Abstract
The existence of walls surrounding targets leads to multipath return at the radar receiver, which provides additional information about the target, and thus can be exploited to strengthen the quality of throughwall radar imaging (TWRI). Based on this, a multipath exploitation method is proposed to identify the location of the multipath ghost. An algorithm that combining the modified Green’s function with back projection algorithm is presented to associate and map the multipath ghosts to the location of real targets. The theoretical analysis is verified according to the simulation results obtained using gprMAX software as well as practical radar measured data, and our proposed method is shown to outperform that in conventional multipath exploitation method (Setlur et al., IEEE Trans. Geosci. Remote Sens. 49:4021–4034, 2011).
Introduction
Throughwall radar imaging (TWRI) has attracted much attention in various fields, due to its numerous civil and military application [1–4]. The targets of the throughwall radar are those objects that are located inside rooms surrounded by walls. This leads to the superimposing of multipath returned signals and the direct returned signal. In the traditional narrowband radar system, it is difficult to separate the original returned signal of target with the those caused by multipath due to the poor range resolution and the timedelay resolution [5]. Alternatively, ultrawide band (UWB) throughwall radar systems [6] can provide sufficient range resolution to differentiate the direct and the multipath returned signal. In the meantime, the multipath return caused by walls can in turn enlarge the investigation domain and enhance the number of the effective array aperture, if sufficiently separated from the direct return, which motivates the multipath exploitation method [7].
In [8, 9], the algorithm of back propagation (BP) and delaysumming beamforming were proposed to realize the target refocusing, respectively. However, the interference produced by multipath cannot be suppressed by the above approaches, which results in the emergence of multipath ghosts in the image region. In order to solve these problems, several methods were proposed to remove ghosts produced by multipath [10, 11]. An adaptive CLEAN algorithm was proposed to remove the false targets or ghosts based on the BP imaging algorithm [10]. From the perspective of multisensor fusion, a distributed algorithm which restrains the ghosts introduced by multipath was proposed in [11] based on the algorithm of fully polarimetric beamforming. The analysis was verified by using data collected from multiple polarimetric orientations. However, the methods proposed in [10–12] regarded multipath return as clutter, and multipath was not particularly exploited or analyzed.
Recently, a group sparse compressive sensing approach was used to reconstruct stationary scenes [13, 14]. The returned signal from the target and walls were separated and shown in the image by using a sparse reconstruction approach, which jointly uses the wall and target models. Moreover, the sparse reconstruction and representation methods are also widely used in various fields of image processing [15–19]. On the basis of the local maximum values extract method and the 1D Kalman filter, the authors extracted the 1D trajectories of the real target and the multipath ghosts in each receiving channel in [20, 21]. Then, based on the principle of the firstorder multipath echoes, the position of the sidewall was computed in each frame, and the multiple frame average value is used to improve the detection accuracy. Besides, instead of using multipleinput multipleoutput (MIMO) antenna array, a limited number of transceivers mounted on robot are used [22, 23].
In this paper, the multipath effect is exploited to reduce false positives and signaltoclutter ratio (SCR) at the target location by mapping the ghost targets back to the real target position. We propose a calculation model of modified Green’s function based on the nonlineofsight (NLoS) propagation path. After that, Back Projection (BP) imaging combined with modified Green’s function is used to obtain the composite image. In order to associate and map the ghosts, an association matrix is constructed, which has deep nulls at the assumed ghosting locations and a large peak value at the real target location, which helps removing the ghosts and ensuring the accuracy of target location. Comparing to the existing literature [7], our simulation results are shown to achieve about 3 dB SCR gain, which demonstrates the superiority of our proposed method.
The remainder of the paper is organized as follows. The multipath propagation model is introduced in Section 2. In Section 3, we present a multipath exploitation algorithm by using the modified Green’s function imaging. Section 4 reports representative simulation results of the proposed method, and concluding remarks follow in Section 5.
Through this paper, matrix is denoted by boldface capital letters, \(\mathcal {C}\) denote the complex number set, and \(\mathbf {A} \in \mathcal {C}^{N \times M}\) denotes a complexedvalue matrix of dimension N×M. Moreover, A(x,y) denotes A’s (x,y)th element. The vector is denoted by boldface lowercase letters x and x=(x,y) denotes the position vector of object with x and yaxis coordinates being x and y, respectively. A triangle with vertices A,B, and C is denoted by △ABC with edges AB, BC, and AC. The symbol ⊥ means perpendicularity between two edges. The operator ⌊x⌋ denotes the largest integer smaller than x.
