
基金项目
 Supported by the National Natural Science Foundation of China (No.61471006)

通信作者
 Li LianLin. Email: lianlin.li@pku.edu.cn
作者简介
Wang Libo was born in 1990. He received his B.Eng. degree in physical electronics from the University of Electronic Science and Technology of China in 2013. He is currently working toward the M.S. degree in Peking University. His research interests are superresolution electromagnetic imaging and new imaging methods. Email: 1301214165@pku.edu.cn
Li Lianlin was born in 1980. He received his Ph.D. degree from the Institute of Electronics, Chinese Academy of Sciences in 2006. He is currently a hundred talented program professor with Peking University. His research interests are superresolution imaging, microwave imaging, sparse signal processing and ultrawideband radar systems.Email: lianlin.li@pku.edu.cn

文章历史

收稿: 20150527
改回: 20151010
网络优先出版：20151106
Radar has become a powerful tool in inferring the location, shape and reflectivity of targets under investigation by transmitting an electromagnetic signal with transmitter(s), and postprocessing the echo of transmitting signal after experiencing the interaction with probed targets^{[1, 2]}. Such technology has found its valuable applications in many application fields including subsurface sensing, security management, biomedical imaging, to name a few.
Over past decades, an amount of excellent imaging algorithms for postprocessing GPR/TWI data has been developed^{[3, 4]}, for instance, the filtered backpropagation algorithm^{[5]}, the phaseshift imaging algorithm^{[6]}, the timereversal imaging algorithm^{[7]}, the resolutionenhanced imaging algorithm based on compressive sensing^{[8]}, and so on. Almost all of methods mentioned above rely on the use of Born approximation^{[9, 10]}, which leads to a linearized imaging approach, since it does not account for the multiplescattering effect between the electromagnetic wavefield and objects under investigation. Therefore, the Bornbased imaging approach is theoretically guaranteed to be used for treating weak scattering objects. Consequently, when the probed object is not very weak, the radar image produced by performing the Bornbased imaging approach usually exhibits some multipath ghosts (i.e., the false target)^{[11, 12, 13, 14, 15, 16, 17]}, and thus impairs the reliability of identifying real targets at the same time. Recently, it has been observed that the multipath ghost is aspect dependent^{[18]}, which implies that the shape and position of ghosts will be different with different measurement aperture. Inspired by this observation, several methods of mitigating the multipath ghost have been proposed, which compounds, in the manner of multiplication, images achieved from different subapertures^{[18, 19]}.
This letter presents a novel concept of mitigating the multipath ghost and enhancing the resolution of real targets at the same time, involved in the problem of GPR (Ground Penetrating Radar) and TWI (ThroughWall Imaging) imaging. To this end, we model for the first time the reflectivity of probed objects as a probability function with a normalized factor, which means that the radar imaging is reformulated into the one consisting of retrieving the probability function. Furthermore, the concept of the random subaperture is incorporated into our imaging procedure. The random subaperture is formed by randomly picking up some measurement locations from the complete observation aperture, which implies that multiple random subapertures can be created, and thus multiple probability functions (i.e., radar images) can be produced. Apparently, it is natural that the final radar image can be regarded as a joint probability distribution, which is the product of radar images achieved from multiple random subapertures. We would like to emphasis that the random subaperture adopted in this letter is advantageous over traditional overlapped subaperture used in Ref. ^{[18, 19]}, in the sense that (a) it is more flexible in creating the subaperture configuration, and (b) it is capable of avoiding the drawback of the traditional subaperture^{[2, 18, 19]} that some real targets will be missing when they are missing in some radar image corresponding to some radar subaperture, due to the multiplication operation of multiple radar images, as demonstrated in Section 4.
The rest of this letter is organized as follows: Section 2 discusses the probability model for the problem of GPR and TWI imaging, and illustrates the operational principle of proposed imaging methodology. Afterwards, a selected number of numerical experiments are conducted to demonstrate the stateofart performance of proposed methodology in Section 3, where the simulation data is achieved from running a commercial software GPRMax 2.5, a fullwave solver to the Maxwell’s equations. Finally, Section 4 summarizes this letter.
