Open Access
Add to BibSonomy
Abstract
A novel scheme is proposed in this paper to model the complex scattering pattern of radar target with a small training data set. By employing the ideal equivalent scattering center as transfer function, the frequency domain response can be represented by series of parameters so that the aspect and frequency domain dependency can be decoupled, and modeled, independently. In specific, neural network is employed to model the aspect dependency considering the complexity. To maintain the continuity of transformed parameters, a parameter extraction algorithm based on the Orthogonal Matching Pursuit is designed. With the same amount of training set, the proposed scheme exhibits a much better performance than the existing representative modeling techniques such as Geometrical Theory of Diffraction (GTD)-based model, the polynomial scattering center model and so on. At the same time, the training speed of the proposed model is also faster than those techniques.
© 2022 Optica Publishing Group under the terms of the Optica Open Access Publishing Agreement
1. Introduction
The scattering response of radar target plays a vital role in the field of modern radars because it contains abundant information about the size, shape, structure and material of targets. Such information is necessary for the design, test and the simulation of radar system [1–3]. For example, the Radar Cross Section (RCS) determines whether the target can be detected by the radar and thus influence the design of radar transmitting power.
In general, the scattering responses of the targets are obtained based on the computation electromagnetic such as method of the moment and ray tracing technique. To construct the complete pattern of the scattering response, a dense sampling is usually required, therefore, the electromagnetic solver needs to run many times which leads to a huge computation burden especially for electrically large target. To solve such problems, surrogate modeling are widely employed to reduce the high computation burden by decreasing the training size. The concept of surrogate model has been successfully applied in the field microwave component, antenna modeling, optimization design and so on [4–10]. Some surrogate modeling techniques about the scattering response of radar targets has also been proposed in [11–15]. The equivalent scattering center concept has been widely employed to model the scattering response of radar target and proven to be effective [16–26], and have been widely used for the scattering response modeling. In general, the aspect and frequency dependencies are formulated based on approximate electromagnetic theory such as Geometrical Theory of Diffraction. However, due to the complexity of electromagnetic process, it is too hard to use several specific analytical forms to model scattering response, especially for targets with complex structures. To overcome these drawbacks, the Polynomial Scattering Center (Poly-SC) model is proposed by employing polynomial as the aspect and frequency dependency with dynamic orders accounting for different kinds of scattering centers [12]. The artificial intelligence is also widely used for scattering modeling [27–34]. The Gaussian Process Regression (GPR) which can be improved with a specific covariance function derived from the Physical Optics is employed for the RCS modeling [15,35]. The Support Vector Machine can also be employed with a Physical Optics inspired kernel function [34]. But, the setting of covariance function in GPR or kernel function in SVM need to varies with different kind structures to model different kinds of pattern. Therefore, the generality of such methods is limited. The neural network has been proven to be very effective when dealing with the multi-dimension problem, and widely employed in microwave component modeling [13,36–39]. However, the methods based on neural network usually require a large amount of training data and consume tremendous time to train the weights and bias of neurons.
This paper proposes a scheme to combine the neural network with the equivalent scattering center model so that the broad frequency and aspect band scattering response can be modeled with limited samples. The equivalent scattering center model are employed to model the frequency domain. Employing the designed OMP algorithm, the frequency domain response can be represented by sparse parameters which keep continuous over the aspect domain. Based on the fact that the scattering response usually have a much more complex pattern over the aspect domain, the neural network is chosen to model variation of the parameters with respect to aspect due to strong ability for non-linear process. By employing the equivalent scattering model which is derived by the electromagnetic theory, the scheme can make use of prior information about the scattering pattern in frequency domain to some extent. Actually, the proposed scheme transforms the two-dimensional(aspect-frequency) modeling problem into two cascade one-dimensional modeling problems so that the modeling complexity can be decreased dramatically.
The rest of paper is organized as: second part presents the method of proposed scheme, third part gives the simulation used to verify the effectiveness of proposed model and last part concludes the paper.