System model
In this section, the multipath system model where we perform throughwall radar imaging is introduced in details. In Section 2.1, we first describe the physical scenario, including the geometric positions of walls and target. Then, under this scenario setup, we obtain the signal model considering the multipath returns.
Scenario description
A twodimensional multipath propagation model is illustrated in Fig. 1, where a target locates inside homogeneous walls, with the relative permittivity, electric conductivity, and thickness of the walls being ε_{1}, σ_{1}, and d_{1}, respectively. The length of front and the back are the same and denoted by D_{1}, while the length of the side walls are D_{2}. It is assumed that the transmitter and receiver are ideal dipoles and centered at the same position R_{n} with position vector r_{n}=(−x_{n},y_{n}),n=1,2,⋯,N, where N is the number of antennas along the wall. The target is located at position x_{t}=(−x_{t},y_{t}) behind the front wall. It is assumed that the bandwidth is sufficient wide such that direct or multipath returns can be resolved.
Multipath signal propagation model
In Fig. 1, both the direct path as well as three indirect paths are considered. The firstorder multipaths are defined as the signal returned through the indirect path together with the path A or vice versa, while the secondorder multipath is defined as the signal that reaches the target and returns back to the transceivers by the indirect way. Let the timevarying transmitted signal denoted by s(t), then the composite received signal at the nth sensor by the superposition of the direct path and multipath returns is given as
where A_{0}, A_{p,1}, and A_{p,2} are the complex amplitudes related to the direct reflection and transmission coefficients for the direct path, firstorder multipath, and secondorder multipath of the pth path, respectively, where p∈{B,C,D}. Moreover, \(\tau _{\mathrm {A}}^{(n)}\) and \(\tau _{p}^{(n)}\) are the singlebounce multipath delays of the returned signals through path A and paths B,C, and D, respectively.
As mentioned in [13], the computation complexity of solving the overdetermined nonlinear equations to calculate the singlebounce path delay is extremely high; regarding this, the Snell theory and the approximation algorithm are combined to transform the overdetermined nonlinear equations into linear equations, leading to considerable decrease of complexity.
As shown in Fig. 1, let A and B be the refraction points on the airwall interface and wallair interface, respectively. For the path R_{n}→A→B, if the dielectric constant of the front wall is equal to the one in freespace, then the signal will transmit along the path R_{n}→M→B, where M is the refraction point on the airwall interface. In the meantime, if the dielectric constant of the front wall is assumed to be infinity large, the signal will transmit along the path R_{n}→A→F, where F is the refraction point on the wallair interface. Denote the true value of dielectric constant by ε_{1}∈[1,∞), then the xaxis coordinates of refraction points A and B satisfy the following equation:
Assuming that BB_{1}⊥xaxis with A_{1} being the intersection point on the airwall interface, then the triangluars ΔR_{n}BB_{1} and ΔMBA_{1} are similar to each other, and according to the Triangle Similarity Law, we have
Similarly, we can obtain the geometrical relations of the refraction points for the other propagation paths. Let xt′=(x_{t},y_{t}) be the position of the virtual target with respect to wall 1, which is shown in Fig. 1, and can be obtained by using the Householder transformation [29]. Then, the oneway path delays for path A and path B are at the nth sensor position are given by
Substituting (4) and (5) into (1), we obtain the received signal.
Throughwall imaging with multipath exploitation using modified Green’s function
In this section, the multipath returned signal, which is termed as “ghosts,” is exploited to further enhance the imaging performance. Similar to the conventional method where multipath returns are not taken into account, we first derive the modified Green’s function. Then, we proposed a method to identical and localize the multipath returned signal from the composite data. Finally, combining with the modified Green’s function, we associated and mapped the multipath signal to obtain the image of the real target.
Throughwall imaging based on modified Green’s function
According to traditional Green’s function of three layered background mediums, the lineofsight (LoS) propagation path in the absence of the wall is considered in the calculation, but the variation of the path propagation and phase are not considered due to the presence of the wall and thus there exists estimation error using the traditional Green’s function in indoor propagation model. Considering the effect of delay inside the wall, a modified Green’s function calculation model based on the nonlineofsight (NLoS) propagation path is proposed.
