1. Introduction

Microwave and millimeter-wave (MMW) imaging systems have been widely used for various applications such as concealed weapon detection [1–3], though-wall imaging [4,5], ground-penetrating radar (GPR) [6,7], non-destructive testing [8,9], and medical diagnosis [10,11]. To obtain sufficient capabilities for target recognition or classification, high resolution images need to be reconstructed by forming large apertures or arrays and by using precise imaging algorithms.

Wide aperture results in high cross-range resolution, and is usually implemented through phased array [12], synthetic aperture radar (SAR) [13], or multiple-input multiple-output (MIMO) array antennas [14]. Both phased array and SAR use collocating transceivers, thus only acquire monostatic measurements. By contrast, MIMO array exploits the spatial diversity of transmit and receive antennas by acquiring multistatic scattered waves. Although from sampling point-of-view, all three techniques could synthesize a large effective (i.e. virtual) aperture, they are not equivalent when imaging targets with strong scattering directivity. In this perspective, only MIMO possess the full potential to collect the complete multistatic scattering information of a complex object, and is more likely to perform imaging with sufficient accuracy.

Existing imaging algorithms for planar and cylindrical geometry show great performances in reconstructing simple objects, or targets with convex surfaces. Many of these algorithms originated from the field of SAR, such as the range doppler (RD) algorithm, the chirp scaling (CS) algorithm, the range migration algorithm (RMA), and their extensions for bistatic or multistatic imaging modes [15–17]. For more complex imaging applications, such as land-mine detection or through wall imaging, time-reversal [18,19] and Kirchhoff migration [20] methods are developed for target reconstruction in homogeneous or inhomogeneous dielectric background. On the other hand, imaging algorithm aiming at boundary extraction are proposed for super-resolution radar imaging by using the reversible boundary scattering transform between the range wavefront and the target boundary [21]. This is done either by calculating the envelope of spheres determined by the observing antenna location, or by directly mapping from observed ranges to target points using range point migration [22,23].

The abovementioned techniques usually encounter difficulties when imaging complex objects with concave surfaces. In this case, the resulting images could contain significant artifacts which leads to misinterpretation of the target. One of the important problems is the high-order artifacts appearing between the legs and/or beneath the arms during security screening in cylindrical MMW portal scanning, which makes it difficult to detect contraband hidden in these body regions [24]. The problem of the concave structure artifacts in near-field imaging is caused by the assumption of Born approximation used in most of the conventional linear inversion algorithms. Under the Born approximation, the incident field is taken in place of the total field to be the driving field at each point in the scatterer [25]. This is accurate if the scattered field is weak comparing with the incident field. However, target with complex shape specifically with high contrast (i.e. mirroring) concave surface structures may cause strong high-order scattering.

The connection between multiple-scattering mechanism and reconstructed image using linear inversion algorithm has been analyzed in [26] for microwave imaging. This type of problems has also been studied in the field of radar cross section (RCS) estimation because multiple reflection affects RCS of the target and lead to difference between far-field and near-field measurements [27,28]. In the field of inverse scattering, full-inversion algorithms are developed to reconstruct the image of complex targets when multiple reflection exists [29,30]. Such theory uses the full wave model to compute the distribution of physical parameters such as the target’s dielectric properties. Due to the ill-posed nature of inverse scattering estimation, issues such as convergence and excessive computational complexity are often challenging problems when facing electrically large objects.

In the field of near-field SAR imaging, the mechanism of artifacts formation caused by multiple reflections between targets has been investigated [31,32]. A bistatic imaging method for reducing artifacts is proposed under cylindrical scanning geometry [33]. The algorithms considering double-scattered signals between targets have been developed to enhance the recoverable range of a target shape and to suppress false images [34,35]. Under the scenario where a dielectric background exists, such as in the field of land-mine detection or through wall imaging, multiple reflections between the dielectric background and the targets are considered in the wave propagation procedure to improve imaging accuracy [36,37]. An efficient algorithm has been proposed to accurately reconstruct concave objects under cylindrical monostatic geometry by analytically including the multiple scattering process in the forward formation for dihedral-type structures [38]. It was demonstrated that such technique can perform high-quality near-field imaging of specific type of targets with strong high-order scattering.