2 Problem Statement and MethodologyTo illustrate the working principle of the proposed methodology, we restrict ourselves into the TwoDimensional (2D) scalar case. However the proposed methodology and associated results can be generalized and applicable to more sophisticated imaging scenarios, such as, 3D and fullvector case in inhomogeneous media. We consider two popular imaging configurations, i.e, the GPR configuration (Fig. 1(a)), and TWI configuration (Fig. 1(b)). Referring to Fig. 1(a) and Fig. 1(b), an antenna is used to sequentially emit TMpolarized pulse
Under socalled Born approximation, the relation between the scattering data
$\begin{array}{*{20}{l}} {{u^{{\rm{sca}}}}({r_{\rm{R}}};\omega ) = k_0^2\int_{{{\rm{D}}_{{\rm{inv}}}}} {G({r_{\rm{R}}}, r';\omega ){u^{{\rm{inc}}}}(r')\chi (r'){{\rm{d}}^2}r'} }\\ {\quad \quad \quad \quad \quad \quad + n({r_{\rm{R}}}, \omega ), \quad r' \in \Gamma } \end{array}$  (1) 
where the contrast function of probed objects is defined by
This letter provides a novel interpretation for Eq. (1) in the context of probability. We would like to treat the location
${\rm{Pr}}({r}')=\frac{\chi ({r}')}{\int{\chi ({r}')}{\rm{d}}{r}'}$  (2) 
To avoid too much notations, we still refer to
$ \begin{array}{l} {{u}^{\rm{sca}}}({{r}_{\rm{R}}};{ω}a )\! = \!{{\left\langle k_{0}^{2}G({{r}_{\rm{R}}}, {r}';{ω} ){{u}^{\rm{inc}}}({r}') \right\rangle }_{\chi ({r}')}} \\ \quad\quad\quad\quad\quad\quad + n({{r}_{\rm{R}}}, {ω} )\end{array}$  (3) 
Now, the original radar imaging problem Eq. (1) is reformulated into the one which aims at retrieving the probability function
It is noted that the probability will be different for different radar viewing angle, since the radar with different viewing angle has different ability of detecting objects. In this sense, the final radar image can be treated as a joint probability function from multiple images associated with different radar view angles. Motivated by this observation, we explore the concept of random subaperture in our imaging procedure. Specifically, M random subapertures can be created, where each one is from randomly picking up observation locations from the entire radar aperture. Accordingly, M probability functions, ${\chi _m}(r')$, m=1, 2, $\cdots $, M, can be produced. Consequently, the final radar imaging $\chi ({r}')$ corresponds to the joint probability function of ${\chi _m}(r')$, m=1, 2, $\cdots $, M, in particular,
$\chi ({r}')=\prod\nolimits_{m=1}^{M}{{{\chi }_{m}}({r}')}$  (4) 
In this section, numerical simulations are conducted in two imaging scenario, i.e, GPR (Fig. 1a) and TWI (Fig. 1b), to evaluate the validity of the proposed method, where the input data for simulations is achieved by running a commercial software GPRMax 2.5, a fullwave solution to the Maxwell equations. For numerical implementations, the 2D investigation domain D_{inv}=1.6 m×1.6 m for GPR application and D_{inv}=1.6 m×1.5 m for TWI application has been discretized into square pixels with size of 0.02 m×0.02 m. The complete measurement line with the length of L=3.2 m is sampled with a uniform step of 0.02 m, which means there are 161 pairs of transmitters and receivers in total. The transmitting signal is a 'ricker' wavelet with center frequency f_{0}=1 GHz, time duration T=40 ns and time interval dt=11.793 ps. Accordingly, the operational frequency range is [100, 3500] MHz, which is sampled with a step of 50 MHz. All simulations are performed in a personal computer with the configuration of 8 GB access memory, I72600 central processing unit, and a Matlab 7.9 environment.
3.1 Simulations for GPR imagingIn this subsection, we examine the performance of proposed imaging methodology in the typical setup of GPR imaging as illustrated in Fig. 1(a), where the sensor is located in freespace while the targets are embedded into the lower part of halfspace medium with the relative permittivity being 6. The distance between the airground interface and the measurement line is 0.1 m, and four metallic cylinders are located at (0.4, 1.0) m, (0.2, 0.5) m, (0.2, 1.3) m, and (0.4, 0.7) m, with the radius being of 0.1 m, 0.02 m, 0.16 m and 0.04 m, respectively. Before starting imaging procedure, the direct wave from the reflection at the airground interface is removed by operating the meanvalue filter on GPR raw data.
