploited to validate the effectiveness of the
proposed algorithm for super-resolution ISAR imaging from
complete and undersampled data acquisition cases.
The remainder of this paper is organized as follows. Section
II introduces the signal model. Section III presents the propo-
sed 1D and 2D-ADMM ISAR imaging methods. Then, simu-
lated and real data are utilized to validate the efficiency and
superiority of the proposed approach in section IV. Eventually,
section V concludes this study.
II. SI GNA L MO DE L
In ISAR imaging, the instantaneous range of a target in-
cludes rotational and translational motions, which the former
provides the essential angular diversity to achieve cross-range
resolution, and the latter is undesired and in this paper, it
is assumed to be compensated completely by conventional
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Fig. 1. Geometry of a rotating target in ISAR model
motion compensation methods such as [2], [21]. Therefore,
after removing the translational motion, it is supposed that the
target is uniformly rotating as shown in Fig. 1.
After range compression, the received signal from the target
can be expressed as follows:
s(fm, tn) =
K
X
k=1
σk·exp −j4πfmRk(tn)
c+z(fm, tn)(1)
where z(fm, tn)is the additive white Gaussian noise after
range compression. fmdenotes the range-frequency and defi-
ned as: fm=fc+ (m−1)∆f,m=−(M−1)/2, . . . , M/2,
fcis the carrier frequency and ∆fis the space between two
samples in the frequency domain. tnstands for discrete slow
time and n= 1,2, . . . , N is the number of pulses. Kis the
number of scattering centers in the target, σkdenotes the radar
cross section (RCS) coefficient of the kth scatterer, cis the
speed of light, and Rk(tn)is the instantaneous distance from
the kth scatterer to the radar. In ISAR imaging, the target
is usually assumed to be in the far field, so Rk(t)can be
approximated as
Rk(tn)∼
=R0+xkcos(Ωtn)−yksin(Ωtn)(2)
where R0and Ωare the target’s initial range distance from the
radar and the rotation rate of the target, respectively. When the
CPI or the rotation rate or both are sufficiently small, Rk(t)
can be approximated as:
Rk(t)≈R0+xk−ykΩtn(3)
By substituting (3) into (1), neglecting the constant phase
term, and compensating the migration through resolution cells
(MTRC) [22], we have:
s(fm, tn) =
K
X
k=1
σk·exp −j4πfmxk
c
·exp j4πfcykΩtn
c+z(fm, tn)(4)
The target scene can be represented as a 2D matrix Xwith the
size of P×Qwhere p= 1,2, . . . , P and q= 1,2, . . . , Q are
the number of pixels along y and x axis, respectively. Then,
equation (4) can be written in matrix form as:
S=FaXFT
r+Z(5)
where (·)Tstands for transpose of a matrix. S∈CN×Mand
Z∈CN×Mrepresent the received signal and the complex
Gaussian noise matrix, respectively. Fa∈CN×Pand Fr∈
CM×Qexpress the partial Fourier matrices in the azimuth and
range directions, respectively, which are given by
Fa=
1 1 . . . 1
1ω . . . ω(P−1)
.
.
..
.
.....
.
.
1ω(N−1) . . . ω(N−1)(P−1)
, ω = exp (−j2π
N)
(6)
Fr=
1 1 . . . 1
1ν . . . ν (Q−1)
.
.
..
.
.....
.
.
1ν(M−1) . . . ν(M−1)(Q−1)
, ν = exp (−j2π
M)
(7)
In order to have supper-resolution ISAR images, N and M
should be smaller than P and Q, respectively.
III. PROP OS ED M ET HO D
In this section, first, an ADMM-based approach for sparse
recovery is derived, then the method is developed for the 2D
model of (5). Finally, the proposed algorithm is generalized
for 2D undersampling.
A. ADMM-based ISAR imaging
Conventional sparsity-driven algorithms cannot be directly
applied to matrix representation, as in (5). Thus, in CS, the
sparse matrix is stacked to a single column vector to recover
uniquely from an underdetermined linear system [3], [4]. By
utilizing Kronecker products, the 2D ISAR model in (5) is
equivalently expressed in vector form as:
s=Φx +z(8)
where s= vect(S),x= vect(X),z= vect(Z),Φ=Fr⊗
Fa,⊗denotes the Kronecker product, and vect(·)represents
stacking columns of a matrix into a vector one after the other.