In this paper, we address the multistatic imaging problem of concave objects and propose an extended algorithm for accurate multistatic near-field image formation. Shooting and bouncing rays (SBR) principle is applied to build an analytical forward model for dihedral type structures under cylindrical multistatic configuration. Based on this, a reconstruction algorithm is formulated to produce accurate images despite of strong high-order scattering within the concave object.

The rest of this paper is organized as follows. Section 2 establishes the range equation of coupling objects under multistatic configurations. This is used to interpret and analyze the artifacts from conventional algorithms. In Section 3, an analytical forward model including the process of multiple reflections is formulated. Based on that, a backward model capable of performing accurate reconstruction is proposed. Numerical and experimental validation of the proposed technique is presented in Section 4. Finally, Section 5 summarizes the results and conclusions of this paper.

2. Analysis of scattering directivity of concave objects and cause of artifacts

2.1 Scattering directivity of concave structures

The scattering directivity of concave targets are firstly investigated. Figure 1 shows the simulated distribution of scattered waves over both rotation (incident) angle and multistatic angles from dihedral structures with 60^o and 30^o opening angles using EM full-wave method of moments (MoM). Signals belonging to different number of reflection times (RT) are marked by boxes with solid lines indicating odd number of RT and dashed lines for even RT. Some interesting trends are observed. For the 60^o dihedral structure, the maximum number of scattering for the EM wave before exiting the structure is 3. The signals within the two vertical yellow boxes (solid lines) at higher rotation angles are the 1 RT signals, and the vertical red box (solid lines) near 0^o rotation angle represent the 3RT signals. The two horizontal orange boxes (dashed lines) mark the 2RT signals.

 

Fig. 1. Distribution of scattered waves from 60° and 30° dihedral structures under different number of reflection times (RT) over a range of rotation (incident) angles and multistatic angles. RT stands for the number of reflection times of the incident wave before exiting the structure.

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In comparison, the maximum reflection time for a 30^o dihedral is much higher, as it reaches 6 RT. This is reasonable since the smaller the opening, the more likely the wave experience higher number of reflections during the interaction with the concave object. The 1RT, 3RT, and 5RT signals are all vertically distributed, ranging from high to low rotation angles. The 2RT, 4RT, 6RT signals are horizontally distributed, ranging from high to low bistatic angles. As we further investigate the scattering distributions for the complete range of dihedral opening angles, it is observed that odd RT signals are always spatially distributed as vertical lines, while even RT signal are distributed horizontally. And the larger the number of reflection times, the closer to 0^o the signal appears.

The vertical distributions in Fig. 1 mean that the odd RT signal only appears at limited ranges of incident angles, but can be received by a large range of bistatic angles. And horizontal distributions mean that the even RT signal appears when the incident angle is near the bisector of the concave structure, and can only be received by certain bistatic angles. Comparing with monostatic measurements, a multistatic array will receive more odd RT signals and some even RT signal that may not be captured under monostatic configuration. As an example, the 2 RT signal can only be acquired by bistatic angles higher than 30^o for the case of a 60^o dihedral. In practice, because different bistatic angles contain different RT signals from different regions of the object, multistatic array measurements are necessary in obtaining an accurate and complete reconstruction of the target.

2.2 Artifact analysis under multistatic configurations

Figure 2 gives the illustration of the cylindrical multistatic imaging geometry and the reconstructed image of a dihedral structure which exhibits strong multiple reflections between its two edges. The measured signals from all transceiver pairs are simulated by EM full wave method of moments (MoM) from 12 GHz to 24 GHz. Then the image is reconstructed by the conventional imaging algorithm used in cylindrical aperture synthesis (CAS) [39]. The reconstructed image splits into three parts, marked as A, B, and C, where A and B are symmetric and dihedral-shaped structure with different opening angles, and C is a strong scattering point at the vertex. Only A corresponds with the actual position of the target. It is obvious that the reconstructed image contains high-order artifacts and many parts of the actual object are missing. Therefore, a new multistatic reconstruction algorithm is needed in order to achieve accurate reconstruction of concave objects.

 

Fig. 2. Illustration of cylindrical multistatic imaging geometry and reconstruction of a dihedral using conventional algorithm. Cylindrical scanning is performed by rotating the object from -90° to 90°with 1° step while the scattered waves are captured by the array consisting of a single transmitter in the center and 9 receivers ranging from -40° to 40° with 10° interval.

