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Abstract
Millimeter-wave (MMW) imaging is becoming an important option in many sensing applications. However, the resulting images are often plagued with artifacts caused by complex target scenarios such as concave structures, hampering applications where precise recognition is emphasized. It has been shown that existing imaging techniques can effectively resolve this issue by considering the multi-reflection propagation process in the forward model of the inverse problem. But the accuracy of such method still depends on the precise separation of reflected signals exhibiting different number of interactions with the target surfaces. In this article, an improved imaging technique based on circular polarizations is proposed for accurate imaging of concave objects. By utilizing circular polarized measurements, the received signal can be divided into odd and even number of reflection times. Then, an iterative reconstruction technique is introduced to automatically separate signal components and reconstruct precise contours of the concave surfaces. Furthermore, a strict observation angle boundary model is derived based on methods of the stationary phase to correct the image deformation of edges existing in previous algorithms. Both numerical and experimental results synthesized from 6∼18 GHz dual-polarized measurements are used to demonstrate the improved accuracy and automation of the proposed method.
© 2021 Optical Society of America under the terms of the OSA Open Access Publishing Agreement
1. Introduction
Microwave and millimeter-wave (MMW) imaging is a novel and promising measurement technique that offers new solutions to short-range detection, such as human security inspection at public places [1–3], ground-penetrating radar (GPR) [4,5], nondestructive testing (NDT) [6,7], medical diagnosis [8,9], and through-wall imaging (TWI) radar [10,11], etc. With the demanding requirement of achieving accurate target recognition and classification, synthetic aperture techniques and fast imaging algorithms have been extensively exploited to obtain high-resolution images, frequently show great performances for targets with convex contours. However, conventional imaging theory regards the object as independent scattering points without considering the effect of multiple reflections among the targets of concave structure [12]. The strong high-order scattering caused by multiple reflections results in artifacts or missing parts in the image, which have been great hindrances for practical MMW imaging application [13,14].
Polarimetric measurement has already proven its effectiveness in several applications from image contrast enhancement to accurate detection [15,16]. This is mainly because the polarimetric backscattering is sensitive to objects’ structure [17,18], as shown in Fig. 1, the transmitted left-handed circular polarized (LHCP) wave changes to right-handed circular polarized (RHCP) wave after incident into a smooth reflecting surface. In the field of Synthetic Aperture Radar (SAR) imaging, circular polarization has been used in terrain classification, especially for scenes with multiple scattering such as urban areas or rainforests [19,20]. Near-field application represented by the TWI radar focused on reducing the artifacts caused by the coupling between the objects and clutters. Using a circular polarized antenna array, the system obtained better imaging quality and detection accuracy [21]. And in [22], by fusing multiple polarization images, the contrast between the target and the interference is enhanced and stabilized. However, these technologies focus primarily on extracting and improving the information of interest, lacking the mechanism interpretation and reconstruction methods for image artifacts caused by multiple reflections, further research on accurate imaging of circular polarization is significant.
Fig. 1. Schematic diagram of the polarization changing when circular polarized waves incident on a smooth reflecting surface.
For human security application, multi-static imaging was applied to suppress high-order artifacts [23,24]. In previous works, Liang et al. in [25,26] studied the dihedral structure as a typical concave structure for its strong multiple reflections. They investigated the artifact formation mechanism under monostatic or multi-static configuration, respectively. The precise segmentation of signals with different reflection times is a key step in the algorithm implementation, and CP measurement can preliminarily separate the odd- and even-bounce signals [27]. In [28], circular polarization imaging technique was used to isolate multipath effects and enriched imaging details. Figure 2(top) gives the illustration of the near-field cylindrical imaging geometry under study and the reconstruction of dihedral structures based on both linear polarization (LP) measurement and CP measurement. As shown in Fig. 2(bottom), the simulated data were reconstructed using a conventional cylindrical aperture synthesis (CAS) algorithm. Compared with LP measurement, we can find that the artifacts caused by odd and even higher-order reflections are effectively separated under CP measurement. Here, the simulated data is calculated by the shooting and bouncing rays (SBR) solver from the applied computational EM (CEM) software ANSYS HFSS. The measurement frequency band is from 6 to 24 GHz with step width 0.2 GHz, the dihedral edge length is 0.2 m and the cylindrical rotation radius is 0.6 m.