2. Methodology
2.1 Formulation of the proposed model
Based on the geometrical optics, the scattering field in the frequency domain can be regarded as the superposition of fields from a series of equivalent scattering centers. This approximation is widely used and presents robust performance when dealing varies of target including the simple or complex structure. The specific analytical expression of the ideal equivalent scattering center model in frequency domain is given as below [16]:
where $K$ is the number of equivalent SC, $f$ is the frequency, $c$ is the speed of light, $E$ is the mono-static scattering field, $A_i$ and $r_i$ are the amplitude and radial distance of $i$-th equivalent SC, respectively. The accuracy and robustness of Eq. (1) have been verified in many researches [16–26]. However, when taking the aspect domain into consideration, the modeling becomes much more difficult, because the scattering response could change rapidly due to the different electromagnetic mechanism within small aspect range. To extend the frequency domain into frequency-aspect domain, there are two problems. First is about the phase term of response determined by the location of equivalent SC. Most parametric models based on GTD just represent the location with 2D coordinate [17,20] so that the phase term presents a form of trigonometric function. However, this assumption is quite inaccurate, because the equivalent scattering centers could be invisible at some aspects when it is sheltered by some structure. Another problem is about the amplitude dependency in aspect domain, most parametric models employ the $sinc$ or exponential function to model it, but for complex target, such form is too simple to describe the complex scattering pattern. Considering such two difficult problems, a completed new form of the frequency-aspect domain model inspired by the concept of equivalent scattering center is proposed. The detailed formulation of the proposed model is presented as below: where $\theta$ is the aspect of excitation signal. Differing from the conventional aspect-frequency domain equivalent SC model, the Eq. (2) directly replacing the amplitude and phase term with a flexible function form. And the function $A_i(\theta )$ and $r_i(\theta )$ can be fitted by any method as long as it can be predicted properly. It abandons the inaccurate assumption of 2D location of equivalent SC and fixed analytical form of amplitude term. Therefore, the proposed formulation can combine the machine learning technique with the equivalent SC concept in a novel way. To fit parameter series $A_i(\theta )$ and $r_i(\theta )$, their continuities should to be guaranteed. Thus, the sensitivity of scattering with respect to the parameter series are derived as below: It is obvious that the absolute value of sensitivity about parameter series $A_i$ is smaller than 1 from Eq. (3) which means that the continuity of $A_i$ is easier to be achieved. Meanwhile, the variation range of parameter series $r_i$ is much larger because of the multiplier based on Eq. (4). Therefore, to keep the continuity of parameters of point SC model, the function $r_i(\theta )$ is set as constant. The effectiveness of such operation will be validated based on the simulation results further. Considering the non-linearity of scattering response in aspect domain, the neural network is used to fit the aspect dependency $A_i(\theta )$. The overall structure of the forward proposed model is presented in Fig. 1. In total, the detailed forward scattering model can be regarded as a combined neural network form inspired by the equivalent SC of which the structure is presented in Fig. 2. Here, the combined neural network has four layers including the input layer, hidden layer, SC model parameters layer and the output layer. The first three layers consists of $K$ independent Multi-Layer Perceptron (MLP) used to fit $K$ parameters of ideal SC model $A_i(\theta ),i=1,2,\ldots,K$. Considering that the $A_i$ is complex-valued, the real part and imaginary part can be modeled by two MLPs, respectively. For simplicity of notation, these two MLPs are regarded as one contributed to each $A_i$. The activation function of the hidden layer is the sigmoid function. The point SC model can be regarded as the activation function in the output layer.
Fig. 1. The structure of proposed forward aspect-frequency domain model combing the neural network and ideal equivalent SC model.
2.2 Training scheme of the combined neural network inspired by the equivalent scattering center
The training procedure can be regarded as an inverse process of the forward scattering model which needs to obtain $K$ well-trained MLPs employing training set generated by the electromagnetic solver. The detailed steps of the proposed scheme are presented in Fig. 3. The first step is the training samples generation with electromagnetic solvers. The distribution of samples is specified and presented in Fig. 4. Such distribution can be generated with two substeps. First, generating the aspect values $\theta _i, i = 1,2,..,N_{\theta }$ located in the aspect range of interest. Then, generating $f_{i,j}, i = 1,2,..,N_{\theta }, j = 1,2, \ldots N_f$ frequency values located in the frequency range of interest for each aspect value $\theta _i$. After these two substeps, the training set in frequency-aspect domain can be obtained and notated with $\{\theta _i,f_{i,j}\}$ which contains $N_\theta \times N_f$ samples in total. For each aspect slice, the point SC model will extract its corresponding parameters. Before parameters extraction, machine learning technique could be performed to reconstruct the frequency domain scattering response to decrease the demand of data amount used to extract the parameters. Here, the Spline method is chosen because of its robust fitting ability for varies of curves. It should be noted that the Spline can be replaced by any other modeling method as long as it can reconstruct the frequency domain scattering response well using sparse samples. Based on the frequency domain model, enough samples can be generated which will be used for parameter extraction. In specific, the second and third steps of training scheme are illustrated in Fig. 5.
