Proceedings of the 7th Annual ISC Graduate Research Symposium
ISC-GRS 2013
April 24, 2013, Rolla, Missouri
Zengli Yang
Department of Electrical and Computer Engineering
Missouri University of Science and Technology, Rolla, MO 65409
A COMPARATIVE STUDY OF SPARSE METHODS FOR 3-D SYNTHETIC
APERTURE RADAR IMAGE RECONSTRUCTION
ABSTRACT
This paper compares four sparse methods for 3-D synthetic
aperture radar (SAR) image reconstruction. The four different
sparse methods are image denoising or compressed sensing
(CS) approach combined with range migration algorithm
(RMA) using Stolt transform or non-uniform fast Fourier
transform (NUFFT). With undersampled measurements, the
sparse methods decrease the load of data acquisition but still
reconstruct the important features of 3-D SAR images. We
adopt the structured similarity (SSIM) index to assess the
quality of the 3-D reconstructed images. Numerical simulations
demonstrate that the sparse methods based on the Stolt-RMA
benefit little from the low-complexity Stolt transform, and the
image denoising approach using the NUFFT-RMA achieves the
best tradeoff between the reconstruction quality and
computational costs.
1. INTRODUCTION
Wideband 3-D synthetic aperture radar (SAR) imaging finds
important applications in the area of nondestructive testing and
evaluation (NDT&E) [1] for its feasibility to acquire highresolution holographic images of specimen under test (SUT).
Microwave and millimeter wave NDT&E techniques have been
applied to diverse applications, such as detection and evaluation
of corrosion under paint and composite laminates, fatigue crack
detection and sizing in metal surfaces, dielectric material
characterization, etc.
However, data collection for wideband 3-D SAR imaging
imposes great challenges on application of microwave and
millimeter wave. Conventional raster scanning takes about two
hours to scan a 50 × 50 mm2 area at 2 mm spacing, hindering
the practical applications of the 3-D SAR imaging system.
Random undersampling can reduce the acquisition time, and
advanced sparse methods using either image denoising or
compressed sensing (CS) approach [2] can reconstruct image
with good quality comparable to full sampling [3]–[7].
Conventional 3-D SAR imaging with planar aperture
usually adopts the range migration algorithm (RMA) [8] with
either Stolt transform [9] or non-uniform fast Fourier transform
(NUFFT) [10]. For brevity, we call these two conventional
RMAs as Stolt-RMA and NUFFT-RMA, respectively. The
recovered image using the NUFFT-RMA usually has less
artifacts and better resolution than that using the Stolt-RMA,
however, the NUFFT-RMA has much higher computational
costs than the Stolt-RMA. The procedure for converting the raw
measurements to the 3-D image is called forward SAR
imaging, and the reverse procedure is called reverse SAR
imaging. Nonlinear transform exists when applying the RMAs
for SAR imaging. With undersampled measurements, the image
denoising or CS approach can be combined with the RMAs for
image reconstruction, yielding four different sparse methods.
For brevity, we denote the four sparse methods as Stolt-DN,
NUFFT-DN, Stolt-CS, and NUFFT-CS, respectively.
Although both image denoising and CS approaches exploit
the sparsity of the 3-D SAR image, differences between these
two approaches exist when combined with RMA for 3-D SAR
imaging. The image denoising approach is to denoise the SAR
image that is obtained by applying RMA directly on the undersampled measurements. However, the CS approach applies the
forward and reverse RMAs during the reconstruction, and finds
the solution to an underdetermined system. For each iteration,
the CS approach has larger computational costs than the image
denoising approach due to its more complicated transforms.
Our prior works investigated the performance of the CS
approach using the Stolt-RMA or NUFFT-RMA separately, and
demonstrated their capability to reconstruct the important
features of 3-D SAR images [4]–[7]. Therefore, it is interesting
to compare the overall tradeoff between the reconstruction
quality and computational costs for the four sparse methods.
In this paper, we compare the four different sparse methods
using a synthesized SUT and its simulated measurements. To
solve the l1-regularized image reconstruction problems, we
adopt the split Bregman framework [2], [11] to achieve fast
convergence and better numerical stability. To be more
consistent with human eye perception, we adopt the structured
similarity (SSIM) [12] index instead of the peak signal-to-noise
ratio (PSNR) to assess the quality of the 3-D reconstructed
images. Numerical simulations demonstrate that the sparse
methods based on the Stolt-RMA benefit little from the lowcomplexity Stolt transform, and the image denoising approach
using the NUFFT-RMA achieves the best tradeoff between the
reconstruction quality and computational costs.
