To evaluate the SR capacity of the proposed scheme, we conducted SR experiments using four target images for three different types of targets: point static, extended static, and extended moving.
3.1. SR Results for Point Static Target
To obtain the target image for a point target, KOMPSAT-5 observed a real corner reflector (CR) located at the KOMPSAT calibration site in Mongolia using spotlight mode and HH polarization. The target image was then extracted to contain only the impulse response function (IRF) of the CR, as shown in
Figure 5.
A preprocessed target image was first generated to investigate the SR capability of the proposed scheme, and the preprocessed (PR) target image was first generated. Then, the spatial resolution of the preprocessed target image was intentionally worsened by reducing the slant-range or azimuth frequency bandwidth, leading to a low-resolution (LR) target image, as described in [
25]. Let the ratio of the adjusted SR to the original SR be denoted as
. In this section,
is set to 1.6. Next, the AR-model-based LP and CS algorithms were applied to the LR target image in both the slant-range and azimuth directions. The slant-range cut and azimuth cut were then obtained by cutting the super-resolved target image at the center pixels in the slant-range and azimuth directions, respectively. Then, the 3 dB bandwidth, peak side-lobe ratio (PSLR), and integrated side-lobe ratio (ISLR) [
35,
36], which are widely used as the quality parameters of SAR images, were computed to quantify the SR capacity. The 3 dB bandwidth is the distance between the points with intensities 3 dB below the maximum intensity of the main lobe peak [
35]. In addition, the PSLR is defined as the ratio of the peak amplitude of the most prominent side lobe to the peak amplitude of the main lobe, as in the following [
35]:
where
denotes the peak amplitude of the most prominent side lobe, and
is the peak amplitude of the main lobe. In addition, the ISLR is the ratio of the total power in all the side lobes to the power in the main lobe, as in the following [
35]:
where
denotes total power in all the side lobes, and
is the power in the main lobe.
Figure 6 shows the slant-range and azimuth cuts of the PR, LR, and super-resolved images.
In
Figure 6, the solid red line denotes the slant-range and azimuth cuts of the PR target image; the dashed green line denotes the slant-range and azimuth cuts of the LR target image generated from the PR target image; the remaining four dotted lines denote the SR results. From
Figure 6, the proposed scheme exhibits remarkable SR performance along both the slant range and azimuth directions. In particular, the main lobes of the super-resolved slant range and azimuth cuts approximately match those of the PR slant range and azimuth cuts. Quantitative comparisons of the quality parameters are summarized in
Table 1 and
Table 2. The results in
Table 1 and
Table 2 are obtained from the average values of 100 independent realizations to provide reliable performance evaluations for the BP and BPDN algorithms. In these tables, the super-resolved slant-range and azimuth cuts show significant improvements in the three quality parameters compared with the LR slant-range and azimuth cuts. In particular, as expected from
Figure 6, all algorithms almost perfectly retrieved the 3 dB bandwidth of the PR target image from the LR target image. Thus, the proposed scheme successfully achieved the objective of SR (i.e., improvement of 3 dB bandwidth) for the KOMPSAT-5 image. In addition, the proposed scheme enhances the PSLR and ISLR of the LR target image. Although they cannot attain PSLR and ISLR equivalent to the PR target image, the qualities of the super-resolved target images are superior to those of the LR target images.
The results in
Table 1 and
Table 2 demonstrate that the proposed scheme can effectively enhance the quality of the target image (i.e., the spatial resolution, PSLR, and ISLR). However, note that
is an important factor affecting SR capability. The capabilities of the four SR algorithms are sensitive to variations in
. To examine the SR performances of the four algorithms in detail, we define the relative error rate of the three quality parameters as follows:
where
denotes three quality parameters of the PR target image, and
denotes those of the super-resolved target images obtained using the four SR algorithms. In addition, the 1D relative errors of the slant range and azimuth cuts are defined as follows:
where
denotes the slant-range or azimuth cuts of the PR target image,
denotes those of the super-resolved target images, and
denotes the summation of all elements in a vector.
and
in Equations (16) and (17) were then computed by varying
from 1.2 to 4 in increments of 0.4, as shown in
Figure 7 and
Figure 8.
In
Figure 7, it can be observed that the
of all algorithms is low over the entire range of
. This indicates that the main lobe of the slant-range cut of the PR target image can be successfully reconstructed from the LR slant-range cut, regardless of the variation in
. In particular, the BP and BPDN provide reliable
, the maximum of which is just lower than 4%. In the case of
,
, and
, SR performance worsens as
increases. For
, the AR-model-based LPs yield better performances than the CS techniques when
, and their performances become similar when
. In the case of
and
, the Burg and MCM algorithms also show better performance than the CS techniques when
. However, CS techniques lead to lower (better)
and
when
. In particular, it is remarkable that the
of the Burg and MCM algorithms rapidly increased during
.
In
Figure 8, as is the case with slant-range cuts, all algorithms exhibit reasonable
. In addition,
,
, and
tend to deteriorate as
increases. For
,
, and
, the AR-model-based LPs exhibit worse results than the CS techniques for
. In particular, the
of the Burg and MCM algorithms significantly increase when
.
In short, in the case of a point static target, all SR algorithms yielded reliable results during . In addition, the CS techniques produced more robust SR results than the AR-model-based LPs during .
3.2. SR Results for Extended Targets
To analyze the SR performance for extended targets, we used two target images (i.e., extended static target and extended moving target) extracted from two different large-scale KOMPSAT-5 images, as shown in
Figure 2. The KOMPSAT-5 image in
Figure 2a was obtained using stripmap mode and HH polarization, whereas the KOMPSAT-5 image in
Figure 2b was obtained using the spotlight mode and HH polarization.