Fig. 5. The Spline modeling process in frequency domain and parameters extraction of the point SC model.
The parameter extraction is actually to optimize objective function as below:
where the $\textbf {f}$ is the complex-valued scattering response measurement vector, the D is the abundant dictionary generated based on Eq. (1), $\sigma$ is a complex-valued sparse vector of which element is the amplitude $A$ of equivalent SC. And the constraint $\textbf {f} = \textbf {D}(\textbf {r})\sigma$ can be represented as the matrix form as below:
Algorithm 1. Determination of the number of SC:K, corresponding rK, and AK(θ) based on aspect slices in the training set
Algorithm 2. Function [A, r,MAE] = OMPmodi (f,D, flag, r,K)
After the parameters $\textbf {A}(\theta )$ for all $N_\theta$ aspect slices located in training set are extracted, the neural network can be used to learn the curve of each $A_i(\theta )$. The mean square error is used as the loss function of neural network when training. The Levenberg-Marquardt back-propagation is employed as the neural network optimizer which is an improved version of the Newton’s optimization method with varying learning rate. Thus, the learning rate doesn’t need to be specified. It should be noted that the proposed combined neural network is different with the conventional neural network. It has different structure and special training scheme. In specific, the amplitude $A_1$ to $A_k$ are intermediate variables. After they are modeled by the MLPs, the final scattering response needs to be calculated further based on the Eq. (2). The proposed scheme actually model the aspect-frequency domain scattering response by two cascade one-dimensional interpolation problems. Compared with the proposed method, the conventional method such as pure neural network needs to model the complex aspect-frequency scattering pattern, while the neural network in the proposed scheme only needs to model $K$ curves of which the complexity has been decreased dramatically.
3. Numerical examples
3.1 Unmanned aerial vehicle (UAV)
A metallic UAV is employed as the first target in the simulation as depicted in Fig. 6, which contains many structures such as motor, holder, battery, propeller and so on. The size of target is 86cm$\times$86cm$\times$35cm, the frequency is set from 8GHz to 16GHz with step size of 40MHz, and the observation aspect is set from $0^\circ$ degree to $40^\circ$ with a step size of $0.02^\circ$. A linearly polarized plane wave of which the electric field along the $\theta$-axis in spherical coordinates frame is illuminated on the target. And only the $\theta$-axis component of scattering field is considered as the response, because the $\phi$-axis component is much smaller and negligible compared with $\theta$-axis component. The maximum wavelength of the excitation signal is 3.75cm with the corresponding minimum frequency 8GHz, thus the UAV can be regarded as an electrically large target which is ten times the maximum wavelength. Considering tiny structures and complex shape, the asymptotic solver such as the Shooting and Bouncing Ray (SBR) could not be accurate enough, which can’t take the resonance, creeping wave into consideration so that the full wave solver Multilevel Fast Multipole Algorithm (MLFMA) is employed. The commercial solver CST which contains the MLFMA algorithm is employed in practice. And the results by the commercial solver are regarded as the ground truth for the further processing. To save overall simulation time, the proposed method is employed to decrease the data amount used to reconstruct full pattern of the scattering response. The full pattern of the frequency-aspect domain scattering response is presented in Fig. 7 as below. It can be found that the RCS varies very rapidly especially along the aspect axis. The goal of proposed scheme is to interpolate the pattern as depicted in Fig. 7 with small datasets. The detailed description of the datasets is presented in Table 1. To decouple the aspect and frequency dependency, the point SC model is employed to transform the frequency-domain scattering response into sparse parameters. The number of SC is determined with a convergence analysis as depicted in Fig. 8. Based on Fig. 8, 30 SCs are used to represent the frequency domain scattering response where the $Error_{\rm conv}$ is pretty small. Then the fixed radial distance $\textbf {r}$ is determined based on the Algorithm 1. As mentioned above, the continuity of extracted parameters influences the quality of model construction a lot. Therefore, the extracted parameter of the first point SC model is presented in Fig. 9 which exhibit a very smooth curve for both amplitude and phase. Next, the neural network is used to fit the parameters of point SC as shown in Fig. 9. 