Abstract
In this letter, an incident field optimization (IO) method integrated with contrast source inversion (CSI) method and multiplicative regularized CSI (MR-CSI) method is proposed to deal with the electromagnetic inverse problems. For the conventional iterative inversion methods, the incident field is a prior knowledge and is usually obtained by the calibration process. Through the synthetic data experiments, it can be found that a small distortion of the incident field has little impact on the inversion results, but a high distortion will lead to fault inversion results or no convergence. In this letter, the incident field optimal factor, being updated during the iterative process, is introduced into the CSI and MR-CSI method, named as IO-CSI and MR–IO–CSI. The new methods have good inversion performance even with high incident field distortion. The proposed methods are applied to the Fresnel experiment data directly without the calibration procedure. Both the single-frequency and multifrequency inversion experiments demonstrate that the IO-CSI and MR–IO–CSI method can achieve good inversion results, indicating the method’s practical applicability.
1. Introduction
For the electromagnetic inverse scattering problem, the task is to reconstruct the permittivity or conductivity distribution in the domain of interest (DOI) by the measured scattered field. The contrast source inversion (CSI) method is one of the outstanding frequency domain iterative methods to solve the inverse problem by minimizing the cost function, which is a linear combination of normalized mismatches between the data equation and the state equation. Since its proposition [1, 2], the CSI method has become very popular and many transformations have been developed. In [3], a multiplicative regularized inversion scheme of CSI was proposed and the artificial weighting parameter is avoided. In [4, 5], a new integral equation was introduced which can solve the inverse problem with high contrast. In [6, 7], a subspace-based optimization method (SOM) was invented, in which the induced current is divided into two parts, major and minor parts, to reduce the computational burden and to improve the inversion result. In [8, 9], the authors added cross-correlated error into the final cost function, forming the cross-correlated contrast source inversion (CC–CSI) method, which can achieve good robustness and higher inversion accuracy. To deal with the nonhomogeneous background inverse problem, the finite-difference frequency domain (FDFD)-based CSI [10, 11] and finite-element method (FEM)-based CSI [12, 13] were proposed, extending the application scenario of the CSI method. Recently, the deep learning (DL) method was introduced to solve the multifrequency electromagnetic inverse problems [14].
Usually, in these methods, the true incident field is known as the prior information or obtained by a certain calibration factor, which is derived from the ratio of the measured incident field to the simulated one at the receiver located at the opposite of the source [15, 16]. Ostadrahimi et al. [17] used the scattered field for calibration. Narendra et al. [18] adopted the source reconstruction method to achieve more stable inversion results. The incident field was represented by a superposition of a few cylindrical waves in [19]. In [20], an incident-modified CSI method is proposed to design the artificial material.
Although it is better to perform the incident field calibration in order to start the iterative inversion algorithm with the most accurate information that is available, in reality the incident field cannot be obtained precisely. Therefore, it is necessary to study the influence of incident field distortion on the inversion results. The incident field optimization method, cooperating with contrast source inversion (CSI) method and multiplicative regularized CSI (MR-CSI) method, is proposed in this paper. The incident field calibration factor is treated as an optimization factor that is updated during each iteration step and the incident field will converge to its true value.
The IO-CSI and MR-IO-CSI methods are applied to the Fresnel-measured data where the initial incident field is the unit line source. It is shown that the methods have a very good performance even though the initial guess of back propagation (BP) inversion result has distortions because of the incorrect incident field information. For the multifrequency inversion experiment, the conventional CSI has failed to reconstruct the targets while the IO inversion methods can still converge to the true value.
The letter is organized as follows: Section 2 is the formulation of the problem. The incident field optimization CSI (IO–CSI) method and multiplicative regularized IO-CSI (MR-IO-CSI) method are introduced in Section 3. In Section 4, the synthetic and measured data inverse experiments at single and multiple frequencies are carried out. Section 5 is the conclusion.
2. Materials and Methods
2.1. Electromagnetic Inverse Problem Configuration
The configuration of a simple two-dimensional (2D) electromagnetic scattering inverse problem is shown in Figure 1. The domain of interest (DOI) D containing unknown scatterers, characterized by , is illuminated by incoming waves generated by nth (n = 1…N) transmitter at kth (k = 1…K) measurement frequency, while the scattered field is measured by M receivers located in the measurement domain S around the domain D.