Fig. 2. Illustration of cylindrical imaging geometry and reconstruction of dihedral (with 30° opening angle) using a conventional CAS algorithm that treats all received signals as single scattering events. (a) both the transceiver antennas are linear polarization (HH), (b) the transmitting and receiving antennas are left-handed and right-handed circular polarized, respectively, (c) both the transceiver antennas are left-handed circular polarized.
In this article, a method for automatic data separation and imaging of concave objects is proposed. In Section II, combining the multiple reflection propagation models with the method of stationary phase (MSP), we extend the previous work and obtain a precise observation angle integral boundary. Based on the formation mechanism of odd and even artifacts, an automatic data separation process utilizing CP measurement is formed. Numerical and experimental results are presented to demonstrate the validity of the method in Section III. Finally, Section IV summarizes the results and conclusions of this paper.
2. Accurate multiple reflection model
In active imaging theory, the transmitting antenna emits electromagnetic waves and the target forms a scattered field under the excitation of the incident field. For targets with complex shapes and materials, the strict description of the scattering process is really complicated. Therefore, the Born approximation is usually used to simplify the scattering model [12]. In passive free space, Maxwell’s equation can be written as
Equation (4) shows that under the Bonn approximation, the total field E is completely excited by the incident field ${E_{inc}}$. For cavity objects, this will lead to an incorrect calculation of the target propagation path. In this section, we summarized the multi reflection process in dihedral, and introduced suitable segmentation and reconstruction methods.
2.1 Forward model and artifacts mechanism
Establishing the real EM wave propagation model corresponding to each number of reflection times (RTs) (e.g., n RT represents the nth time of reflection) is of great significance for understanding the mechanism of artifact formation and further accurate image reconstruction. In near-field cylindrical scheme, the maximum number of reflections (MRT) is determined by the opening angle of the dihedral [29]
where ⌈ ⌉ means rounds to the nearest integer greater than or equal to that element. As shown in Fig. 3, based on the principle of SBR method [25,26], the radiation process of EM waves inside the dihedral is split into two stages, and the forward model can be createdFig. 3. Schematic illustration of the mirror reflection method for estimating the propagation range which is split into two stages. The first stage includes the propagation path from the transmitting antenna to the last reflection point, and the second stage refers to the distance from the last reflection point to the receive antenna.
As shown in the forward model (6), the radiation process of EM waves inside the dihedral is split into two stages. The first stage refers to the propagation path of EM waves from the transmitting antenna to the final reflection point. The propagation path d1 equals the line segment between the mirror reflection point and the last reflection point. The second stage is the process of EM waves radiating from the last reflection point to free space, so the propagation range d2 is equal to the distance between the last reflection point and the receiving antenna (r0, θ0). The formula for two paths can be expressed as
The integral in the formula (6) can be evaluated by the method of stationary phase [30], the point ${\hat{r}_2}$ of stationary phase fulfilling the following equation
The formulation ${\hat{r}_2}$ is presented as follows, and its detailed derivation can be found in Appendix A.
Some important conclusions can be drawn from (10)∼(13). Firstly, for odd n, the point ${\hat{r}_2}$ of stationary phase always exists, which means all odd RTs signals can be received. The low-order reflection signal will appear at the higher observation angle, and as the number of reflection times increases, the observation angle will be closer to the 0° observation angle. On the other hand, when n is even, usually only the highest RT can be received. Especially when the opening angle 2φ of the dihedral can be divided evenly by 180°, like $2\varphi = 90^\circ ,45^\circ ,30^\circ \cdots $, only the dihedral vertex (0,0) satisfies (9), we can consider that the multiple reflection signals are formed in a very small cavity structure near the origin, which describe the formation mechanism of even RT artifacts in Fig. 2(b). Also, since the point of the stationary phase only appears near the vertex when n is even, the received even RT signal does not contain the edge length information of the dihedral.