First, the result achieved by using the wholeaperture data is shown in Fig. 2. It can be observed from this figure that four targets are seriously surrounded by the false targets, especially below the line of y=1.2 m, and it is pretty a risk in distinguishing real targets from false targets. Note that the ghost is a little bit more away from the sensors than real targets and spreads into a big clutter, which makes sense since the multipath of wavefield between targets implies that the wave has some tails after directarrival waves. As for the ghosts spread into a big clutter, this is because that the ghosts' locations are dependent on the measurement configuration. Therefore, for each couple of transmitter and receiver, there is a clutter of ghosts closely around the true object, and the size of clutter depends strongly on the radar viewing angle.
Second, we perform the GPR imaging with the configuration of subarray adopted in Ref. [21], where the whole aperture is divided into seven subaperture which has a length of 0.8 m and the overlap is 0.4 m between adjacent subapertures. The corresponding results are reported in Fig. 3, where Figs. 3(a)3(g) correspond to seven subapertures; while Fig. 3(f) is the product of these seven images. Comparing these results shown in Fig. 2 and Fig. 3, we can draw two conclusions immediately, in particular, (a) the location of the identified real objects is independent on the radar view angle, if they can be observed by GPR. (b) the real object is dragged with a clutter of ghost, and the shape and position of clutter of ghosts depends strongly on the subaperture, as mentioned previously. Invoked by these two conclusions, as guided by Eq. (4), we multiply the images achieved by all subapertures and the result is shown in Fig. 3(h). Comparing Fig. 3(h) with Fig. 2, we can see clearly that the ghost has been significantly impaired while the real targets are still clear. Nonetheless, note that two real targets are missing, which arises from a fact that the multiplication operation of multiple radar images associated with multiple subapertures will suppress those weak targets, even clear them out, when some real targets are missing in some of these images. So, the selection of measurement configuration should be careful, since it plays an important role of getting a good result.
Third, to overcome the drawback mentioned above, we adopt the random subaperture technique. The random subaperture composed of sensors selected randomly or subaperture in the following. Fig. 4 shows the results with five random subapertures and each random subaperture is assigned with a possibility of 0.2. Here, the probability means the one when the antenna element of entire measurement aperture is picked up in the random manner. By multiplying the five images obtained from five random apertures, the result is shown in Fig. 4(f). Comparing Fig. 4 with Fig. 3, we can see that there is no missing real targets due to the use of random subaperture, and thus the multiplication operation can suppress significantly ghosts and enhance real targets very well, without suppressing the real targets.
Finally, we examine the effect of image performance from changing the number of antenna elements for random subapertures or the possibility of random subaperture. The results for different possibility of random subaperture are shown in Fig. 5, from which we can see that the real targets will be enhanced more and more, and the clutter of ghosts will be weaker and smaller with the increasing probability of random subapertures over the range of [0.05, 0.2]. Generally, we can safely say that the random subaperture is more flexible and robust than traditional subarray technique.
Comparing the reconstructed images by traditional method with our method, it is clear that our method can suppress the multipath ghosts noteworthy and enhance the real targets at the same time. To avoid interpretation errors, the normalized mean square error (MSE)^{[22]} was calculated to characterize how the reconstructed images compared to the true object. The normalized mean square error for all pixels is defined as:
${\rm{MSE}}=\frac{\sum\nolimits_{i=1}^{\rm{N}}{{{({{{\hat{x}}}_{i}}{{x}_{i}})}^{2}}}}N$  (5) 
where
This subsection consider the scene of throughwall imaging, as illustrated in Fig. 1(b), where the target and antenna are located at different sides of the wall. The thickness and relative dielectric constant of the wall are specified to be 0.3 m and 6.0 respectively. The complete measurement line is at the distance of 0.1 m away from the wall. Other simulation parameters are the same as above. Note that here the Green's function is for threelayered medium, in contrast to the twolayered medium for GPR imaging.
Analogously, the image obtained by using the whole measurement aperture is shown in Fig. 6, which shows that four targets are severely disturbed by the clutter of false targets, and it is pretty a risk in distinguishing real targets from false targets. The final images obtained by random subapertures with different probability of random apertures are shown in Fig. 7. From these simulations, it can be observed that the ghosts can be significantly impaired by using proposed methodology, and the real targets can be enhanced at same time.
This letter presents a novel probabilistic concept of suppressing the multipath ghosts involved in the problem of GPR and TWI imaging. First, this letter provides a model of the reflectivity of probed objects in a probability function. Second, the concept of the random subaperture is incorporated into the imaging procedure. Thus, the final radar image is regarded as the joint probability distribution, which is the product of retrieved radar images corresponding to multiple random subapertures. Finally, a selected number of numerical experiments is provided to demonstrate the performance of proposed imaging methodology for the application of GPR and TWI imaging.
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