The sparse vector xin (8) can be recovered by solving the
following optimization problem
ˆ
x= arg min
x
1
2ks−Φxk2
2+λkxk1(9)
where k·k2represents the Euclidean norm of a vector. To solve
(9), an algorithm based on ADMM is derived. In ADMM, the
problem must be reformulated by introducing a new variable
ˆ
x,ˆ
b= arg min
x,b
1
2ks−Φxk2
2+λkbk1s.t.x=b(10)
The associated augmented Lagrangian (AL) function is
LA
δ(x,b,u) =1
2ks−Φxk2
2+λkbk1
+<uH(x−b)+δ
2kx−bk2
2(11)
where uand δare the Lagrange multiplier and the penalty
parameter, respectively. ADMM minimizes LA
δover x,b,u
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separately, leading to sub-problems. Therefore, the scaled form
of ADMM reads
xj+1 = arg min
x
1
2ks−Φxk2
2+δ
2kx−bj+vjk2
2(12)
bj+1 = arg min
b
λ
δkbk1+1
2kxj+1 −b+vjk2
2(13)
vj+1 =vj+xj+1 −bj+1 (14)
where (·)Hdenotes the Hermitian transpose of a matrix
and vj=δ−1ujis called the scaled dual variable. The
minimization of the quadratic functions in (12) can easily be
solved by taking the first derivative. The solution of the l1
norm in (13) can be obtained by using proximal operators. A
proximal operator of g with parameter ρcan be defined as
proxρg(z) = arg min
x
1
2kx−zk2
2+ρg(x)(15)
For some choices of g, well-known closed form of the so-
called mforeau proximal mapping (MPM) function exists. For
example, when g(x) = kxk1, then the solution can be achieved
by soft-thresholding function: Sρ(z) = sgn(z)·max(|z|−ρ, 0).
Therefore, the solution for each of the optimization problems
in (12)-(14) can be obtained through the following equations:
xj+1 =ΦHΦ+δI−1ΦHs+δ(bj−vj)(16)
bj+1 =Sλ/δ(xj+1 +vj)(17)
vj+1 =vj+xj+1 −bj+1 (18)
Note that, although (16) has analytical solution, the compu-
tation of matrix inverse is impractical for large values of I.
Therefore, a Conjugate-Gradient (CG) algorithm with warm
starting is exploited in [23] to solve this subproblem. However,
applying CG algorithm is also time consuming. Here, we want
to further simplify (16) based on the matrix inversion lemma.
Faand Frare partial Fourier matrices with N < P and
M < Q, which satisfy the FaFH
a=I, and FrFH
r=I,
respectively. Therefore, ΦΦHcan be written as:
ΦΦH= (Fr⊗Fa)(Fr⊗Fa)H
= (Fr⊗Fa)(FH
r⊗FH
a)
= (FrFH
r)⊗(FaFH
a)
=I.(19)
Using (19) and the Woodbury matrix identity, we have
ΦHΦ+δI−1=1
δI−ΦH(ΦΦH+δI)−1Φ
=1
δI−1
1 + δΦHΦ.(20)
Based on (20), and after simple manipulation, the equation
(16) can be rewritten as:
xj+1 = (bj−vj)−1
1 + δΦH(Φ(bj−vj)−s)(21)
Therefore, the cost of (21) is the products by ΦHand Φ. If
we use the fast Fourier transform (FFT) algorithm instead of
matrix multiplication, the cost is of order O(nlogn)operati-
ons.
Based on above solutions for each step, the proposed 1D-
ADMM ISAR imaging procedure is demonstrated in Algo-
rithm 1.
Algorithm 1 The 1D-ADMM algorithm for ISAR imaging
1: Input: s, δ > 0, λ ≥0
2: Initialization: j= 0,x0=b0=v0=0.
3: while stopping criterion is not met do
4: xj+1 = (bj−vj)−1
1 + δΦH(Φ(bj−vj)−s)
5: bj+1 =Sλ/δ(xj+1 +vj)
6: vj+1 =vj+xj+1 −bj+1
7: end while
8: Output: ˆ
x=xj
It is worth noting that Algorithm 1 is stopped if a predefined
maximum number of iterations is reached, or if the correspon-
ding objective function does not decrease significantly any
more, i.e. when kxj+1 −xjk2/kxjk2< , where is a small
positive number.
The computational complexity of the 1D-ADMM algorithm
is dominated by step 4 in Algorithm 1. The matrix-vector
multiplications with the assumption of P=Qrequire
O(P4MN +P4+P2MN )computations. Since M , N < P ,
the complexity is in the order of O(P4MN). Thus, 1D-
ADMM is feasible only when P,M,N are fairly small.