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To understand why certain artifacts are generated, we first derive the range equation that describes the dependency of propagation distance at certain transceiver pair on the target geometry and observation angles. This is formulated with the help of mirror reflection method applied in [38], in which consecutive mirrored points with respect to the edges of the object are used to simplify the estimation of the propagation distance. Here high frequency approximation is assumed where EM wave propagation can be simplified as ray traces [40]. Figure 3 gives the schematic illustration of the mirror reflection method of a dihedral with 2φ opening angle (from –φ to +φ). Considering the EM wave propagating between a certain transceiver pair from transmitting antenna at point $A({{r_t},{\theta_t}} )$ to the object, and after multiple reflections scattered back to the receive antenna at point $D({{r_r},{\theta_r}} )$, where $({{r_t},{\theta_t}} )$ and $({{r_r},{\theta_r}} )$ are the polar coordinates of these antennas. Under the mirror reflection principle, the wave reflected by edge $+ {\varphi }$ are equivalent to the wave radiated from point B, where B is the mirror point of A corresponds to edge $+ {\varphi }$. If the EM wave is reflected by edge $- {\varphi }$ for the second time, the equivalent mirror point becomes point C at $({{r_t},{\theta_a}} )$. Due to the principle of reflection, every time we make a mirror point, the distance from the mirror point to $({0,0} )\; $equals to the distance from the original point to $({0,0} )$, meaning all the mirror points are on a circle with dihedral vertex as the center. As a result, the length of propagation path equals to the distance between receiving point D and the last mirror point, as point C in the presented scenario, and labeled as $Rang{e_{CD}}$. The number of reflection times in this scenario is two, but it could be an arbitrary number of times. Therefore, the propagation path equals to the line section whose endpoints are the incident point and its mirror point from the last reflection. The formulation of $Rang{e_{CD}}$ for a certain bistatic transmit/receive pair within the MIMO array is presented in (1), and its detailed derivation can be found in Appendix-A.

 

Fig. 3. Schematic illustration of the mirror reflection method for estimating the propagation range of concave objects at multistatic transceiver pairs.

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This range equation can help understand the formation of artifacts. In the conventional aperture synthesis algorithm under cylindrical geometry, the focusing procedure identifies scattering points at midpoint position of the propagation range. Thus, the image consists of equivalent reflection points, as hypothesized under Born approximation. When the number of reflections equals to one ($n = 1$), the equivalent reflection points are on the edge of dihedral, so it coincides with the actual position of a part of the target, such as ‘A’ in Fig. 2(b). When $n > 1$, it should divide into two cases. If n is an odd number, the artifacts form image as several isolated dihedrals sharing one vertex, and the opening angle is triple or five times large as the original opening angle, such as ‘B’ in Fig. 2(b). This kind of artifacts is named as type I. If n is even, the propagation range does not depend on the transmitting angle ${\theta _t}$ or receiving angle ${\theta _r}$ since $({{\theta_t} - {\theta_r}} )$ is constant for a fixed transceiver pair, meaning it will appear at a constant arrival time in the received data over the observation angles, and therefore is reconstructed into a point located at the vertex of the dihedral, such as ‘C’ in Fig. 2(b). This kind of artifacts is named as type II.

It should be noted that because the cosine functions in (1) are insensitive to positive/negative symbols, there exists two $\theta $ values which can fulfill the same range equation. And the two $\theta $ values keep the relationship as shown below

When ${\theta _t} - {\theta _r} = 0$, meaning a monostatic measurement, the two $\theta $ values are the same.

3. Formulation

In this section, we first establish the relation between the outline of the concave object and the acquired multistatic signals. Similar to the monostatic method [38], shooting and bouncing rays (SBR) method is utilized to build the analytical forward model and then a backward model can be formulated to perform accurate reconstruction of the target.

3.1 Forward formulation

To obtain an accurate forward model, we apply the concept of SBR which was proposed for calculating the RCS of cavity structures [41]. It involves tracing a dense grid of rays originating from the incident plane wave into the concave structure using geometrical optics, and after multiple bounces among the interior walls the exit rays are used within an integral formula to compute the total scattered field from the structure. In this work, we use the same underlying principle but with analytically derived propagation ranges and near-field spherical source model to formulate the forward process under multistatic imaging geometry.