1
2. CONVENTIONAL
3-D
SAR
IMAGE
RECONSTRUCTION WITH FULL SAMPLING
Consider a wideband monostatic stripmap 3-D SAR imaging
system where data are collected by scanning the SUT over a 2D planar aperture denoted as XY -plane. A point target is
characterized by its reflectivity function g ( x ¢, y ¢, z ¢) . In 3-D
Cartesian space, we define the dimension perpendicular to the
2-D aperture plane as the Z -dimension and its origin is located
at the 2-D aperture plane. Without the far-field assumption, the
wavefront curvature is no longer negligible. Without plane
wave approximation for spherical wave, the received spherical
waveform at position (x, y) with temporal angular frequency w
is given by
r ( x, y, w) =
- j 2 k ( x - x ¢)2 + ( y - y ¢)2 + z ¢2
ò ò ò g ( x ¢, y ¢, z ¢) ´ e
which is known as the 3-D Stolt-RMA [13]. Here F 2D denotes
2-D cross-range fast Fourier transform (FFT) along the XY plane, F 3D- 1 denotes the 3-D inverse FFT (IFFT), and denotes
the 1-D Stolt transform with the nearest neighbor interpolation.
Note that the distinction between the primed and unprimed
coordinate systems is now dropped because the coordinate
systems coincide after the FFT and IFFT.
Let P(k x , k y , k ) represent the 2-D cross-range Fourier
transform of r ( x, y, w) , and Q(k x , k y , k z ) represent the 3-D
Fourier transform of the reflectivity function g ( x, y, z ) , where
the
wavenumbers in the x, y, and z
dimensions, respectively, then k z =
SPARSE METHODS FOR 3-D SAR IMAGE
RECONSTRUCTION
To take the advantage of sparse methods for 3-D SAR
imaging, the radar probe measures only at a small percentage of
randomly-selected positions on the uniform XY grid. The
dx ¢dy ¢dz ¢
(1)
where w = 2p f with f being the temporal frequency, k = w / c
is the wavenumber with c being the propagation speed of
microwave, and g(x′, y′, z′) denotes the 3-D reflectivity function
of the SUT. With plane wave decomposition for spherical wave
and Stolt transform [9], the 3-D reflectivity function of the SUT
is give by [13]
g ( x, y, z ) = F 3D- 1 {Θ{F 2D [r ( x, y, w) ]e- jkz z }},
(2)
k x , k y , and k z are
3.
4k 2 - k x2 - k y2
due to
curvature in radar wave propagation. It is necessary to obtain
equispaced Q(k x , k y , k z ) in k z for 3-D IFFT operation under
uniform measurement grid, and direct interpolation is known as
Stolt transform. Alternatively, nonuniform spaced Q(k x , k y , k z )
in k z can be transformed to the reflectivity image by applying
NUFFT,
- 1
- 1
g ( x, y, z ) = F 2D
{F 2D [r ( x, y, w) ]e- jkz z }},
(3)
{F NUFFT
which is known as the 3-D NUFFT-RMA [10]. Here F 2D- 1
- 1
denotes 2-D cross-range IFFT, and F NUFFT
is the 1-D inverse
NUFFT along k z . With more complicated interpolation, the
NUFFT-RMA has larger computational costs than the StoltRMA. The complete procedures for the conventional 3-D SAR
image reconstruction methods are summarized in Fig. 1.
Fig. 1. Conventional 3-D SAR image reconstruction methods: StoltRMA and NUFFT-RMA.
backscatter data at these positions are collected and saved as
the raw data. Image can be reconstructed from the
undersampled measurements by two iterative sparse
approaches: one is image denoising based on the full image
data g ( x, y, z ) and another is CS based on the undersampled
raw measurement r ( x, y, w) . Both approaches exploit the
sparsity of the 3-D SAR image and rely on the l1-norm and
total variation (TV) minimizations [2]. For each approach, the
Stolt-RMA or NUFFT-RMA can be applied yielding four
different sparse methods: Stolt-DN, NUFFT-DN, Stolt-CS, and
NUFFT-CS.