From the formula (10)∼(13), the propagation distance of multiple reflection signal is related to the number of reflections n. In addition, the odd and even reflection times signals have different propagation laws. On the one hand, utilizing circularly polarizations can physically separate odd and even signals. On the other hand, the separated even signals can also be used as a priori information to further automatically separate each order signal. The accuracy of the separated signal is much higher than the method that relies on manual segmentation.
2.2 Backward model with precise integral limit
Based on the concept of aperture synthesis, the response at the transceiver will simply be the superposition of each point on the target compensates the roundtrip phase in frequency-domain, the backward formula for the reconstruction can be derived from the forward model (6) by exchanging the position of the signal and target’s reflectivity function. Following [25], a reflectivity image of the object can be formulated in a discrete form as follows:
According to (10)∼(13), we can derive the relation between observation angle θ0 and the stationary points ${\hat{r}_2}$, these conditions are formulated as follows
Following (15)∼(18), the observation angle θ0 for n RT is limited by the length of the dihedral’s edge L and the distance from the antenna to the rotation center r0. It is worth noting that, due to the different characteristics of the signal propagation path, the highest odd or even RT signals may be observed simultaneously. For the case that ${r_0} = 0.6m$ and $L = 0.2m$, with the same imaging geometry and frequencies as demonstrated in Fig. 1, observation angle θ0 range corresponding to 1RT of 90° dihedral is $( - 45^\circ , - 25.5^\circ )\cup(25.5^\circ ,45^\circ )$, and θ0 range of 2RT is $( - 45^\circ ,\textrm{45}^\circ )$. The time-domain signal measured by linear polarization (LP) is shown at the top of Fig. 4(a), a considerable part of the 1RT and 2RT signals will overlap. The same situation will happen at 45° dihedral, where θ0 range corresponding to 3RT is $( - \textrm{22}\textrm{.5}^\circ ,\textrm{ - 8}\textrm{.9}^\circ )\cup(\textrm{8}\textrm{.9}^\circ ,\textrm{22}\textrm{.5}^\circ )$, and θ0 range of 4RT is $( - \textrm{22}.5^\circ ,\textrm{22}\textrm{.5}^\circ )$.
For the reconstruction of measured data, signal separation with different reflection times is the key step. In [25], a method for isolating data by different RT has been proposed. According to the local minimum of the projection of time-domain signal in angle, signals with different RTs are roughly separated. However, as mentioned above, for some observation angles, signals with different RTs can be observed simultaneously, leading to distortion of the reconstructed image. As shown in Fig. 5(a), based on the method mentioned in [25], 1RT signal from 90°, 70°, 60° dihedral measured by LP are separated and reconstructed using a conventional CAS algorithm. Especially for the reconstruction of 90° dihedral, since a considerable part of the observation angle range belongs to both the 1RT and 2RT, a large image distortion occurred. For 70° and 60° dihedral, inaccurate segmentation is mainly manifested in the curvature of the inner edge of the dihedral. This is because the observation angle should be limited by the formula (15)∼(18), and the compensation path at this time satisfies the law of ray tracing. Otherwise, the near-field scattering near the vertex of the dihedral will cause the inner edge image to shrink to the vertex.
Fig. 4. Comparison between LP and CP measurement in time-domain of 90° dihedral (top) and 45° dihedral(bottom). (a)results from LP measurement, (b) results from CP measurement, where the polarization of transceiver is different, (c) results from CP measurement, where the polarization of transceiver is the same.
Fig. 5. Comparison of reconstruction of 1RT signal using a conventional CAS algorithm. (a) Obtained by the separation method mentioned in [25], (b) obtained according to the formula (15)∼(18).
2.3 Automatic imaging method for multiple reflections
The rotational handedness will be reversed when circularly polarized waves incident on smooth reflecting surfaces. For example, left-hand circular polarization (LHCP) is reflected to become right-hand circular polarization (RHCP). This means that when the polarizations of the transceiver are the same, the received signals are even RTs, and when the polarizations are different, the received signals are odd RTs.