By dividing the whole rectangle surrounded by walls into N_{x}×N_{y} grids, the modified Green’s function at arbitrary pixel point x_{p} can be derived as [24, 25]
where k is the wavenumber, T(k_{x}) denotes the wall’s transmission coefficient, which is calculated as
where
and k_{1} and k_{2} are the wavenumbers of freespace and the wall, respectively. Moreover, k_{x} is the horizontal component of the wavenumber vector, k_{1y} and k_{2y} are the vertical components of the wavenumber vector k_{1} and k_{2}, respectively, and Γ_{12} is the local reflection coefficient at the air/obstacle interface. In order to solve (6) efficiently, the method of saddle point is applied here. More specifically, let
then the modified Green’s function can be rewritten as
According to the stationary phase method, we have
And the stationary phase point \(k_{x_{0}}\) can be obtained by solving (10), given as
Using the Taylor series expansion of Φ(k_{x}) at \(k_{x} = k_{x_{0}}\phantom {\dot {i}\!}\) and ignoring the high order terms, the phase function Φ(k_{x}) can be approximated as
Since the function F(k_{x}) only has value near the stationary phase point, whereas has zero value on the other locations, the modified Green’s function could be further simplified as
where \(F(k_{x_{0}})\) is readily given by
Taking all the antennas into account, then the composite modified Green’s function at x_{p} can be given as
Multipath identification and localization
In this subsection, a method for localizing the multipath ghosts is described. Based on the beamforming algorithm, when Δτ_{n}(x_{p},x_{t})=τ_{n}(x_{p})−τ_{n}(x_{t})=0,n=1,2,⋯,N holds for the pth image pixel located at x_{p}, which depicts the locations of target or ghost in the beamformed image and the pixel x_{p} becomes focus [26]. It is noted that τ_{n}(x_{t}) represents the twoway path delay for the direct path or the delay for the firstorder multipath at the nth sensor’s position. The firstorder ghost location is linked with L(r_{n},x_{t}) and L(r_{n},xt′), where L(r_{n},x_{t}) and L(r_{n},xt′) are the singlebounce propagation distances of path A and path B, respectively. Consider the ghost located at x_{p}, which satisfies the following equation
The firstorder ghost location is relatively fixed in the original beamformed image. We can determine the travel distance from the nth antenna to the pixel along the direct or indirect way, which indicates that L(x_{n},x_{p}) is a known parameter. If we provide a circular contour with radius L(r_{n},x_{p}) centered at R_{n}, the ghost must lie on it. Similarly, Fig. 2 shows that using two circular contours with the size of L(r_{n},x_{p}) by L(r_{m},x_{p}) centered at R_{n} and R_{m}, respectively, the ghost must appear in the place where the circular contours cross. Nevertheless, two positions, which are crossed by the contours, are presented: one is in front of the array and the other is behind it. Because of the imaging scene locate directly in front of the array, it is reasonable to suppose that the location of the ghost is in front of it. The equations for both contours can be interpreted by
In the realworld radar system, the array contains N elements. Then, there exist \(\lfloor \frac {N}{2} \rfloor \) possible pairs if each element is considered only once. Providing two circular contours with the size of L(r_{n},x_{p}) and \(L(\boldsymbol {r}_{n+ \lfloor \frac {N}{2} \rfloor }, \boldsymbol {x}_{p})\) which are centered at R_{n} and \(\mathrm {R}_{n+\lfloor \frac {N}{2} \rfloor }\) respectively, the procedure for identifying the ghost locations x_{pn} is as follows.
For all \(n = 1, \cdots, \lfloor \frac {N}{2} \rfloor \), we obtain the intersection points x_{pn} of the two circular contours by letting their analytical expression be equal. Then, taking expectation with respect to n, we have the final value of ghost location as per
For the ghosts associated with walls 2 and 3, the location of them can be obtained in a similar way.
Multipath association and mapping
After we localize the multipath ghosts, we can then associate and map these signals back to the position of the real target. To enhance the signaltoclutter ratio and remove the false targets, a composite image can be constructed by using Hadamard product of back propagation (BP) image and modified Green’s function as follows:
where I^{BP}(·) is Back Projection imaging matrix [10], \(\mathbf {G}(\cdot) \in \mathcal {C}^{N_{x} \times N_{y}}\) is the modified Green’s function matrix, whose element for arbitrary pixel is obtained in Section 3.1, and \(\bigotimes \) denotes the Hadamard product operation. Combining the strong shadow region of the target with the multipath ghost associated with wall 2, the image I^{BP}(·) has a strong false target at the same crossrange as the target and at a downrange position equal to the back wall. The target location can be identified under the simple threshold operation.