3.1. Image Denoising Approach
For the image denoising approach, we first obtain the full
image data g ( x, y, z ) by applying the RMA on the undersampled measurements as preprocessing. Then the following
denoising procedure is performed [2],
(4)
min J (gˆ ) subject to ‖ gˆ - g‖ 22 < s 2 ,
gˆ
where ‖ ×‖ 2 is the l2 -norm, g Î £ n´ 1 is the vectorized 3-D SAR
image of the interrogated scene obtained directly from the
undersampled measurements, gˆ Î £ n´ 1 is the vectorized
estimated 3-D image, and s 2 is the tolerance. The cost function
J (gˆ ) represents some l1 regularization term with respect to ĝ
and is selected as
J (gˆ ) =‖ Ψgˆ‖ 1 + ‖ Dgˆ‖ 1 ,
2
(5)
where ‖ ×‖ 1 denotes the l1 -norm, Ψ Î £ n´ n is the linear operator
that transforms from voxel representation into a sparse
representation, and D is the discrete 3-D isotropic TV operator.
Depending on the type of the RMA in the preprocessing, we
denote the two sparse methods using image denoising
technique as Stolt-DN and NUFFT-DN.
3.2. Compressed Sensing Approach
The CS approach always values the consistency between the
estimated measurements and actual received data during the
reconstruction. For the 3-D SAR imaging, the CS approach can
be interpreted as [7]
(6)
min J (gˆ ) subjectto ‖ Φgˆ - r‖ 22 < s 2 ,
gˆ
where J (gˆ ) has the same definition as (5), and Φ Î £ m´ n (m <
n) is the measurement matrix, which reflects the acquisition of
the vectorized received measurements r Î £ m´ 1 for the scene’s
reflectivity. For the Stolt-CS and NUFFT-CS, we adopt the
reverse Stolt-RMA and NUFFT-RMA for Φ , respectively.
According to (2) and (3), we can write the measurement
operator as
ΦStolt = UF 2D- 1 {Θ† [F 3D (·)]},
(7)
(8)
ΦNUFFT = UF 2D- 1 {F NUFFT [F 2D (·)]},
if omitting the phase compensation term for brevity. Here †
represents the pseudoinverse 1-D Stolt transform with nearest
neighbor interpolation, F 3D denotes 3-D FFT, and U denotes
the binary matrix, which is to select estimated received
waveform at these random positions.
Although (4) and (6) have the same objective function, (6)
is actually an underdetermined system because of its constraint.
Also, nonlinear forward and reverse RMAs are involved when
finding the solution to (6). Apparently, tradeoff between the
reconstruction quality and computational complexity exists for
the image denoising approach and the CS approach.
4. SIMULATION AND RESULTS
Our previous publications reported that undersampled
measurements have been successfully performed with the 3-D
SAR imaging system, and experimental results have
demonstrated good image qualities [4]–[7]. However, in this
paper, we use simulations to compare the difference of the four
sparse methods because simulation provides ground truth
image.
Suppose a uniform measurement grid in the XY-plane
being 2 mm in both X and Y directions, and the steppedfrequencies ranging from 35.04 GHz to 44.64 GHz in the Qband with a step-size of 0.64 GHz. We simulated the
backscatter data over the square area with 128×128 mm2 in the
noiseless and electromagnetic interference (EMI)-free
environment. The SUT contains three objects square-pad,
cross-profile, and circle-profile distributed at different depths of
−28 mm, −58 mm, and −88 mm, respectively. Since the
maximum range Rmax = c / (4d f ) with d f being the frequency
interval, Rmax » 118 mm for our benchmark. We set the
sampling interval along the Z-dimension as 2 mm, so the data
cube of the 3-D image had the dimension of 64×64×59. The
raw measurements were generated from the simulated SUT
according to (1), and the undersampled measurements were
randomly selected from the raw measurements at the XY
positions for the scanning area.
When solving (4) or (6) for the four sparse methods, we
adopt the split Bregman framework [2] to achieve fast
convergence and better numerical stability. Proper parameters
and stopping criteria for the split Bregman algorithm were
selected. Also, we chose the sparse transform as Haar wavelet
transform in (5). All the conventional methods and sparse
methods were implemented by MATLAB R2011a (x86) on a
computer with Intel(R) Core(TM)2 Quad CPU Q9400 @ 2.66
GHz and 8.00 GB RAM. The SSIM [12] has been considerably
used for 2-D image quality assessment, and is considered more
consistent with human eye perception than the PSNR. Hence,
we adopt the mean SSIM (MSSIM) averaged over 3-D
windows other than 2-D windows to measure the similarity
between the 3-D reconstructed image and ground truth image.