For transverse magnetic (TM) cases, the forward problem is described by two equations:where the first one is the state equation and the second one is the data equation, is the total electric wave field, is the incident field, is the wavenumber of the homogeneous background medium at kth frequency, and is Green’s function. The contrast is defined aswhere . For the inverse scattering problems, the task is to reconstruct by the measured scattered field at receiver locations.
2.2. Contrast Source Inversion Method
The cost function of CSI method is
The first term is the normalized data error, contributing from the mismatch of the data equation, and the second term is the normalized state error due to the mismatch of the state equation. and denote the Euclidian length of the function defined in domain S and domain D. The induced current and the contrast are updated alternatively during each iteration.
Usually, the back propagation method is used for the initial guess, the initial contrast source iswhere denotes the inner product of functions in domain S and the initial contrast iswhere .
2.3. Incident Field Optimization Method
For the conventional iterative inversion methods, the calibration factor is derived from the calibration procedure by data preprocessing. For the IO-CSI method, the calibration factor was treated as an optimization factor , which is updated during each iteration. So, the cost function, with the change of in the second term of (3), can be rewritten as follows:
For the two-dimension (2D) case, the initial guess of the incident field can be chosen from the line sources parallel to the z-axial or plane waves, . The initial optimization factor is . For the TM case, the incident field can be modelled as follows:where is the zeroth-order Hankel function of the first kind, is the source location. The optimization factor is updated at each iteration by minimizing the numerator of the second term of the (6):
The implementation steps of the IO-CSI method are shown in Algorithm 1.
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The method is also applied to the multiplicative regularized CSI, named MR-IO-CSI, which has the same update steps as the IO-CSI method. The cost function is given bywhere is defined aswhere is the area of the domain D and is the mesh size of the discretized domain D. The optimization will be terminated if the termination condition (the maximum iteration is reached or the residual value does not change obviously) is satisfied.
3. Results and Discussion
In this letter, the scatterers are lossless so that their permittivity is positive and conductivity equals zero. The background is the free space, so its relative permittivity equals 1.
3.1. The Synthetic Experiment of Austria Profile
As shown in Figure 2(a), the ‘Austria’ profile consists of two discs and one ring. In the electromagnetic simulation, 16 line sources and 32 line receivers are evenly placed on circles with a radius of 3 m, using the method of moment (MoM) to simulate the scattered field at frequency 400 MHz. 100 × 100 grid mesh is used in the forward problem to generate the scattered data and the grid mesh is 64 × 64 in the inverse problem. Figure 2(b) gives the original CSI result with the true incident field.
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To show the influence of the incident field distortion on the performance of the CSI method, the scattered field was obtained under different incident field amplitudes as , 2, −2, 0.5j, 1 + 1j, and 2j in the forward simulation processes. In the inversion stage of CSI, the calibration factor was set to without optimization. Some inversion results are shown in Figure 3. It can be seen that the shape and relative permittivity of the ‘Austria’ profile were reconstructed with some noise in Figure 3(b) () and Figure 3(c) (), while the inversion results of Figure 3(a) () and Figure 3(d) () have big distortion, making them unrecognizable.
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Figure 4 is the inversion result of the IO-CSI method with 1000 iterations. Figure 4(a) is the curve of the optimization factor α under different incident field amplitude . Starting from , the optimization factor is converged to the corresponding true value of the incident field under all conditions. Figure 4(b) is the IO-CSI inversion result, which is very similar to Figure 2(b) that is the original CSI result with the true incident field and the exact scatterers.
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3.2. The Fresnel Measurement Data
The profile of the Fresnel experiment [16] is shown in Figure 5. The small circle represents a plastic cylinder (berylon) with a diameter of 31 mm and relative permittivity of . The large circle represents a foam cylinder (SAITEC SBF 300) with a diameter of 80 mm and relative permittivity of . The 8 transmitting antennas and 241 receiving antennas are all wideband ridged horn antennas with a frequency range from 1 to 18 GHz. The source-object center and object center-receiver distances are 1.67 m. Every target can be included in a 0.15 × 0.15 m2 square centered at the center of the receiver circle. In this letter, only the TM data at 3 GHz and 5 GHz has been considered and the size of domain D is 0.18 × 0.18 m2 which is discretized into pixels.