Figure 4 illustrates the imaging geometry. The propagation range at certain transceiver pair is divided the into two parts. The first part denoted as ${d_1}$ represents the wave propagation from the transmit antenna at $({{r_t},{\theta_t}} )$ to the last reflection point $({{r_2},\varphi } )$ before exiting the concave structure. The second part denoted as ${d_2}$ is the propagation path from the last reflection position to the receiving antenna at $({{r_r},{\theta_r}} )$. The sum of both parts $({d_1} + {d_2})\; $is the total propagation range between the transmit and receive antennas within the array. Applying the same mirror reflection approach, ${d_1}$ equals to the line section between the mirror reflection point $({{r_t},{\theta_a}} )$ of the transmit antenna and the last reflection point $({{r_2},\varphi } )$.

 

Fig. 4. Schematic illustration of the multistatic imaging geometry and the propagation range which is split into two sections in the forward formulation.

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The total scattered field at the receiver can be expressed as the integral of the scattered fields of the last reflections on the inner surface of the structure according to the Rayleigh Sommerfield diffraction formula [42]. Here we formulate the received signal ${S_n}({{\theta_t},{\theta_r},f} )$ at receiver ${\theta _r}$ corresponding with transmitter ${\theta _t}$ for different number of reflection times, n, in an integral equation in the following, and its derivation is given in details in Appendix B.

The forward formulation in (3) establishes the generalized form between the target’s reflectivity distribution $D({r,\varphi } )$ and the received signal ${S_n}({{\theta_t},{\theta_r},f} )$ at different reflection times n under the cylindrical multistatic configuration. It can be reduced to the monostatic case ([38], Eq. (5)) when the transceivers are collocated.

3.2 Reconstruction algorithm

Based on the concept of aperture synthesis that compensates the phase shift in frequency-domain, the backward formula for reconstruction can be derived from the forward model in (3) by exchanging the position of the acquired multistatic signal and target’s reflectivity function. The reflectivity map of the target $D({r,\theta } )$ can be formulated in a discrete form as follows.

Signals corresponding to different reflection times should be processed accordingly due to different range equations. The resulting images need to be further integrated in order to obtain the complete multistatic reconstruction of the target.

As we mentioned in the forward formulation, there are corresponding bounds for the transmitting and receiving angles ${\theta _{t/r}}$ for different reflection times n. These conditions are formulated as follows:

3.3 Algorithm implementation

The proposed algorithm can be implemented in five major steps which is illustrated in Fig. 5. The first step is to isolate received raw data by different number of reflection times (RT). This is necessary due to ambiguities introduced by the multiple reflections from the concave structure. The range equation in (1), for both odd and even number of reflection times n, contains a $n\theta $ term that may result in the same propagation distance under different combinations, e.g. 1 RT of 30° dihedral will have the same range as 3 RT of 10° dihedral, or 5RT of 6° dihedral, etc. Consequently, if the ${n^{th}}$ RT data is not isolated and reconstructed only by the ${n^{th}}$ range equation, artifacts can be introduced.

 

Fig. 5. Flow chart of proposed multistatic reconstruction algorithm considering high-order scattering.

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Performing step1 is obvious for simulated data since all scattering events during the forward simulation can be recorded and separated. But when dealing with measured data, signals of different reflection times are already integrated at the receiver. In our current implementation, this separation step is performed by utilizing the feature that signals belonging to different reflection times are centered on distinct observation angles, which will be demonstrated in details in session 4. This step is similar to the monostatic condition and was described in details in [38] (Appendix C).

After separating the data, the isolated n RT data are processed accordingly. The second step involves calculating the range equations, and compensating phase shift according to (4) for different number of reflection times. Then the third step is integrating the signals for all frequencies f and transmitting and receiving angle pairs ${\theta _{t/r}}$ which can fulfill the conditions in (5)-(8). This produces an image for every reflection times.

Before integrating results from all reflection times, some special treatment on the even RT images are required. The fourth step involves limiting the region of even RT image by the corresponding bistatic angle. This is a necessary step due to the ambiguity of even RT signals to the opening angle of the structure, as indicated in Eq. (2). Fortunately, we could avoid this type of artifact because the region that can receive even RT reflection signal will not vary with the dihedral rotation angle. This means if a certain bistatic transmit/receive pair could receive the even RT signal, it must come from only the opposite edge of the dihedral. Therefore, if bistatic angle $b{i_{angle}} > 0$, it can only receive even RT signal coming from $\varphi < 0$, and vice versa.

After this treatment on the even RT images, the final reconstruction of the object is obtained by integrating results from all reflection times.