Following (12) and (13), although the precise edge length of dihedral cannot be sensed from the even RT signal, we can still utilize the even RT image to separate the odd RTs data, and finally realize the automated imaging process. Figure 6 illustrates the flowchart of the proposed data segmentation and image reconstruction algorithm. Here we define the received odd RTs signal and even RTs signals under circular polarized measurement as ${S^{odd}}$ and ${S^{even}}$, respectively. Through the previous analysis, the received even RTs signal ${S^{even}}$ only contains the highest number of reflections echoes. Therefore, the first step is obtaining the highest even reflection number Ne. This step is described in detail in Appendix B.
The second step is gradually separating each odd RT data according to the highest even RT image ${I_{{N_e}}}$. The received odd data ${S^{odd}}$ is firstly reconstructed with n=1. Since the contour of the dihedral has been reconstructed in ${I_{{N_e}}}$, the reflection map of nRT is the overlapping region between ${I_n}$ and ${I_{{N_e}}}$. The third step is calculating the propagation range and derives the reflection map from scattering field ${S_n}$ based on the forward model (6) with observing angles bounds (15)∼(18). And in the fourth step, the isolated nRT data is removed from the odd signal ${S^{odd}}$. Repeat this process until $n > Ne$, ensure each number of reflection data are separated. The illustration of the method is shown in Fig. 7.
It should be noted that the final treatment is different from [25] and [26]. According to the analysis of MSP, only the opening angle can be determined from the highest even RT signal. The highest even RT signal will form a longer and complete dihedral regardless of the length of the target edge or whether it is missing or not. Therefore, even RT images should not be included in the final integral to avoid covering the specific information on both edges of the dihedral.
3. Experimental verification
In this section, simulation and experimental results are used to demonstrate the superiority of the proposed reconstruction technique.
3.1 Numerical simulations
Figure 8 compares different imaging methods for dihedral shape targets with missing parts. The same simulation parameters and measurement geometry as demonstrated in Fig. 2 are used. For the 90° dihedral, 2RT signal forms a strong scattering point at the dihedral vertex of the structure from the conventional CAS method. Only when the missing part is near the dihedral vertex, the energy of the received 2RT signal will be reduced, but it still causes blurring of the object’s contour. Due to 1RT and 2RT signals can be obtained in some observation angles simultaneously, the method proposed in [25] will lead to the separated 1RT signal containing part of the 2RT component, so the intensity of the dihedral edges gradually decreases from the vertex. The reconstruction of 80° dihedral also faces the same problem. As the observation angles belong to both 1RT and 2RT signal are less than 90° dihedral, the target contours are more uniform in Fig. 8(c). For the 45° dihedral, the maximum number of reflections increases to 4, so there are two types of artifacts in the CAS images. The first one caused by the 3RT signal is a new dihedral with an opening angle of 135°, and the other artifact is a strong scattering point at the vertex formed by the 4RT signal. It is worth noting that when the missing part is in the middle of the dihedral, the 1RT signal can also incident into the dihedral through the missing region. Therefore, the left edge of the dihedral is not smooth under all three methods. Besides, imprecise segmentation also results in decreased contour contrast, and blurring of missing parts can lead to false recognition, as shown in Fig. 8(c). In contrast, the proposed method in this article reached outstanding results regardless of the dihedral opening angles and missing part positions.
Compared with Fig. 8(b), the proposed method can expand the dynamic range more than 15 dB in most cases. To further quantify the performance of the three methods, all the cases in Fig. 8 were evaluate according to relative mean error (RME) [31] which is given by
where $x_j^o$ and $x_j^r$ denote the nth pixel of the normalized ground-truth image and the image for comparison, respectively. And the ground-truth used for comparison is generated by the real object shapes. Table 1 lists the RME of the examples in Fig. 8 in order. It can be seen that with decreasing opening angle of the dihedral, the RME of conventional CAS algorithm rises rapidly. The main reason is the high-order odd reflection data will form artifacts in wrong position. On the other hand, limited to the segmentation accuracy, the performance of the method proposed in [25] is weaker than the proposed method especially when the opening angle is small.