B. 2D-ADMM Algorithm
In this section, we want to generalize Algorithm 1 to
obtain the 2D-ADMM method for sparse matrix recovery. The
equivalent matrix form of the l1minimization problem of (9)
is as follows:
ˆ
X= arg min
X
1
2
S−FaXFT
r
2
F+λkXk1(22)
where λis a regularization parameter and k·kFis the
Frobenius norm. It is worth mentioning that, the matrix
Φ∈CMN ×P Q has much more elements than Faand Fr.
Thus, solving the 2D problem in (22) which requires much
smaller memory is more efficient than the 1D optimization in
(9).
In order to obtain the 2D-ADMM algorithm, first, the
equivalent matrix form of step 4 in Algorithm 1 is derived.
Assume that b= vect(B)and v= vect(V), then we have:
(ΦHΦ)(b−v)−ΦHs
=(FH
r⊗FH
a)(Fr⊗Fa)vect(B−V)−(FH
r⊗FH
a)vect(S)
=(FH
rFr)⊗(FH
aFa)vect(B−V)−(FH
r⊗FH
a)vect(S)
=vect(FH
aFa(B−V)FT
rF∗
r−FH
aSF∗
r)
=vect FH
a(Fa(B−V)FT
r−S)F∗
r.(23)
where (·)∗stands for complex conjugate of a matrix. Therefore
the equivalent matrix form of step 4 in Algorithm 1 is obtained
as:
Xj+1 = (Bj−Vj)−1
1 + δFH
a(Fa(Bj−Vj)FT
r−S)F∗
r
(24)
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Then, the matrix representation of step 5 in Algorithm 1 is
given by:
Bj+1 =Sλ/δ(Xj+1 +Vj)(25)
where Sρ(Z)denotes an element wise operation with Sρ(Z) =
sgn(Znm)·max(|Znm |− ρ, 0) for all indices n, m of the N×
Mmatrix Z. These steps lead to the 2D-ADMM algorithm,
as a generalization of 1D-ADMM with matrix inputs, which
is given in Algorithm 2.
Algorithm 2 The 2D-ADMM algorithm for ISAR imaging
1: Input: S, δ > 0, λ ≥0
2: Initialization: j= 0,X0=B0=V0=0.
3: while stopping criterion is not met do
4: Xj+1 = (Bj−Vj)−1
1 + δFH
a(Fa(Bj−Vj)FT
r−S)F∗
r
5: Bj+1 =Sλ/δ(Xj+1 +Vj)
6: Vj+1 =Vj+Xj+1 −Bj+1
7: end while
8: Output: ˆ
X=Xj
Similar to Algorithm 1, the stopping criteria of Algorithm
2 is met when either kXj+1 −XjkF/kXjkF< ξ, where
ξis a small positive number, or the number of iterations
reaches a predefined limit. The computational complexity of
Algorithm 2, is also dominated by step 4. The matrix-matrix
multiplication at step 4 in Algorithm 2 with the assumption
of P=Qis performed with O(P2(P+N+ 3M) + P M N )
computations. Since M, N < P , the worst case complexity
is O(P3). Recall, that 1D-AMM has worst case complexity
of O(P4MN). Thus, there is a O(P M N )gain in the matrix
version compared to the vector approach. Consequently, Al-
gorithms 1 and 2 have the same output, however, Algorithm
2 is more efficient.
Let us compare the computational complexity of Algorithm
2 with 2D-SL0 and 2D-GP-SOONE. The most expensive
computational operation of both 2D-SL0 and 2D-GP-SOONE
in each iteration, is to compute the following equation [12],
[17]:
Xj+1 =Xj−F†
a(FaXjFT
r−S)(F†
r)T(26)
where (·)†denotes the pseudo-inverse of a matrix and is
computed as:
F†
a=FH
a(FaFH
a)−1=FH
a(27)
In a similar manner we have: F†
r=FH
r, and consequently
(26) is simplified to:
Xj+1 =Xj−FH
a(FaXjFT
r−S)F∗
r(28)
By comparing (28) with step 4 in Algorithm 2, it is observed
that 2D-SL0, 2D-GP-SOONE and 2D-ADMM have the same
worst-case complexity i.e. O(P3). However, the run time of
the algorithms are not necessary the same, since the other steps
and the number of iterations required for convergence of each
algorithm are different.
C. Generalizing to 2D-CS
Let us consider the sparsely sampled signal reconstruction.
