Quantitative and Qualitative Comparison of SAR
Images from Incomplete Measurements Using
Compressed Sensing and Nonuniform FFT
Hamed Kajbaf, Joseph T. Case, Yahong Rosa Zheng, Sergey Kharkovsky, and Reza Zoughi
Electrical and Computer Engineering Department
Missouri University of Science and Technology
Rolla, Missouri 65409
Email: {hamed.kajbaf, j.t.case, zhengyr, sergiy, zoughir}@mst.edu
Abstract— In this paper the performance of two wideband
synthetic aperture radar (SAR) imaging methods from incom-
plete data sets are compared quantitatively and qualitatively. The
first approach uses nonuniform fast Fourier transform (NUFFT)
SAR to form images from nonuniform spatial and frequency
data points. The second approach benefits from the emerging
compressed sensing (CS) methodology to recover raw data from
undersampled measurements. The results of our experimental
tests show that CS has a better performance in terms of error
and image contrast while NUFFT SAR has lower computational
complexity.
I. I NTRODUCTION
Microwave synthetic aperture radar (SAR) imaging is a high
resolution nondestructive testing and evaluation (NDT&E)
technique which can be exploited to detect discontinuities in
critical structures by raster scanning using a single antenna
probe (e.g. an open-ended waveguide) [1]–[3]. Wideband
SAR is capable of determining depth of discontinuities and
providing 3D images of specimens under test (SUT) such
as spacecraft tiles, airplane coating, bonding of adhesive or
composite materials. However, the drawback of microwave
SAR imaging for NDT&E applications is the time needed
for scanning the region of interest, which for a relatively
large SUT might be hours of data acquisition. Reducing the
number of spatial samples significantly helps in decreasing the
acquisition time.
The emerging compressed sensing (CS) theory have in-
troduced one method of reducing the number of samples
by sampling below the conventional Nyquist rate [4], [5].
Nonuniform fast Fourier transform (NUFFT) SAR is another
method for forming SAR images from incomplete data with
low computational requirements. In this paper the performance
of sample reduction in SAR imaging using CS and NUFFT
SAR are compared in terms of metrics indicating the quality
of the SAR images.
II. NUFFT SAR
Suppose that an antenna probe located at (
′
,
′
,
0
) illu-
minates a target and a general point on the target, (, , ),
reflects back the pulse. The same probe receives the backscat-
tered coherent signal, (
′
,
′
,), which is the superposition
of reflection from all points in the illuminated area
(
′
,
′
,)=
∫∫∫
(, , )
−
d d d (1)
where is the range between the probe and the target
point =
√
( −
′
)
2
+( −
′
)
2
+( −
0
)
2
, =
is
the wavenumber, is the propagation speed, and (, , )
is the reflectivity function of the target, which is the ratio
of the reflected field to the incident field. Decomposing the
spherical wave propagation into a superposition of plane wave
components, we can rewrite (1) in 3D Fourier transform form
[6]
(
′
,
′
,) =
∫∫ [∫∫∫
(, , )
−(
′ +
′ +)
d d d
]
×
(
′
′
+
′
′
+0)
d
′ d
′ (2)
where
′ and
′ are Fourier transform variables correspond-
ing to
′
and
′
, respectively, and the one corresponding
to is
=
√
4
2
−
2
−
2
. The triple integral in (2) is
the 3D Fourier transform of (, , ). Solving this equation
and dropping the distinction between primed and unprimed
coordinates, the 3D image is formed by [6]
(, , )= ℱ
−1
3
{
ℱ
2
{ (, , )}
−
√
4
2
−
2
−
2
0
}
(3)
where ℱ{.} and ℱ
−1
{.} are the Fourier and inverse Fourier
transform operators, respectively. Since in (3) the frequencies
are uniformly sampled in the frequency band and the probe
scans at uniform step size along X-Y dimensions,
’s are
nonuniformly distributed and Stolt interpolation is normally
used to interpolate the uniform points in
[7]. It has been
shown that NUFFT can be exploited to improve the accuracy
of SAR imaging [8] since it is a good approximation of
the nonuniform discrete Fourier transform (NDFT) [9], [10].
Using NUFFT, (3) becomes
(, , )=
ℱ
−1
2
{
ℱ
−1
{
ℱ
2
{ (, , )}
−
√
4
2
−
2
−
2
0
}}
(4)
978-1-4244-8902-2/11/$26.00 ©2011 IEEE 592