Table 1. Quantitative evaluation of the cases in Fig. 8 using RME
The running time of the method is an important issue. Owing to the iteration number is directly related to the maximum number of reflections, for a small opening angle, more reconstruction time is needed. For example, if the polar coordinate image region is 1∼360° with an angle step 1°, the imaging radius is 0.3 m, and the resolution is 0.01 m, the image matrix size is 360×31. For 90° and 45° dihedrals, the reconstruction time is 9.45s and 24.31s, respectively. The 45° dihedral needs to be compared three times to determine the highest number of reflections in the first step, while 90° only needs two times. And 45° dihedral needs to separate the signals and reconstruct the high-order reflection data. The methods proposed in [25] and [26] require much less computational cost because each reflection time data relies on manual segmentation, it also leads to the loss of imaging accuracy. For 90° and 45° dihedrals, the reconstruction time is 5.04s and 9.22s. The methods were both implemented by Matlab 2017b in a computer, equipped an i7-8750H CPU and 8.00GB of RAM.
As shown in Fig. 9(a), a number of objects of different orientations are simultaneously placed in a scene to further clarify the feasibility of the proposed method. The same settings as Fig. 2 are used in this case. As shown in Fig. 9(b), the image reconstructed by conventional imaging algorithm contains two kinds of artifacts. One is the strong scattering point at the vertex formed by the even reflection signals of 90°, 80°, and 45° dihedrals, and the other is the two edges formed by the 3RT signal of 45° dihedral. Applying the proposed method to reconstruct the image of the complex object, the strong scatting point artifact can be removed, so the dynamic range of the dihedral courter can be enlarged by about 20 dB, as shown in Fig. 9(c). In addition, the two edges of the 45° dihedral can be reconstructed correctly. Due to the segmentation error caused by the overlap of the signals of the 45° and 90° dihedrals, the right edge of the 90° dihedral is slightly weakened near the vertex.
Fig. 9. Complex target composed of three dihedrals with different orientations and their imaging results from both the conventional and proposed algorithms. (a) Target models. (b) Imaging results from conventional algorithm. (c) Imaging results using proposed technique.
The proposed method is effective for dihedral targets and does not depend on the material. Figure 10 shows the imaging results of a dihedral composed of PEC and FR-4. As shown in Fig. 10(a), part of the right edge has a different dielectric constant. In the reconstruction of conventional algorithm, strong scattering point formed by even order signal appears at the vertex. And because the materials are inconsistent, the two artifact edges are intermittent. On the contrary, our method clearly shows the expected structure and strength of the target. But it is worth noting that the intensity of the region formed by 3RT in left edge, as shown in Fig. 10(c), is weaker than other position. The attenuation is caused by the weak reflectivity of FR-4 in multiple reflections. Applying the proposed method to reconstruct the image, the strength dihedral edge is enlarged from -12.3 to 0 dB, and the part of FR-4 is -11.9 dB. In another example, we used a series of circular saw teeth to create a rough surface, as shown in Fig. 11(a), and the height of the protrusion is 1 mm. After removing the strong scattering point formed by 2RT signal, the intensity of the dihedral edges is enlarged from -18.8 to 0 dB.
Fig. 10. Comparison among imaging results target with nonmetallic materials. (a) Target model. (b) Imaging result from CAS algorithm. (c) Imaging result using proposed method.
Fig. 11. Comparison among imaging results target with rough surface. (a) Target model. (b) Imaging result from CAS algorithm. (c) Imaging result using proposed method.
The proposed method is not satisfactory for non-dihedral targets. Figure 12 shows the imaging results from conventional algorithm and our method. The measurement frequency band is from 6 to 24 GHz with step width 0.2 GHz, the cylinders diameter is 0.13 m and the rotation radius is 1.2 m. As shown in Fig. 12(b), in this case, the artifacts formed by 2RT signal are two curves between the two cylinders. Circular Polarizations can effectively remove the artifacts in Fig. 12(c). But due to the proposed method is based on dihedral shape, the reconstruction result cannot recover the complete cylinder structure. As shown in Fig. 12(d), new artifacts appeared in the image.