4. Imaging results

In this section, simulation and experimental results are presented to verify the proposed multistatic reconstruction technique for concave objects.

4.1 Numerical simulations

Figure 6 illustrates multistatic reconstructions of 30° and 60° dihedral objects from both conventional and proposed algorithms. Cylindrical multistatic measurements are acquired from 0° to 40° multistatic angles with 10° steps. It is clearly visible that the proposed technique can eliminate high-order artifacts that often appears in conventional algorithms. By utilizing the proposed reconstruction algorithm and by collecting signals from a complete range of bistatic angles, we can perform accurate imaging of concave objects despite of strong high-order scattering.

 

Fig. 6. Comparison of multistatic imaging results of both 60° and 30° dihedral objects from (a) conventional CAS algorithm and (b) the proposed algorithm.

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4.2 Measurement results

The proposed reconstruction technique is further verified with experimental data. Near-field cylindrical imaging experiments are carried out in an anechoic chamber with vector network analyzer (VNA) and mechanical turntable. The setup is shown in Fig. 7(a). Data acquisition has been performed in the frequency domain with VNA connected with multistatic array consisting of antipodal Vivaldi antennas [43] through a multi-port RF switch. The measurement frequency band was from 12GHz to 24GHz, and the cylindrical rotation radius is 0.6m with a rotation step of 1^°. The multistatic array was 40cm wide with the transmit antenna at the center resulting in multistatic angles from 0 ^° to 20 ^°.

 

Fig. 7. Measurement setup and concave objects under imaging.

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Multistatic imaging measurements on both 30° and 80° dihedral structures are acquired as shown in Fig. 7(b). The edge length of the dihedrals is 0.2 m in both cases. The measured signals in time-domain are shown in Fig. 8. It is visible that the complete data can be separated into different reflection times by their observation angles and arrival time. Figure 9 shows the multistatic imaging results using both conventional CAS and proposed algorithms. The advantage of the proposed algorithm is clearly illustrated showing no high-order artifacts and accurate reconstruction of the concave contours. In the case of 30° dihedral, both type I and II artifacts described in Section 2 are corrected. On the other hand, the scattering from 80° dihedral primarily contains double bouncing between its two edges, and therefore displaying a strong type II artifact with a strong point-like artifact located at the vertex of the CAS image. This is also resolved by the proposed technique with complete recovery of the target surfaces.

 

Fig. 8. Measured multistatic signals in time-domain for (a) 60° and (b) 30° dihedral structures. Measurements at 0°, 10° and 20° bistatic angles are shown (from left to right). Different number of reflection times can be separated and are marked accordingly.

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Fig. 9. Comparison of experimental multistatic imaging results of both 30° and 80° dihedral objects from (a) conventional CAS algorithm and (b) the proposed algorithm.

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The proposed algorithm is further tested under a typical human body imaging scenario with more complex concave structures. The picture of the tested mannequin and imaging results from conventional and proposed techniques are presented in Fig. 10. The opposing surfaces between the legs of the body cause strong coupling during imaging, and leads to high-order artifacts appearing as phantom objects in the results from conventional algorithm. In contrast, the proposed algorithm can significantly reduce these artifacts between the legs. It was assumed the vertex of dihedral structures are located between the center of the thigh area in order to perform accurate separation of signals from different reflection times. Comparing with monostatic results, the multistatic one shows the advantage of delivering more complete reconstruction of the inner contours of the body.

 

Fig. 10. Comparison of experimental imaging results of the lower part of a human-sized manikin. (a) photo of the manikin during test, (b) reconstruction from conventional CAS algorithm using monostatic data, (c) reconstruction from the proposed algorithm using monostatic data, (d) reconstruction from conventional CAS algorithm using multistatic data, and (e) reconstruction from the proposed algorithm using multistatic data. Comparing with monostatic images, multistatic results give more complete reconstructions of the inner contours of the legs.

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5. Conclusions

This paper presents an extended study of multistatic imaging of dihedral-type concave objects based on previously established theories on monostatic array configurations. High-order artifacts that often appear in the results from conventional imaging techniques under multistatic configurations are analyzed and classified based on the odd and even number of reflection times. An analytical forward model including the process of multiple scattering propagations is formulated. And based on this, a reconstruction algorithm is proposed to perform accurate multistatic imaging. Both numerical and experimental results demonstrate the improvements of the proposed method in terms of artifact reduction and complete surface reconstruction of dihedral-type concave structures. This method can be the building block for the development of more generic reconstruction techniques for accurate multistatic near-field imaging of complicated high-contrast objects.

