In this section, several simulations are carried out to verify the effectiveness of IPCA.
The imaging distance is set to 1 km in the simulations and the target is a B727 airplane with its point scattering model shown in Figure 4
a. The size of the imaging plane, approximately parallel to the radar antenna array, is
m. The radar consists of
transmitters and 4 receivers and all of the antennas are located on the same transceiver plane, as is illustrated in Figure 4
b. Besides, the size of the radar array is
m and the radar works at C-band. The detailed simulation parameters are given in Table 1
4.1. Imaging Simulation
To verify the effectiveness of the proposed method, the following simulation was carried out. System parameters in the simulations are shown in Table 1
. The parameters of the center and pose angles of the imaging plane both have errors. The real parameters are given in Table 2
According to the derivation in Section 2
and the deviation parameters of the imaging plane, the estimated and real fulfilling region of the spatial spectrum are illustrated in Figure 5
a,b respectively. It can be seen that when there are deviations of the imaging plane parameters, the estimated fulfilling region of the spatial spectrum is quite different from the fulfilling region of the real spatial spectrum. The recovered image is illustrated in Figure 5
c when deviations of the imaging plane parameters are not known, and the adopted algorithm is the IFFT algorithm. It can been seen that even when there are slight deviations of the imaging plane parameters, e.g., the deviations of the posture angles of the imaging plane are within
, the image is badly defocused, and the target’s contour is not clear.
In the following, the proposed IPCA is taken to search out the six parameters, namely , and then the calibration operation is taken to obtain the final image. The number of the particles, I, in IPCA is set to 100 and the maximum number of iterations, , is set to 100 too. Meanwhile some other parameters in PSO are set: , .
According to Figure 6
, the image has clearer target outlines and better focusing performance with fewer noisy points around the target. In the final image, each scattering point of the target can be clearly identified. Meanwhile, the image entropy is smaller using IPCA than that using IFFT, as is shown in Table 3
, which means better focusing performance.
shows the value of the objective function and the image entropy during the IPCA iterations. It can been seen that when the number of iterations exceeds 80, the value of the objective function tends to be stable, that is, the search results gradually converge.
From what has been discussed above, the method proposed in this paper can effectively converge. Meanwhile, the entropy of the inversion image is lower and the image quality is higher.
a shows the estimated
during the iterations. It can be seen that, with the increase of the number of iterations,
gradually approaches the real value, and finally converges to the real value. The values of
are still deviated from the real values. This is because the deviations
of the image will only make the image translation in the imaging plane, which have no effect on the image focus effect and target sparsity. Therefore,
do not affect the imaging quality and IPCA cannot guarantee
convergence to the real value. Likewise, Figure 8
b shows the estimated imaging plane posture angles
with the number of iterations. It can be seen that as the number of iterations increases, the value of
converges to the real value while
not. This is because the estimation error of the rotation angle
, whose rotating axis parallel to the line of sight, will make the image rotate in the imaging plane, which has no influence on the image entropy and target sparsity. Therefore
does not affect the imaging quality and IPCA cannot guarantee
convergence to the real value.
Meanwhile, since the PSO algorithm is a metaheuristic optimization algorithm, the computation time of IPCA is hard to predict. Therefore, 10 Monte Carlo trials are taken to count the computation time. The computation times of 10 Monte Carlo trials vary from 954 s to 2017 s with average computation time is 1617 s.
To sum up, the algorithm proposed in this paper can obtain more accurate imaging plane parameters, resulting in better image focusing performance and better imaging quality.
4.4. Simulations under Different Parameter Ranges
In order to verify that the proposed method has a certain degree of tolerance for the deviations of the imaging parameters, the following simulations are conducted. According to the first subsection in this section, the center deviation parameter , of the imaging plane only cause the image to shift in the imaging plane, and has no influence on the imaging quality and focusing performance. In addition, the posture angle makes the image rotate in the imaging plane, while the focusing performance is not affected. So in the following simulations, only , and are considered.
In the simulations, , and are uniform random selected in , and . are set to , while are set to whereas are set to . For each , and , 10 Monte Carlo trials are taken. Meanwhile, when the value , and are changing, , and are set to be zeros.
The errors between the estimated
and the real values and the image entropy obtained by 10 Monte Carlo trials with different
are illustrated using boxplot in Figure 11
. On each box, the central mark indicates the median, and the bottom and top edges of the box indicate the 25th and 75th percentiles, respectively. The whiskers extend to the most extreme data points not considered outliers, and the outliers are plotted individually using the red ’+’ symbol. According to Figure 11
a–c, the errors between the final estimated
and the real values are within
m, and the errors between the final estimated
and the real value are within
is chosen little than
. Moreover the errors between the final estimated
and the real values are within
is chosen little than
. In addition, according to Figure 11
d–f, the image entropy are all lower than 4 when
are chosen within 1 m,
In short, IPCA can obtain the final estimated , and with errors at most m, and when , , are chosen within 1 m, , respectively, and the image entropy is smaller than 4.