The observation matrix for the 1D model has the form ΨΦ,
where Ψis an L1L2×MN binary matrix, where L1(L1< M)
and L2(L2< N)are the sparsely sampled numbers in the
range and cross-range directions, respectively, and Ψcan be
obtained by taking a subset of rows of an identity matrix.
Due to its particular structure, this matrix satisfies ΨΨT=I.
Similar to (20), based on this fact and using the Woodbury
matrix identity, we have
ΦHΨHΨΦ +δI−1=1
δI−1
1 + δΦHΨTΨΦ.(29)
Therefore, according to (29), step 4 in Algorithm 1, is modified
as:
xj+1 = (bj−vj)−1
1 + δΦHΨT(ΨΦ(bj−vj)−s)
(30)
For the matrix form in Algorithm 2, similarly step 4 can be
derived as:
Xj+1 = (Bj−Vj)−
1
1 + δFH
aΨT
a(ΨaFa(Bj−Vj)FT
rΨT
r−S)ΨrF∗
r(31)
where Ψawith the size of L2×Nand Ψrwith the size of
L1×Mare the sensing matrices in the cross-range and range
directions, respectively, and Ψ=Ψr⊗Ψa.
IV. EXP ER IM EN TAL RESULTS
A. Simulated data
In this section the performance of the proposed 2D-ADMM
algorithm is evaluated using synthetic data. The radar parame-
ters are shown in Table I. The scene size is 50 ×50, with 11
ideal point-like targets. The 2D-FFT, 2D-SL0, 2D-GP-SOONE
and 2D-ADMM algorithms have been applied to the simulated
data to form the ISAR images. The target scene is displayed in
Fig.2a. The normalized mean square error (NMSE) between
the estimated ISAR image and the target scene defined as
NMSEX= 10log10kˆ
X/|ˆ
X|max −X/|X|maxk2
F, is used to
quantitatively compare the different algorithms.
In order to achieve super-resolution ISAR images, we set
P= 2Nand Q= 2M, (i.e. P=Q= 100) in this simulation.
By adding complex Gaussian noise, raw data is generated for
signal-to-noise ratios (SNR) from −10 to 30dB. For example,
in SNR=-10 dB, the acquired ISAR images by the different
methods are shown in Figs. 2b-2e. Furthermore, the NMSEX
of the aforementioned algorithms for the different SNR levels
is shown in Fig. 2f. Observing the obtained ISAR images
(Figs. 2b-2e) and the NMSEXdiagram (Fig. 2f), it can be
understood that the proposed approach outperforms the other
ones in all SNR levels.
In the next simulation, we compare the algorithms for 2D-
CS. Fig. 3 demonstrates the ISAR images obtained by 2D-
GP-SOONE, 2D-SL0 and 2D-ADMM in udersampling ratio
(M N/P Q) of 0.1 for SNR =-5 and 5dB. Some artificial points
exist in the imaging results of 2D-GP-SOONE and 2D-SL0.
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TABLE I
RADA R PARA MET ER S FOR S IM ULATE D DATA
Carrier frequency fc10 GHz
Bandwidth B500 MHz
Pulse Repetition Frequency P RF 50 Hz
Number of range cells M50
Number of pulses N50
Original scene
-6 -4 -2 0 2 4 6
Range (m)
-6
-4
-2
0
2
4
6
Cross-range (m)
(a)
2D-FFT
-6 -4 -2 0 2 4 6
Range (m)
-20
-10
0
10
20
Doppler (Hz)
(b)
2D-GP-SOONE
-6 -4 -2 0 2 4 6
Range (m)
-20
-10
0
10
20
Doppler (Hz)
(c)
2D-SL0
-6 -4 -2 0 2 4 6
Range (m)
-20
-10
0
10
20
Doppler (Hz)
(d)
2D-ADMM
-6 -4 -2 0 2 4 6
Range (m)
-20
-10
0
10
20
Doppler (Hz)
(e)
-10 -5 0 5 10 15 20 25 30
SNR [dB]
-25
-20
-15
-10
-5
0
5
10
15
20
NMSE [dB]
2D-FFT
2D-GP-SOONE
2D-SL0
2D-ADMM
(f)
Fig. 2. (a) Original target scene. The obtained ISAR images using (b) 2D-
FFT, (c) 2D-GP-SOONE, (d) 2D-SL0 and (e) 2D-ADMM for SNR =-10 dB.
(f) NMSEXof different methods versus different SNR levels.