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In this inversion problem, the termination condition is the value of the cost function below the . The iteration times are shown in Table 1, the MR-IO-CSI and IO-CSI need more iteration steps to meet the condition compared to the MR-CSI and CSI because of the introduction of the optimization factor.
The inversion results of ‘FoamDielExtTM,’ ‘FoamDielIntTM,’ and ‘FoamTwinDielTM’ at 3 GHz and 5 GHz are shown in Figures 6 and 7. Figures 6(a) and 7(a) is the BP results with the incident field initialized by equation (13) as . The BP results nearly have no similarity with the original profile of the targets. After several iterations, with the optimization factor converging to the true amplitude of the incident field, the MR-IO-CSI method successfully reconstructed the targets with a good match of shape and relative permittivity, as shown in Figures 6(b) and 7(b) at 3 GHz and 5 GHz. Figures 6(c) and 7(c) are the IO-CSI results also at 3 GHz and 5 GHz. It can be seen that the reconstructed images have some distortion, but each cylinder is still clearly shown in all cases.
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The experiments show that both MR-IO-CSI and IO-CSI can solve the electromagnetic inverse problem without the prior knowledge of the incident field, and the MR-IO-CSI method has better performance than that of the IO-CSI method.
3.3. Multifrequency Inversion
In this part, the multifrequency datasets of ‘FoamDielExtTM,’ ‘FoamDielIntTM,’ and ‘FoamTwinDielTM’ at 5 GHz, 6 GHz, 7 GHz, and 8 GHz are processed by the multifrequency IO-CSI method and multifrequency MR-IO-CSI method. The iterations stop after 1500 steps.
The computation times of each method are shown in Table 2. Because of the incident field optimization process, the computation times of IO-CSI and MR-IO-CSI are longer than those of CSI and MR-CSI.
The CSI (the first row) and IO-CSI (the second row) results are shown in Figure 8. The CSI results show that the reconstructed relativity permittivity of small target is higher than the true value and the shape of the big target is neglected in ‘FoamDielIntTM’ and ‘FoamTwinDielTM’ cases where the two types of target are overlapping. However, the IO-CSI method retrieve the shape and relativity permittivity of the targets more correctly in all cases, even though the initial guess is distorted because of the incorrect initial incident field.
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The MR-CSI (the first row) and MR-IO-CSI (the second row) inversion results are shown in Figure 9. Both methods have good performance in the ‘FoamDielExtTM’ case (Figure 9(a)). In the right two cases (Figures 9(b) and 9(c)), the MR-CSI method has failed to retrieve the targets, but the MR-IO-CSI method successfully reconstructs the shape and relativity permittivity of all targets.
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4. Conclusions
In this letter, the incident field optimization factor is used to replace the calibration factor in the electromagnetic inverse problems. Through the synthetic data inverse experiment, it is shown that the optimization factor is converged to the true value in the proposed IO-CSI and MR-IO-CSI method, and the inversion results are as good as those of the conventional CSI and MR-CSI method without incident field distortion. The measured data inversion results of Fresnel experiment demonstrate the ability of the single and multifrequency IO-CSI and MR-IO-CSI methods to solve practical inverse problems. This method can be applied to situations where the calibration factor is hard to obtain.
Data Availability
Previously reported [TXT] data were used to support this study and are available at (https://doi.org/10.1088/0266-5611/21/6/S09). The datasets are used in part (Results and Discussion—the Fresnel measurement data) in this manuscript to verify the proposed inversion method. Also, the datasets are cited in this manuscript (Results and Discussion—the Fresnel measurement data) within the text as references [16].
Conflicts of Interest
The authors declare that they have no conflicts of interest.
Acknowledgments
This work was supported by the National Natural Science Foundation of China (grant nos. U1633202 and 61731007).