Appendix A: Derivation of range equation among multistatic transceiver pairs

According to Fig. 3, the propagation distance between a multistatic transceiver pair, $Rang{e_{CD}}$ equals to the length of the line segment $CD$,

(9)$$Rang{e_{CD}}^2 = r_t^2 + r_r^2 - 2{r_t}{r_r}cos({{\varphi_{CD}}} )$$

where ${\varphi _{CD}}$ is the angle corresponding to the line segment $CD$, and it’s expressed as

(10)$${\varphi _{CD}} = {\hat{\theta }_a} - {\theta _r}$$

where ${\hat{\theta }_a}$ is decided by the number of reflection times n and the angle of transmitting position ${\theta _t}$. Using the principle of reflections, if the ray first illuminates $\varphi $ edge, the 1 RT mirror point ${\hat{\theta }_{a1}}$ of ${\theta _t}$ corresponds to edge $\varphi $, and the sum of ${\hat{\theta }_{a1}}$ and ${\theta _0}$ is $2\varphi $; the 2 RT mirror point ${\hat{\theta }_{a2}}$ of ${\hat{\theta }_{a1}}$ corresponds to edge $- \varphi $, and the their sum is $- 2\varphi $, and so on. It should be noted that $\varphi $ could be either positive or negative representing one edge of the dihedral. Table 1 shows the relation between ${\hat{\theta }_a}$ and ${\theta _t}$, and based on the mathematical induction the expression of ${\hat{\theta }_a}$ can be derived as

(11)$${\hat{\theta }_a} = {( - 1)^{n - 1}}({2n\theta - {\theta_t}} )$$

where $\theta ={\pm} \varphi $. Putting (10) and (11) into (9) results in the range equation presented in (1).

Table 1. The relationship between ${\hat{\theta }_a}$ and ${\theta _t}$

View Table

Appendix B: Derivation of the forward model

In the second step of the forward model, the scattered field are obtained by the Rayleigh Sommerfield’s diffraction formula.

(12)$${S_n}(P )= \frac{{j{A_n}}}{\lambda }\mathop {\int\!\!\!\int }\nolimits_{{A_n}} \frac{{{e^{jk({{d_1} + {d_2}} )}}}}{{{d_1}{d_2}}}cos (\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} ,{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over d} _2})ds$$

where in our case P is receiving position, ${A_N}$ is the integration aperture of the final interaction with the concave structure, $\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} $ is the normal vector of the integration aperture, ${d_1}$ is the length of propagation of the first $n - 1$ times of reflections, ${d_2}$ is the distance between receiving position and the aperture, and $\cos \left( {\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over n} ,{{\mathord{\buildrel{\lower3pt\hbox{$\scriptscriptstyle\rightharpoonup$}} \over d} }_2}} \right)$ is the tilting factor of the last scattering event.

Because amplitude change plays relatively little effect in coherent imaging, we only keep the exponential term and ignore the amplitude decay with range and tilting factor. Under polar coordinate system the scattered signal can be rewritten as

(13)$${S_n}({{\theta_t},{\theta_r},f} )= {\int\!\!\!\int }\delta ({\theta \pm \varphi } )D({r,\theta } )exp [ - j\frac{{2\pi f}}{c} \cdot ({{d_1} + {d_2}} )]drd\theta $$

where n is the number of reflections, ${\theta _t}$ is the observation angle of the antenna, f is the frequency, c is the speed of light, and $D({r,\theta } )$ is the spatial distribution of reflectivity of the target under the polar coordinate system.

In the case of a dihedral-type structure of two opposing edges, a bound exists for both the length r, the transmitting angle ${\theta _t}$ and the receiving angle ${\theta _r}\; $of the integration aperture on the physical structure in (13), resulting in the following equation

(14)$${S_n}({{\theta_t},{\theta_r},f} )= \mathop {\int\!\!\!\int }\nolimits_{{r_{min}}}^L \delta ({\theta \pm \varphi } )\cdot D({r,\theta } )\cdot exp\{{ - j2\pi f/c \cdot ({{d_1} + {d_2}} )} \}drd\theta $$

where L is the length and ${\pm} \varphi $ are the angles of the two edges of the dihedral, and ${r_{min}}$ is the starting point of the aperture on the inner aperture of the object that can be visible for illumination or reception due to possible blocking effects, as illustrated in Fig. 11. And this can be derived as