12525MB memory and 6.15s time are consumed in total for the $K$ MLPs training. The average training error is 5.43e-6, the average testing error is 2.11e-5, and the average epochs required are 30. The training curve of MLP corresponding to the amplitude of $A_1(\theta )$ is presented in Fig. 10. Once the neural network trained, the proposed model is established. Part of the detailed results are presented. The frequency domain slices are presented in Fig. 11 of which the proposed model shows a very accurate prediction for both the RCS and corresponding phase information within a broad frequency range. It can be found that the RCS presents a relatively mild variation over such a broadband range which proves that it is a right choice to employ the Spline to fit the curve. Also, it should be noted that the phase of the scattering response in frequency domain shows a quasi-phase pattern if it is unwrapped, this is in accordance with the phase term of the point SC model: $\frac {-j4\pi f}{c}r$. It proves that the point SC model could achieve a high reconstruction accuracy for the frequency domain scattering response. Next, it is about the aspect domain slices as shown in Fig. 12. It is obvious that aspect domain slices show a more complex pattern compared with frequency domain slices for both the RCS and phase. Especially, the phase of aspect domain scattering response exhibits a high degree of non-linearity which means that it is not proper to model the aspect domain response using the concept of equivalent SC, because its phase term presents a quasi-trigonometric function with respect to aspect. By employing neural network, the complex pattern of scattering response in aspect domain can be well modeled. The proposed model matches very well with the ground truth for both the RCS and phase. The reconstruction error and computation time of the modeling process with same amount of training data are concluded in Table 2 and compared with representative existing modeling methods. It is apparent that the proposed model exhibits a much better accuracy with less modeling time. The basic MLP models the relationship between scattering response and $(f,\theta )$ directly which is pretty complex so that the modeling accuracy and time are both worse compared with the proposed method. It can also be found that the traditional GTD-SC model presents the largest error with the highest computation time, because the fixed analytical forms cannot fit the response patter well but bring an extremely large dictionary which increases computation complexity. The Poly-SC model which is also a hybrid method combing polynomial with equivalent SC model presents a good performance, but the phase term of the 2D coordinate still not accurate enough and polynomial could not account for the aspect domain well because too high orders are required in this case. Besides, the results of SVM and GPR are much worse compared with the other method especially for the phase calculation. This because that the setting of kernel function in SVM and covariance function in GPR influence the performance a lot, when the kernel function and covariance function don’t match the pattern of scattering response, it completely doesn’t work. The proposed model proposes a new scheme to combine the physical principle with the machine learning technique which lead to the best performance.
Fig. 11. The comparison of aspect domain slices between the proposed method and ground truth for UAV.
Fig. 12. The comparison of aspect domain slices between the proposed method and ground truth for UAV.
Table 1. Statistical analysis of the datasets for UAV.
Table 2. Statistical analysis of the error and computation time for UAV.