In the extreme case, if the reconstructed image is exactly the
same as the ground truth image, then the MSSIM would be 1.
In other words, a larger MSSIM index corresponds to better
reconstruction quality, and vice versa.
Figure 2 shows the comparison of the ground truth image
and the reconstructed images from 40% undersampled
measurements using the conventional methods and the sparse
methods. Left, middle, right columns are 3-D view, top view,
and side view, respectively. The ground truth image in Fig. 2(a)
shows the geometric features and reflectivity of the three
objects, and Fig. 2(b) and Fig. 2(c) show the directly
reconstructed images using the Stolt-RMA and the NUFFTRMA, respectively. Without applying the image denoising and
CS techniques, the shadow of the targets and artifacts caused by
the RMAs and random undersampling are dominant in the 3-D
images. Specifically, the Stolt-RMA blurred the SUT to a large
extent, and the NUFFT-RMA disrupted the shape of the squarepad. In contrast, Fig. 2(d) to Fig. 2(g) show the cleaner
reconstructed images when using the sparse methods. Not
surprisingly, the sparse methods based on the NUFFT-RMA
recovered the 3-D images of the SUT with better resolution and
less background noise than those based on the Stolt-RMA.
Specifically, the Stolt-DN had worse denoising capability than
the Stolt-CS, and the NUFFT-CS recovered the image with
even less horizontal shadow of the cross-profile and circleprofile than the NUFFT-DN. When comparing to the ground
truth image, however, both NUFFT-DN and NUFFT weakened
the reflectivity of the square-pad and failed to remove the
vertical shadow of the SUT. Since the vertical resolution of the
3-D SAR image depends on the bandwidth of the imaging
3
system [13], the sparse methods demonstrated incapability to
make up the bandwidth deficiency. The running time for each
method with 40% undersampling rate is also given in Fig. 2.
Note that in our simulations, since the undersampling rate has
little influence on the running time, we omit showing the
running time for each method with different undersampling
Fig. 2. Comparison of the ground truth image and the reconstructed
images from 40% undersampled measurements using the conventional
methods and sparse methods. Left, middle, right columns are 3-D
view, top view, and side view, respectively. The running time for each
method is also given.
rates for brevity. The NUFFT-RMA took five times the running
time of the Stolt-RMA, while the four sparse methods increased
the running time by one to two orders due to their iterative
reconstructions. Since the RMA only served as preprocessing
for the image denoising approach, the Stolt-DN and NUFFTDN took almost the same running time. Involved the forward
and backward RMAs, the CS approach indeed brought larger
computational load than the image denoising approach.
Interestingly, the running time of the NUFFT-CS was only
slightly more than that of the Stolt-CS. Thanks to the better
accuracy of the NUFFT-RMA, the NUFFT-CS took much less
iterations to reach the stopping criteria than the Stolt-CS .
Therefore, the low-complexity Stolt transform actually cannot
decrease the overall computational load for the sparse methods
due to its inaccuracy.
Figure 3 shows the MSSIM of the reconstructed 3-D SAR
images using the conventional methods and the sparse methods
with undersampling rates varying from 10% to 90%. For each
undersampling rate, The MSSIMs were averaged over the
results of 8 independent trials. Consistent with Fig. 2, the
NUFFT-RMA outperformed the Stolt-RMA especially at high
undersampling rate, and the sparse methods based on the
NUFFT-RMA had much higher MSSIM than the other
methods. With lower computational costs, the NUFFT-DN had
comparable MSSIM with the NUFFT-CS when the undersampling rate was greater than 25%. Therefore, the NUFFT-DN
should be considered as the sparse method with the best
tradeoff between the reconstruction quality and computational
costs for 3-D SAR image reconstruction.
Fig. 3. MSSIM of the reconstructed 3-D SAR images using the
conventional methods and the sparse methods with undersampling
rates varying from 10% to 90%.
4
5
CONCLUSIONS
Four sparse methods for 3-D SAR image reconstruction
have been studied comparatively. Numerical simulations
demonstrate that the sparse methods based on the Stolt-RMA
benefit little from the low-complexity Stolt transform, and the
image denoising approach using the NUFFT-RMA achieves the
best tradeoff between the reconstruction quality and
computational costs.
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9, pp. 1581–1592, Sep. 2001.
8.
ACKNOWLEDGMENTS
The authors wish to thank the support by the Intelligent
Systems Center of Missouri University of Science and
Technology.
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