With regard to the focused multipath ghosts, the association and mapping of them are considered. Let \(\mathcal {X}\) and \(\mathcal {Y}\) be the index of the crossrange and downrange, respectively, and we define the twodimensional space as \(\mathcal {W}= \mathcal {X} \times \mathcal {Y}\). The region we are interested in, I_{G}(·), consists of N_{x}×N_{y} grids, where the pixel is located at the center of each grid. Without loss of generality, we assume that the target is located at \(\boldsymbol {x} = (x, y) \in \mathcal {W}\), and we denote the focused multipath ghost from walls 1, 2, and 3 as \(\boldsymbol {x}_{k}^{\text {wall}} = (x_{k}^{\text {wall}}, y_{k}^{\text {wall}}), k = 1, 2, 3\), \((x_{k}^{\text {wall}}, y_{k}^{\text {wall}}) \in \mathcal {W}\), respectively. Then, the composite image values at the target can be represented in a N_{x}×N_{y} complexedvalue matrix form as \(\mathbf {I}_{G}^{\boldsymbol {x}} (\cdot)\in {\mathcal {C}^{N_{x} \times N_{y}}}\) with the elements being I_{G}(x_{p};x). Similarly, the composite image values are evaluated at \(\boldsymbol {x}_{k}^{\text {wall}}, k = 1, 2, 3 \in \mathcal {W}\) and are denoted as the matrices \(\mathbf {I}_{Gk}^{\text {wall}}(\cdot)\in {\mathcal {C}^{N_{x}\times {N_{y}}}}\) with the elements being \(I_{G}(\boldsymbol {x}_{p}; \boldsymbol {x}_{k}^{\text {wall}})\). To prevent the differences in intensity, the images are normalized such that the elements of these matrices are within the interval [0,1]. For the composite image matrix elements, a threshold operation is considered, with the threshold β_{th}, i.e.,
where I(x,y) is either \(I_{Gk}^{\text {wall}}(x,y)\) or \(I_{G}^{\boldsymbol {x}}(x,y)\), and \(\widehat {\mathbf {I}}_{Gk}^{\text {wall}}(\cdot)\) contains many peak values in and near the locations of the ghosts and \(\hat {\mathbf {I}}_{G}^{\boldsymbol {x}}(\cdot)\) contains many peaks value in the vicinity of the target location. Moreover, we define \(\tilde {\mathbf {I}}_{G}^{\boldsymbol {x}}(\cdot) = \mathbf {I}  \hat {\mathbf {I}}_{G}^{\boldsymbol {x}}(\cdot)\), where I is the identity matrix. If the pixel x_{p} is located at the location of the focused multipath ghost, \(\mathbf {I}^{\text {BP}}(x_{k}^{\text {wall}}, y_{k}^{\text {wall}})\) is assumed to be a large value. In that case, \(\tilde {\mathbf {I}}_{G}^{\boldsymbol {x}}(\cdot)\) can be used to alleviate the multipath pixels that are very close to the genuine target if they are at position \(\boldsymbol {x}_{k}^{\text {wall}}\). It can also be used to reject the pixel location being in the vicinity of the true target location. In order to associate and map the ghost, an association matrix I_{inter} can be constructed, and the element of I_{inter} is calculated as
Repeating (25) for all possible pixel locations, matrix \(\mathbf {I}_{\text {inter}} \in {\mathcal {C}^{N_{x}\times {N_{y}}}}\) can be obtained. Assuming deep nulls at the region of ghost and the strongest peak at the region of target, the final image of the multipath exploitation is obtained by
where the element of I_{f} is calculated as
and λ_{th}∈[0,1] is a predefined threshold.
Results and discussions
In this section, numerical simulations are given to evaluate the effectiveness of the proposed modified Green’s function. The reference scenario and measurement configuration are best defined in the preceding section. The image domain is x=2.0 m and y=1.48 m in width and length, respectively. The transceiver array scans the region of interest from the distance between the front 0.3 m from the front wall, along a line parallel to the wall in x direction from 0 m to − 2 m with a step of 0.05 m. The accurate scattered field is calculated in time domain by gprMax software, we define the relative permittivity, conductivity, and thickness are ε_{1}=6.0, σ_{1}=0.01 S/m, and d_{1}=0.1 m for the front wall in the software, respectively. For a line source, its radiates Ricker wavelet at 900 MHz center frequency and with a bandwidth of 900 MHz. The intensities of the image are described in dB, unless noted otherwise.
Single target
The numerical results for the imaging of one target behind a homogeneous wall are presented in this subsection and the square target whose size is 0.04 m is centered at (− 0.9,0.75) m. The composite image in Fig. 3 is obtained through Hadamard operation.Since the reflection and transmission coefficients for singlebounce trip along with path B, path C, and path D are smaller than that along with path A, the pixel values at the region of ghosts are significantly smaller than that at the region of target. Then, the target location and the focused multipath ghosts’ locations can be identified by the threshold operation. Figure 4 is an intermediate image; it shows the existing deep nulls at the locations of ghosts and the strongest peak can identify the location of the target. Figure 5 shows the result after association and mapping. This figure shows the target and the ghosts that have been mapped back to the target location. Since the method of background subtraction can not filter out the interference produced by the target, the extraneous target is retained, which is located at (− 0.9,0.46) m.