Fig. 12. Cylinder-shape targets and their imaging results from conventional and proposed method. (a) Imaging geometry. (b) Imaging result of linear polarization (HH) from conventional method. (c) Imaging result of 1RT signal from proposed method. (d) Imaging result of 1RT and 2RT signals from proposed method.
3.2 Measurement results
The performances of the proposed method are further verified with experimental data. As shown in Fig. 14(a), the measurement system is established based on the vector network analyzer (VNA) and mechanical turntable in an anechoic chamber. The measurement frequency band is from 6 to 18 GHz, the frequency step is 0.02 GHz and the transmitted power during the experiment is 10dBm. To synthesize the required CP data, a two-port LP antenna was used, where each port can independently transmitting and receiving linear and orthogonal electromagnetic waves. Before the starting of measurement, full dual-port calibration procedures of the VNA first perform, and then calibrate the frequency response by measuring the metal ball to ensure the accuracy of the synthesized data. Following [32], the relationship between the synthesized CP signal and the measured LP S-parameters is
A complex-shaped object containing 60° and 90° dihedrals, shown in Fig. 13, was tested with the rotation step of 1°. The cylindrical rotation radius is about 0.65 m, although the 2RT signal of 60° dihedral is weak in this situation, it can still be used for signal segmentation. The distance between the two parallel planes of the object is 20 cm. The edge length of the 60° dihedral is 10 cm, and the edge length of two 90° dihedrals is 7 cm. Figure 14 compares the imaging results of the conventional CAS technique, the method proposed in [25] and our method. As shown in Fig. 14(a), the 3RT signal from 60° dihedral forms two false edges in ±90°. And 2RT signals form two strong scattering points at the vertices of the two 90° dihedrals, the strength of the strong scattering points is about 18 dB higher than the target contour. In Fig. 14(b), because part of the 2RT signal was incorrectly separated, the intensity of the vertex of the 90° dihedral is about 9.5 dB higher than the edge. In contrast, as shown in Fig. 14(c), it can be seen that the effectiveness of the proposed imaging technique. The complete structure of the object is accurately reconstructed, while the intensity of the entire contour is quite consistent. The running time of the method proposed in [25] and our method are 18.70s and 46.29s in this case. We further estimated the faithfulness of the results by RME, and the ground-truth is generated from the model contour. The RME of Fig. 14(a) and Fig. 14(b) are 2.602 and 1.326, respectively. And the RME of the proposed method is 0.865.
Fig. 13. Picture of the experimental setup and a complex object. (a) Picture of the experimental setup, (b) picture of the complex object made by foam covered with aluminum foil.
Fig. 14. Experimental imaging results of the complex shape object using (a) conventional CAS algorithm, (b) method proposed in [25] and (c)proposed method in this article.
Another non-ideal dihedral object was tested to further verify robustness of the proposed method. As shown in Fig. 15(a), a convex part was added to the right edge of the dihedral. The edge length and the opening angle of the object is 0.30 m and 70°, respectively. The cylindrical rotation radius is about 0.7 m and the other settings were the same as in the previous experiment. As we can see in Fig. 15(b), the artifacts in the imaging results consist of two parts. One is a strong scattered point at vertex formed by 2RT signal, and the other part is composed of two edges in ±105° which formed by 3RT signal. The imperfect target shape reduced the accuracy the method proposed in [25]. In Fig. 15(c), the error image between two edges was located at ±23°. This is because part of the 1RT signal was incorrectly divided into 3RT part. As we can see in Fig. 15(d), quantitatively the intensity of the dihedral is enlarged from -14.6 dB to 0 dB. These results prove the effectiveness of our method even if the target is not composed of absolute straight edges. In this case, the radius of the imaging area is 0.4 m, so the calculation time of the method proposed in [25] and proposed method in this article are increased to 14.17s and 32.18s, respectively.
Fig. 15. Experimental imaging results of the 70° dihedral with a convex part using (a) conventional CAS algorithm, (b) method proposed in [25] and (c)proposed method in this article.