3.2 Missile
To validate the effectiveness of proposed model further, another example about the missile is performed of which the specific structure is presented in Fig. 13. The overall size is about $0.8\times 1 \times 3 \rm {m}$ and the frequency ranges from $1.2\rm {GHz}$ to $1.5\rm {GHz}$ with a step size of $\rm {50MHz}$. The $\phi$ is fixed at zero, and $\theta$ sweeps from $0^\circ$ to $90^\circ$ with a step size of $0.2^\circ$. A linearly polarized plane wave of which the electric field along the $\theta$-axis is illuminated on the target and the $\phi$-axis compoent of the mono-static scattering field is considered as the response in this simulation. The information about the dataset division is concluded in Table 3. With a convergence analysis, $62$ equivalent scattering centers are employed to represent the frequency domain scattering response because of the large aspect variation. One aspect slice and frequency slice are presented in Fig. 14 in which the proposed method matches very well with the ground truth for both RCS and phase. For the frequency domain slices in Fig. 14(a), the phase still shows a regular linear shape which verified the rationality of point SC model in frequency domain further. For aspect slices in Fig. 14(b), the variation between the $0^\circ$ and $35^\circ$ is milder than the variation between $35^\circ$ and $90^\circ$, because the main scatterer changes with different aspect range. From $0^\circ$ to $35^\circ$, the excitation signal mainly hit on the head part of the missile, a hemisphere structure, while from $35^\circ$ to $90^\circ$ mainly hit on the body of missile, a cylinder structure. Obviously, the neural network can fit both situation well.The overall comparison between the proposed model and representative methods about accuracy and computation time are concluded in Table 4. It is obvious that the proposed method presents a much better performance in both accuracy and computation time. Compared with basic MLP and Poly-SC model, the proposed model almost has two- or three- times accuracy improvement with less time. The GTD-SC model still have a bad performance because it is limited by the fixed analytical form for aspect and frequency dependency, especially for the complex target. The basic MLP presents a good performance when fitting RCS of the missile, however, when accounting for the phase it exhibits a relatively large error because the rapid variation of phase pattern. It also means that the concept equivalent SC is efficient for modeling the phase dependency with the term $\frac {-j4\pi f}{c}r$. The results based on SVM and GPR becomes even worse compared with the first case because the scattering pattern becomes more complex.
Fig. 14. The comparison of frequency and aspect domain slices between the proposed method and ground truth for missile.
Table 3. Statistical analysis of training samples for missile.
Table 4. Statistical analysis of the error and computation time for missile.
4. Conclusion
This paper presents a novel scheme to model the frequency-aspect domain scattering response of radar target. The point SC model is employed as the transfer function so that the frequency and aspect dependency can be modeled, independently. The modeling method in frequency and aspect domain can be replaced by any other method, which exhibits the flexibility of the proposed scheme. Compared with the existing representative modeling techniques, the proposed method shows a better fitting accuracy with even less training time. The proposed scheme also shows a general method to model the multi dimension scattering response efficiently, and it can be extended with more dimensions included such as the bi-static scattering response, solid angles and so on. Besides, the transfer function could be investigated further to find a robust way to keep the continuity of parameters and decrease the number of SC so that the modeling efficiency can be improved further.
Disclosures
The authors declare that there are no conflicts of interest related to this article.
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..
References
1. B. Persson and M. Norsell, “On modeling rcs of aircraft for flight simulation,” IEEE Antennas Propag. Mag. 56(4), 34–43 (2014). [CrossRef]
2. C. Uluisik, G. Cakir, M. Cakir, and L. Sevgi, “Radar cross section (rcs) modeling and simulation, part 1: a tutorial review of definitions, strategies, and canonical examples,” IEEE Antennas Propag. Mag. 50(1), 115–126 (2008). [CrossRef]
3. B. Persson and M. Norsell, “Conservative rcs models for tactical simulation,” IEEE Antennas Propag. Mag. 57(1), 217–223 (2015). [CrossRef]
4. S. Koziel and A. Pietrenko-Dabrowska, “Reduced-cost surrogate modelling of compact microwave components by two-level kriging interpolation,” Engineering Optimization 52(6), 960–972 (2020). [CrossRef]