Multiple targets
The configuration of radar system is equal to the previous case. One target is located at (− 0.7,0.75) m and the other at (− 1.25,0.65) m. As shown in Fig. 6, it reveals not only the original beamforming image, but some false targets are also shown from the figure. This is probably due to the interactions with the wall or the targets themselves. It is noted that proposed algorithm can not eliminate erroneous targets. The intermediate image is shown in Fig. 7, which are existing deep nulls at the regions of ghosts and the strongest peak at the locations of targets. In Fig. 8, it reveals the result of association and mapping. After exploitation, we can distinctly see two targets and map ghosts back to real targets. Figure 8 shows the compound interactions between the walls and targets, leading to some remnants persisting.
As noise was not taken into account in our scenario, we consider the signal SCR, defined as the ratio of the power in the target regions to the power in the rest of the image as the performance metric [27]. Through multipath exploitation, the case of the SCR about single target and multiple targets are reported in Table 1 by using the method in [7] and the proposed method. It can be seen that both methods can enhance the SCR after the procedure. According to the exploitation procedure, for single target case, the SCR increases by about 9.9722 dB at the location of the target by using the method in [7] and about 13.549 dB growth by the proposed method. For multiple targets case, the SCR component is 9.4965 dB and 6.7006 dB for the proposed method and the method in [7], respectively.
Results of real measured dataset
An experimental dataset comprising of a single target in an enclosed structure is collected to demonstrate the improvement from multipath exploitation. A throughwall radar system is used for signal synthesis and data collection. The system is operated at a central frequency of 800 MHz. The human is stationary in the imaging scene and the standoff distance to the back side of the front wall is 1.4 m. The size of the imaging scene is 4.0×3.2 m^{2}, consisting of 125×100 grids and the size of each grid is 0.032×0.032 m^{2}. Assuming that the walls are singlelayered and homogeneous with permittivity ε_{1}=6.0 and thickness d_{1}=0.1 m, which can be estimated through the approach proposed in [28].
Figure 9 shows the raw radar data with heavy clutter in the time domain. Due to the complex imaged environment, the electromagnetic interactions among the targets, the unidentified walls, and the furnitures distort the received signals, leading to severe interference to the exploitation results. Compared with the curve presented by target, the multipath returns are relatively weak due to multiple reflections occurring. Using BP imaging algorithm, the result is shown in Fig. 10. There exist so many ghosts or false targets that the location of the target can not be identified with a simple operation. Figure 11 is the result of association and mapping; the majority of ghosts are removed, and we can ensure the target location.
Conclusion
In this paper, a modified Green’s functionbased method is proposed to remove the focused multipath ghosts and enhance the SCR by exploiting multipath returned signals on the basis of the knowledge of geometric model. An algorithm combining the modified Green’s function of throughwall radar with the algorithm of BP imaging can be built, which associates and maps the ghosts to their real target. Simulation and real radar experiment results confirm that the proposed method can strengthen the quality of TWRI and enhance SCR at the target location.
Availability of data and materials
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Abbreviations
 BP:

Back Projection
 LoS:

Lineofsight
 MIMO:

Multipleinput multipleoutput
 NLoS:

Nonlineofsight
 SCR:

Signaltoclutter ratio
 TWRI:

Throughwall radar imaging
 UWB:

Ultrawide band
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Funding
The research in this article is supported by the National Natural Science Foundation of China (No.61561034, No.61261010, and No.71461021) and the Natural Science Foundation of Jiangxi Province (2015BA B207001).
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SW proposed the algorithm and developed the mathematical derivations as well as the majority of simulation results. HZ helped to check the simulation codes and supported the real data measurements. The paper was written by SW and revised by HZ and RD. After receiving the comments of reviewer and editor, SL helped to revise the paper. All authors read and approved the final manuscript.
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Correspondence to Huilin Zhou.
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Wu, S., Zhou, H., Liu, S. et al. Improved throughwall radar imaging using modified Green’s functionbased multipath exploitation method. EURASIP J. Adv. Signal Process. 2020, 4 (2020). https://doi.org/10.1186/s1363402006602
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Keywords
 Modified Green’s function
 Multipath association and mapping
 Throughwall radar imaging