(15)$${r_{min}} = \left\{ {\begin{array}{l} {0,\quad\quad\quad\quad\quad\quad\quad - \varphi \le {\theta_t} \le \varphi \; }\\ {\frac{{{r_t}L\sin ({{\theta_t} - \theta } )}}{{{r_t}\sin ({{\theta_t} + \theta } )- L\sin ({2\theta } )}},\; \begin{array}{l} {\theta = \varphi ,{\theta_t} > \varphi }\\ {{\; }\theta ={-} \varphi ,{\theta_t} < \varphi } \end{array}} \end{array}} \right.$$

 

Fig. 11. illustration of possible limit for the length of integration aperture on the concave object. When the absolute value of the observation angle is larger than the opening angle of the structure $|\mathrm{\theta } |> \mathrm{\varphi }$, only the outer part of the integration aperture (marked as A1) on the target is visible for the transceiver.

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The expression of ${d_1}$ and ${d_2}$ in (14) are given as

(16)$${d_1} = \sqrt {{r_t}^2 + {r_2}^2 - 2{r_t}{r_2}cos ({\varphi _1})} $$

(17)$${d_2} = \sqrt {{r_2}^2 + {r_r}^2 - 2{r_2}{r_r}cos ({\varphi _2})} $$

where ${\varphi _1}$ and ${\varphi _2}$ are the corresponding angles labeled in Fig. 3, and their expressions are given as follows.

(18)$$\left\{ {\begin{array}{c} {{\varphi_1} = |{{\theta_a} - \theta } |}\\ {{\varphi_2} = |{\theta - {\theta_r}} |} \end{array}} \right.$$

Because ${\theta _a}$ is the mirror point of the last time reflection corresponding to edge $\theta $, its expression could be derived from (11), in which $\theta $ represents the first reflection edge. When n is odd, the first and last reflection edge is the same, ${\theta _a} = {\hat{\theta }_a} = 2n\theta - {\theta _t}$. When n is even, the first and last reflection edge are different, if we specify the last reflection edge as$\; \theta $, the first reflection edge will be $-{-}\theta $. Then ${\theta _a} = {\hat{\theta }_a} ={-} ({2n({ - \theta } )- {\theta_t}} )= 2n\theta + {\theta _t}$. Consequently, ${\theta _a}$ can be expressed as

(19)$${\theta _a} = 2n\theta + {( - 1)^n}{\theta _t}$$

Putting (19) into (18), we arrive at the expression of ${\varphi _1}\; $and ${\varphi _2}$

(20)$$\left\{ {\begin{array}{l} {{\varphi_1} = |{({2n - 1} )\theta + ({ - 1{)^n}{\theta_t}} )} |}\\ {{\varphi_2} = |{\theta - {\theta_r}} |} \end{array}} \right.$$

Now we can insert (20) into (14). Because the cosine functions in (16) and (17) are even functions, the absolute signs in (18) can be removed. Therefore, we arrive at the forward model as shown in Eq. (3) Section 3.1.

It’s necessary to note that there are several additional bounds for the forward formulation. Firstly, because ${\theta _a}$ is the mirrored angle of the last reflection point, this point should only exist on the opposite side of the dihedral structure. Thus, when the last reflection point is on the positive edge $\theta > 0$, the mirror point $({{r_t},{\theta_a}} )$ must be located on the other side of the plane marked as line I in Fig. 4. Similarly, when the last reflection point is on the negative edge $\theta < 0$, the mirror point $({{r_t},{\theta_a}} )$ must be located on the other side of the plane marked as line II in Fig. 4. Therefore, this leads to the conditions in the following.