5. S. Koziel and A. Pietrenko-Dabrowska, “Design-oriented computationally-efficient feature-based surrogate modelling of multi-band antennas with nested kriging,” AEU-International Journal of Electronics and Communications 120, 153202 (2020). [CrossRef]
6. A. Pietrenko-Dabrowska and S. Koziel, “Surrogate modeling of impedance matching transformers by means of variable-fidelity electromagnetic simulations and nested cokriging,” International Journal of RF and Microwave Computer-Aided Engineering 30(8), e22268 (2020). [CrossRef]
7. S. Koziel and A. Pietrenko-Dabrowska, “Reliable data-driven modeling of high-frequency structures by means of nested kriging with enhanced design of experiments,” Engineering Computations 36(7), 2293–2308 (2019). [CrossRef]
8. S. Koziel and A. Pietrenko-Dabrowska, “Low-cost performance-driven modelling of compact microwave components with two-layer surrogates and gradient kriging,” AEU-International Journal of Electronics and Communications 126, 153419 (2020). [CrossRef]
9. A. Pietrenko-Dabrowska and S. Koziel, “Reliable surrogate modeling of antenna input characteristics by means of domain confinement and principal components,” Electronics 9(5), 877 (2020). [CrossRef]
10. A. Pietrenko-Dabrowska, S. Koziel, and A. Hasan, “Cost-efficient bi-layer modeling of antenna input characteristics using gradient kriging surrogates,” IEEE Access 8, 140831–140839 (2020). [CrossRef]
11. A. N. O’Donnell, J. L. Wilson, D. M. Koltenuk, and R. J. Burkholder, “Compressed sensing for radar signature analysis,” IEEE Trans. Aerosp. Electron. Syst. 49(4), 2631–2639 (2013). [CrossRef]
12. J. Halman and R. J. Burkholder, “Sparse expansions using physical and polynomial basis functions for compressed sensing of frequency domain em scattering,” IEEE Antennas and Wireless Propagation Letters 14, 1048–1051 (2015). [CrossRef]
13. H. Chen, C. Yang, and Y. Du, “Machine learning-assisted analysis of polarimetric scattering from cylindrical components of vegetation,” IEEE Trans. Geosci. Remote Sensing 57(1), 155–165 (2019). [CrossRef]
14. A. C. Austin and M. J. Neve, “Efficient field reconstruction using compressive sensing,” IEEE Trans. Antennas Propag. 66(3), 1624–1627 (2018). [CrossRef]
15. D. Xiao, L. Guo, W. Liu, and M. Hou, “Improved gaussian process regression inspired by physical optics for the conducting target’s rcs prediction,” IEEE Antennas and Wireless Propagation Letters 19(12), 2403–2407 (2020). [CrossRef]
16. L. C. Potter, D.-M. Chiang, R. Carriere, and M. J. Gerry, “A gtd-based parametric model for radar scattering,” IEEE Trans. Antennas Propag. 43(10), 1058–1067 (1995). [CrossRef]
17. J. A. Jackson, B. D. Rigling, and R. L. Moses, “Canonical scattering feature models for 3d and bistatic sar,” IEEE Trans. Aerosp. Electron. Syst. 46(2), 525–541 (2010). [CrossRef]
18. Z. Li, K. Jin, B. Xu, W. Zhou, and J. Yang, “An improved attributed scattering model optimized by incremental sparse bayesian learning,” IEEE Trans. Geosci. Remote Sensing 54(5), 2973–2987 (2016). [CrossRef]
19. M. Hurst and R. Mittra, “Scattering center analysis via prony’s method,” IEEE Trans. Antennas Propag. 35(8), 986–988 (1987). [CrossRef]
20. K.-Y. Guo, X.-Z. Han, and X.-Q. Sheng, “Scattering center models of backscattering waves by dielectric spheroid objects,” Opt. Express 26(4), 5060–5074 (2018). [CrossRef]
21. R. Carriere and R. L. Moses, “High resolution radar target modeling using a modified prony estimator,” IEEE Trans. Antennas Propag. 40(1), 13–18 (1992). [CrossRef]
22. A. Anghel, G. Vasile, R. Cacoveanu, C. Ioana, S. Ciochina, and J.-P. Ovarlez, “Scattering centers detection and tracking in refocused spaceborne sar images for infrastructure monitoring,” IEEE Trans. Geosci. Remote Sensing 53(8), 4379–4393 (2015). [CrossRef]
23. E. J. Hughes and M. Leyland, “Using multiple genetic algorithms to generate radar point-scatterer models,” IEEE Trans. Evol. Computat. 4(2), 147–163 (2000). [CrossRef]
24. E. K. Miller and D. L. Lager, “Inversion of one-dimensional scattering data using prony’s method,” Radio Sci. 17(1), 211–217 (1982). [CrossRef]
25. Q.-Y. Qu, K.-Y. Guo, and X.-Q. Sheng, “An accurate bistatic scattering center model for extended cone-shaped targets,” IEEE Trans. Antennas Propag. 62(10), 5209–5218 (2014). [CrossRef]
26. K.-Y. Guo, Q.-Y. Qu, and X.-Q. Sheng, “Geometry reconstruction based on attributes of scattering centers by using time-frequency representations,” IEEE Trans. Antennas Propag. 64(2), 708–720 (2016). [CrossRef]
27. H. M. Yao and L. J. Jiang, “Machine learning based neural network solving methods for the fdtd method,” in 2018 IEEE International Symposium on Antennas and Propagation & USNC/URSI National Radio Science Meeting, (IEEE, 2018), pp. 2321–2322.