4. Conclusion
In this paper, we present an effective data separation and reconstruction technique based on circular polarizations for accurate millimeter wave imaging of concave objects. Combined the analysis of MSP, we investigate the mechanism of the artifact formation analytically, then extend the multiple reflection model proposed in previous work with a precise observation angle boundary model. Based on this, circular polarized measurements are applied and an iterative imaging technique is proposed to perform automatic data separation and image reconstruction. Both simulated and measurement results demonstrate the effectiveness of the proposed method. This technology has the potential to be further extended to achieve fully-automated and accurate imaging of arbitrary concave objects in practical applications.
Appendix A: evaluation of the forward model based on MSP
The forward model (6) can be approximately evaluated by the MSP, which states that the main contribution of the integral results from the point ${\hat{r}_2}$ of the stationary phase. Combining (9) with (7) and (8), then simplifying the formula, we obtain the following equation
The solution to (21) can be written in the following form
When the wave transmits into or reflects from the concave structure, certain observation angles could be blocked by the structure. If the number of reflections n is odd, the observation angle cannot be located outside the edge of the last reflection. And when the number of reflections n is even, the observation angle should be within the opening angle (from -φ to φ). Therefore, this leads to the conditions on θ0 as follows
Combining the conditions on θ0, we can get the following simplified result. For odd n, when the last reflection point is on the positive edge θ > 0, φ1 and φ2 can be expressed as
Then substitute (25) and (26) into (22), we can arrive at the expression of the point ${\hat{r}_2}$ of stationary phase
Similarly, for even n, when the last reflection point is on the positive edge θ > 0, φ1 and φ2 can be expressed as
So, the point ${\hat{r}_2}$ of the stationary phase can be derived as follows
Appendix B: determine the highest number of even reflections Ne
According to [33], for two 2-D images x and y, the structural similarity (SSIM) can be calculated as follows
In formula (33), the three terms are the luminance term $l(x,y)$, the contract term $c(x,y)$ and the structural term $s(x,y)$
where ${\mu _x}$, ${\mu _y}$, ${\sigma _x}$, ${\sigma _y}$ and ${\sigma _{xy}}$ are the means, standard deviations, and cross-covariance for the images, respectively. And C1, C2, C3 are constants determined based on the dynamic range of the pixel value. Here, ${C_1} = {(0.01 \ast L)^2}$, ${C_2} = {(0.03 \ast L)^2}$ and ${C_3} = {C_2}/2$, where L is the specified dynamic range value. $\alpha $, $\beta $ and $\gamma $ are used to adjust the relative importance of the three components. In this paper, $\alpha = \beta = \gamma = 1$.To obtain the highest number of even reflections Ne, firstly perform image reconstruction on the odd data with n=1, and the resulting image is marked as I1. Here, only 1RT signal will appear in the correct position, and the other odd high-order RT components will cause artifacts appearing at ${\pm} 3\varphi , \pm 5\varphi \ldots $. For even RTs data, only the highest RT signal can be received, and an important property is that the length of the structure weakly influences the range of observation angles over which the even RT signals can be detected [25]. The even RTs data could be reconstructed with $ne = 2,4,6\ldots $ through (14), and the imaging results are recorded as ${I_{ne}}$. Next, the SSIM within the same dynamic range of I1 and ${I_{ne}}$ is computed. As the edges of the dihedral reconstructed in ${I_{ne}}$ will appear at ${\pm} \varphi \cdot ne/Ne$, it is obviously the summation of SSIM between I1 and ${I_{ne}}$ takes the maximum only when $ne$ is equal $Ne$. At the same time, the overlapping region of I1 and ${I_{ne}}$ can be extracted utilizing an SSIM map. For example, the process of determining the maximum number of even reflections $Ne$ of 45° dihedral is shown in Fig. 16.
Funding
Equipment Development Department of the Central Military Commission (61404130314).
Disclosures
The authors declare no conflicts of interest.
Data availability
Data underlying the results presented in this paper are not publicly available at this time but may be obtained from the authors upon reasonable request.
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