(21)$$\varphi \left\langle {{\theta_a} \le \varphi + \pi {,\; }\theta } \right\rangle 0$$

(22)$$- \varphi - \pi \le {\theta _a} < - \varphi {,\; }\theta < 0$$

These can be further derived as the conditions on ${\theta _t}$ as follows

$$({2\textrm{n} - 1} )\varphi - \pi \le {\theta _t}\left\langle {({2\textrm{n} - 1} )\varphi {,\; }\theta } \right\rangle 0\; \textrm{and}\; \textrm{n}\; \textrm{is}\; \textrm{odd}$$

$$({ - 2\textrm{n} + 1} )\varphi < {\theta _t} \le ({ - 2\textrm{n} + 1} )\varphi + \pi {,\; }\theta < 0\; \textrm{and}\; \textrm{n}\; \textrm{is}\; \textrm{odd}$$

$$({ - 2\textrm{n} + 1} )\varphi \left\langle {{\theta_t} \le ({ - 2\textrm{n} + 1} )\varphi + \pi {,\; }\theta } \right\rangle 0\; \textrm{and}\; \textrm{n}\; \textrm{is}\; \textrm{even}$$

(23)$$({2\textrm{n} - 1} )\varphi - \pi \le {\theta _t} < ({2\textrm{n} - 1} )\varphi {,\; }\theta < 0\; \textrm{and}\; \textrm{n}\; \textrm{is}\; \textrm{even}$$

Furthermore, when the wave transmits into or reflects from the concave structure, certain observation angles could be blocked by the structure. Assuming the distance from the antenna to the rotation center is longer than the edge of the dihedral, ${r_t},{r_r} > L$, which should represent majority of the near-field imaging conditions, these scenarios further add conditions on ${\theta _t}$ and ${\theta _r}\; $as follows

(24)$$\begin{aligned} &- \frac{\pi }{2} \le {\theta _t} < \varphi {,\; } - \frac{\pi }{2} \le {\theta _r}\left\langle {\varphi {,\; }\theta } \right\rangle 0\; \textrm{and}\; \textrm{n}\; \textrm{is}\; \textrm{odd}\\ & - \varphi < {\theta _t} \le \frac{\pi }{2}{,\; } - \varphi < {\theta _r} \le \frac{\pi }{2}{,\; }\theta < 0\; \textrm{and}\; \textrm{n}\; \textrm{is}\; \textrm{odd}\\ &- \varphi < {\theta _t} \le \frac{\pi }{2}{,\; } - \frac{\pi }{2} \le {\theta _r}\left\langle {\varphi {,\; }\theta } \right\rangle 0\; \textrm{and}\; \textrm{n}\; \textrm{is}\; \textrm{even}\\ &- \frac{\pi }{2} \le {\theta _t} < \varphi {,\; } - \varphi < {\theta _r} \le \frac{\pi }{2}{,\; }\theta < 0\; \textrm{and}\; \textrm{n}\; \textrm{is}\; \textrm{even} \end{aligned}$$

Combining conditions in (23) with the ones in (24), we arrive at the following bounds.

For odd $\textrm{n}$:

(25)$$\begin{aligned}&max\left[ {({2n - 1} )\varphi - \pi ,{\; } - \frac{\pi }{2}} \right] \le {\theta _t} < [\varphi \; \textrm{and}\;- \frac{\pi }{2} \le {\theta _r} < \varphi \quad \textrm{when} \;\theta > 0 \\ &- {\; }\varphi < {\theta _t} \le min\left[ {({ - 2\textrm{n} + 1} )\varphi + \pi {,\; }\frac{\pi }{2}} \right]{\; }\textrm{and}\;- \varphi < {\theta _r} \le \frac{\pi }{2}\quad \textrm{when} \;\theta < 0 \end{aligned}$$

For even n:

(26)$$\begin{aligned}&- \varphi \le {\theta _t} \le min\left[ {({ - 2\textrm{n} + 1} )\varphi + \pi {,\; }\frac{\pi }{2}} \right]\; \textrm{and}\; - \frac{\pi }{2} \le {\theta _r} \lt \varphi \quad \textrm{when} \;\theta \gt 0 \\ & max\left[ {({2\textrm{n} - 1} )\varphi - \pi {,\; } - \frac{\pi }{2}} \right] \le {\theta _t} \le \varphi \;\textrm{and}\; - \varphi \lt {\theta _r} \le \frac{\pi }{2} \quad \textrm{when}\; \theta \lt 0\end{aligned}$$

Equations in (25)–(26) together give the valid intervals of the transmitting angle ${\theta _t}$ and receiving angle ${\theta _r}$ under different reflection times $\textrm{n}$ when calculating the forward equation in (3).

Funding

National Natural Science Foundation of China (41531175, 61731001).

Acknowledgments

The authors would like to thank Rohde and Schwarz China for their collaboration and support to our imaging experiments and Weikang Si of Beihang University for his technical support during the measurement campaign.

Disclosures

The authors declare no conflicts of interest.

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