28. T. Song, L. Kuang, L. Han, Y. Wang, and Q. H. Liu, “Inversion of rough surface parameters from sar images using simulation-trained convolutional neural networks,” IEEE Geosci. Remote Sensing Lett. 15(7), 1130–1134 (2018). [CrossRef]
29. Y. Sun, Z. Xia, and U. S. Kamilov, “Efficient and accurate inversion of multiple scattering with deep learning,” Opt. Express 26(11), 14678–14688 (2018). [CrossRef]
30. Z. Wei and X. Chen, “Deep-learning schemes for full-wave nonlinear inverse scattering problems,” IEEE Trans. Geosci. Remote Sensing 57(4), 1849–1860 (2019). [CrossRef]
31. I. T. Rekanos, “Neural-network-based inverse-scattering technique for online microwave medical imaging,” IEEE Trans. Magn. 38(2), 1061–1064 (2002). [CrossRef]
32. F. Xu and S. Fu, “Modeling em problem with deep neural networks,” in 2018 IEEE International Conference on Computational Electromagnetics (ICCEM), (IEEE, 2018), pp. 1–2.
33. S. Caorsi and P. Gamba, “Electromagnetic detection of dielectric cylinders by a neural network approach,” IEEE IEEE Trans. Geosci. Remote Sensing 37(2), 820–827 (1999). [CrossRef]
34. D. Xiao, L. Guo, W. Liu, and M. Hou, “Efficient rcs prediction of the conducting target based on physics-inspired machine learning and experimental design,” IEEE Trans. Antennas Propag. 69(4), 2274–2289 (2021). [CrossRef]
35. J. P. Jacobs and W. P. Du Plessis, “Efficient modeling of missile rcs magnitude responses by gaussian processes,” IEEE Antennas Wirel. Propag. Lett. 16, 3228–3231 (2017). [CrossRef]
36. H. M. Yao and L. J. Jiang, “Machine learning based neural network solving methods for the fdtd method,” in 2018 IEEE International Symposium on Antennas and Propagation USNC/URSI National Radio Science Meeting, (2018), pp. 2321–2322.
37. V. K. Devabhaktuni, M. C. Yagoub, and Q.-J. Zhang, “A robust algorithm for automatic development of neural-network models for microwave applications,” IEEE Trans. Microwave Theory Tech. 49(12), 2282–2291 (2001). [CrossRef]
38. J. Jin, C. Zhang, F. Feng, W. Na, J. Ma, and Q.-J. Zhang, “Deep neural network technique for high-dimensional microwave modeling and applications to parameter extraction of microwave filters,” IEEE Trans. Microwave Theory Tech. 67(10), 4140–4155 (2019). [CrossRef]
39. F. Feng, C. Zhang, J. Ma, and Q.-J. Zhang, “Parametric modeling of em behavior of microwave components using combined neural networks and pole-residue-based transfer functions,” IEEE Trans. Microwave Theory Tech. 64(1), 60–77 (2016). [CrossRef]
40. A. N. O’Donnell, Sparsity and Compressed Sensing for Electromagnetic Scattering and Radiation Applications (The Ohio State University, 2014).
41. H. Liu, B. Jiu, F. Li, and Y. Wang, “Attributed scattering center extraction algorithm based on sparse representation with dictionary refinement,” IEEE Trans. Antennas Propag. 65(5), 2604–2614 (2017). [CrossRef]