As in the case of the point target, after the two target images were preprocessed according to
Figure 1, the spatial resolutions of the two PR target images were deliberately degraded using
. The proposed SR scheme was then applied to the LR target image in both the slant-range and azimuth directions.
Figure 9 shows the PR, LR, and super-resolved target images of the extended static target. As shown in
Figure 9b, the quality of the LR target image is much lower than that of the PR target image. This is because (1) 3 dB bandwidths of IRFs corresponding to scattering centers deteriorate (widen) and (2) interference among IRFs increases [
16]. Then, the target response of the ship is focused on improving the 3 dB bandwidths of the IRFs and reducing the interference among the IRFs, as shown in
Figure 9c–f.
In addition,
Figure 10 shows the PR, LR, and super-resolved target images of the extended moving target. Note that in the case of the moving target, the PR target image in
Figure 10a differs considerably from the original target image in
Figure 2b. This is because the refocusing method was applied to
Figure 2b to achieve an intact response of the moving target. As shown in
Figure 10, the proposed scheme successfully improves the quality of the LR target image. In particular, it appears that the qualities of
Figure 10c,d are comparable to those of the PR target image.
Unlike in the case of the point target, the SR performance of the extended target cannot be quantitatively evaluated using SAR quality parameters. This is because the extended target consists of a large number of scattering centers, leading to arbitrary interference among the IRFs, which impedes the exact computation of the quality parameters.
Thus, as alternatives, we adopted Shannon entropy (SE) and image contrast (IC), which are widely used to evaluate the focus quality of SAR image [
23,
24,
28]. The SE and IC can be written as follows [
28]:
where
denotes a 2D image,
denotes the summation of all the elements in a matrix, and
. In addition,
and
are the mean and standard deviation.
Generally, a lower SE and higher IC imply better focus quality of the SAR image. In addition, the improvement in focus quality can imply the improvement of 3 dB bandwidths of IRFs and a reduction in interference among IRFs, provided that other SAR imaging parameters are the same, as in the case of the PR and LR target images.
Table 3 and
Table 4 show the SEs and ICs of the target images in
Figure 9 and
Figure 10, respectively. The results in
Table 3 and
Table 4 are obtained from the average values of 100 independent realizations to provide reliable performance evaluations for the BP and BPDN algorithms. As expected, the LR target images have much higher SEs and lower ICs than those of the PR target images. Meanwhile, the SEs and ICs of the super-resolved target images were effectively improved. For both case of static and moving targets, the Burg algorithm shows outstanding SEs and ICs, which are comparable to those of the PR target images.
In addition, the computation time (CT) for each super-resolved target image was measured to investigate the applicability of our scheme in real situations. For this, MATLAB programs and a PC with its CPU clock speed of 3.7 GHz were used (the MATLAB program is not optimized to obtain its best computation speed).
Table 5 and
Table 6 show the CTs for super-resolved target images in
Figure 9 and
Figure 10. The results in
Table 5 and
Table 6 are obtained from the average values of 100 independent realizations to provide reliable performance evaluations for the BP and BPDN algorithms. As seen in
Table 5 and
Table 6, the proposed scheme has reliable CTs. In particular, the CT is just 0.05 s for the extended static target, when the Burg algorithm is chosen as the SR technique. Considering that our equipment and software are not optimized for data processing, the proposed scheme has large potential to be used for real systems.
Furthermore, the 2D relative error is defined to compare the SR performances of the four algorithms, as follows [
22]:
where
denotes the PR target image,
denotes the super-resolved target images, and
denotes the summation of all the elements in a matrix. Then,
was computed for target images of extended targets, varying
from 1.2 to 4 in steps of 0.4.
Figure 11 shows
versus
for target images of extended targets. In
Figure 11, all algorithms yield similar performances in the entire range of
, showing continuous growth of
. Among them, the Burg algorithms show the best performances versus
.
From
Table 3 and
Table 4 and
Figure 11, we can observe that the SR performances of the Burg algorithm are better than those of the other algorithms in the case of the extended target used in this study.
In our study, we adopted two super-resolution techniques, namely the AR-model-based LP algorithms and the compressive sensing algorithms. In the case of the AR-model-based LP algorithms, the sparsity of the target image does not affect the super-resolution performances theoretically. Meanwhile, in the case of compressive sensing algorithms, the sparsity can affect the super-resolution performances. Actually, most target images can be sparsely representable, because the target response is concentrated in a small part of the target image. However, the sparsity of the target images can be varied depending on target detection algorithms, which determine the region of interest containing the target response in different ways. Thus, we additionally investigated the super-resolution performances of the proposed scheme using CS algorithms in the case of different image sparsity. Let the ratio of the number of pixels corresponding to target response to the total number of pixels in the target image,
, be the image sparsity. The pixels of the target responses are determined using the constant false alarm rate (CFAR) detector [
1]. For experiments, we first extracted another target image of a specific extended static target from the large-scale KOMPSAT-5 image, as shown in
Figure 12a. The
of
Figure 12a is 6.18%. Next, the
of the target image was artificially adjusted by cropping the target image, as if the region of interest (ROI) for target response was changed. For example,
Figure 12b is the resulting target image whose
is 20.6%. Then,
for CS-based SR procedure versus
were computed when
, as shown in
Figure 13. In
Figure 13,
were 6.18, 15.96, 20.6, 30.33, 40.45, and 45.21%. As seen in
Figure 13, the relative errors do not seriously change until the image sparsity reaches 45.21%. Consequently, it is expected that the proposed scheme can give stable performances in real situations that the image sparsity can be varied by the target detection algorithms.