However, compared to them, our algorithm can achieve better
image quality in both SNR levels. Moreover, the NMSEX
of different methods versus undersampling ratio of 0.1 to
1, is depicted in Fig. 4 for SNR levels of 10, 5 and 0
dB. It is seen that NMSEXof 2D-ADMM in all cases is
lower than that of the other methods. When the sampling
ratio approaches 1, the performance of 2D-SL0 and 2D-GP-
SOONE are reduced and in sampling ratio of 1, the MSEs
of them are equal to that of 2D-FFT. However, for 2D-
AMM, by increasing the udersampling ratio the performance
is improved significantly which again validates the superiority
of the proposed algorithm.
2D-SGP-SOONE
10 20 30 40 50
Range (m)
10
20
30
40
50
Doppler (Hz)
(a)
2D-SL0
10 20 30 40 50
Range (m)
10
20
30
40
50
Doppler (Hz)
(b)
2D-ADMM
10 20 30 40 50
Range (m)
10
20
30
40
50
Doppler (Hz)
(c)
2D-SGP-SOONE
-6 -4 -2 0 2 4 6
Range (m)
-20
-10
0
10
20
Doppler (Hz)
(d)
2D-SL0
-6 -4 -2 0 2 4 6
Range (m)
-20
-10
0
10
20
Doppler (Hz)
(e)
2D-ADMM
-6 -4 -2 0 2 4 6
Range (m)
-20
-10
0
10
20
Doppler (Hz)
(f)
Fig. 3. The ISAR images obtained by 2D-GP-SOONE, 2D-SL0 and 2D-
ADMM in udersampling ratio of 0.1 for SNR =-5 dB ((a)-(c)) and SNR =5
dB ((d)-(f)), respectively
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
undersampling ratio
-25
-20
-15
-10
-5
0
5
10
15
20
NMSE [dB]
2D-FFT
2D-GP-SOONE
2D-SL0
2D-ADMM
(a)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
undersampling ratio
-25
-20
-15
-10
-5
0
5
10
15
20
NMSE [dB]
2D-FFT
2D-GP-SOONE
2D-SL0
2D-ADMM
(b)
0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9
undersampling ratio
-25
-20
-15
-10
-5
0
5
10
15
20
NMSE [dB]
2D-FFT
2D-GP-SOONE
2D-SL0
2D-ADMM
(c)
Fig. 4. NMSEXobtained from different methods as undersampling ratio is
varied for (a) SNR = 10, (b) 5, and (c) 0 dB.
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TABLE II
ENT ROPY O F TH E ISAR IMA GE FO R DI FFER EN T ALG OR ITH MS
Algorithm SNR = 10dB SNR = 5dB SNR = 0dB
2D-FFT 6.86 7.72 8.98
2D-GP-SOONE 5.24 5.93 7.20
2D-SL0 4.70 5.32 6.60
2D-ADMM 4.27 4.30 4.31
B. Real data
In this subsection, the measured data set of the Yak-42
aircraft, is utilized to further validate the effectiveness of the
proposed algorithm. The measured data set is collected by
a radar with a central frequency, bandwidth and duration of
5.52 GHz, 400 MHz and 25.6 µs, respectively. The target is
a Yak-42 aircraft sized 24m×24m. The complete radar echo
contains 256 pulses, and each pulse consists of 256 samples.
In Fig. 5 the ISAR images of Yak-42 exploited from
different methods are demonstrated. The number of pulses
(N), and the range cells (M) are 64 and 256, respectively.
In order to have super-resolution ISAR images, we set P=2N
and Q=2M. The first, second and third rows of Fig. 5 are
obtained under SNR levels of 10, 5 and 0 dB, respectively.
From Fig. 5, it is obvious that the images obtained from
the 2D-FFT have the lowest quality, and 2D-ADMM have
the best performance in all SNR cases. Although, 2D-SL0
and 2D-GP-SOONE have good results in SNR=10 dB, their
performance reduced dramatically in low SNR conditions.
Table II shows the entropy values of the ISAR images in Fig. 5
to quantitatively compares the performance of the algorithms.
The image entropy is defined as [2]
IEx=−
P
X
p=1
Q
X
q=1
|Xp,q|2
Elog |Xp,q|2
E(32)
where E=PP
p=1 PQ
q=1 |Xp,q|2is the image energy. Gene-
rally, a well-focused image has a low entropy value. As it can
be seen, the image entropy of the proposed algorithm in all
cases is the lowest compared to the others.
For the last experiment, we consider 2D-CS, i.e. the al-
gorithms are compared for different sampling rates of 12.5,
25, 50 and 75%. In this experiment, we set P=Q=256, and
the performance of algorithms is considered for SNR=10 dB
(Fig. 6) and 0 dB (Fig. 7). The undersampling is achieved
by randomly selecting the rows of the Fourier matrices in
both range and cross-range directions. Fig. 6 shows that in
low sampling rates the performance of all approaches are near
together, but in the higher sampling rates 2D-ADMM yields
better results comparing to 2D-SL0 and 2D-GP-SOONE. Fig.
7 displays similar results for SNR=0 dB. As expected in low
SNR conditions, the proposed method outperforms the others
in all sampling rate schemes, i.e. is more robust to noise
compared to the 2D-SL0 and 2D-GP-SOONE algorithms.
Furthermore, in order to have a quantitative comparison, the
entropy values of the ISAR images in Figs. 6 and 7 are
demonstrated in Table III. Clearly, the proposed method has
the lowest entropy in all conditions.
TABLE III
ENT ROPY O F TH E ISAR IMA GE OB TAIN ED F ROM D IFFE REN T AL GOR IT HMS
FO R DIFF ER ENT S AMP LI NG RAT ES
Scheme Sampling rate 12.5% 25% 50% 75%
2D-GP-SOONE 5.92 6.11 6.36 6.50
SNR= 10 dB 2D-SL0 5.32 5.65 6.15 6.46
2D-ADMM 4.40 4.69 4.97 5.19
2D-GP-SOONE 6.74 7.41 8.05 8.52
SNR= 0 dB 2D-SL0 6.49 7.04 7.89 8.48
2D-ADMM 4.45 4.70 5.04 5.40
Consequently, the results obtained from both simulated and
real data, validate the superiority of the proposed algorithm.
V. CO NC LU SI ON
In this paper, a 2D-ADMM method for sparse matrix reco-
very and high resolution ISAR imaging was proposed, which
is more computationally efficient compared to solving the 1D
optimization and requires lower memory during computation.
Moreover, compared to 2D-FFT, 2D-GP-SOONE and 2D-SL0,
the performance of the proposed approach is better in different
SNR levels and undersampling ratios. Both simulations and
real data were utilized to show the superiority of the proposed
method.
In this paper the phase error was not considered. Thus,
future works will focus on developing the algorithm for ISAR
autofocusing.
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Authorized licensed use limited to: University of Massachusetts Amherst. Downloaded on July 03,2020 at 19:30:28 UTC from IEEE Xplore. Restrictions apply.
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Journal
IEEE , VOL. ?, NO. ?, 7
2D-FFT
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
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10
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Doppler (Hz)
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2D-GP-SOONE
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Doppler (Hz)
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10
20
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Range (m)
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Doppler (Hz)
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-20
-10
0
10
20
30
Range (m)
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Doppler (Hz)
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-20
-10
0
10
20
30
Range (m)
2D-SL0
-40 -20 0 20 40
Doppler (Hz)
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-20
-10
0
10
20
30
Range (m)
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Doppler (Hz)
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-20
-10
0
10
20
30
Range (m)
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Doppler (Hz)
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10
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Range (m)
2D-ADMM
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
Fig. 5. ISAR images of Yak-42 obtained from different methods for N=64, M=256, P=2N and Q=2M. From top to bottom SNR = 10, 5 and 0 dB, respectively.
12.5% 25% 50% 75%
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
Fig. 6. From top to bottom ISAR images of Yak-42 obtained from 2D-GP-SOONE, 2D-SL0 and 2D-ADMM for different sampling rates. (SNR =10 dB and
P=Q=256)
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Journal
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-40 -20 0 20 40
Doppler (Hz)
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-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
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30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
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30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
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10
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30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
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30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
-40 -20 0 20 40
Doppler (Hz)
-30
-20
-10
0
10
20
30
Range (m)
Fig. 7. From top to bottom ISAR images of Yak-42 obtained from 2D-GP-SOONE, 2D-SL0 and 2D-ADMM for different sampling rates. (SNR =0 dB and
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Hamid Reza Hashempour was born in 1987. He
received the B.S., M.S., and Ph.D. degrees in electri-
cal engineering from Shiraz University, Shiraz, Iran
in 2009, 2011, and 2017, respectively.
His current research interests include radar signal
processing, radar imaging (SAR/ISAR), and com-
pressive sensing.
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