SOM-Net: Unrolling the Subspace-based Optimization for Solving Full-wave Inverse Scattering Problems

The configuration of ISPs is shown in Fig. 1. For convenience, a 2-D transverse magnetic (TM) [32] wave is considered, where the longitude direction is along $\hat{z}$. The background is free-space with permittivity $\epsilon_{0}$ and permeability $\mu_{0}$. The unknown scatterers are supposed to be lossless by default and located in the domain of interest (DOI) $D$. The $N_{i}$ transmitters located at $r_{l}$ with $l=1,2,...,N_{i}$ sequentially illuminate the unknown scatterers, and the $N_{r}$ receivers located at $r_{q}$ with $q=1,2,...,N_{r}$ simultaneously measure the scattered field.

The computational complexity of the SOM-Net is mainly divided into three parts, preparing input data, feed-forward calculation of SOM-Net, and the estimation of the scattered field and the permittivity.

To prepare input data, it needs to obtain the raw permittivity image $\bar{\bar{\epsilon}}_{r}^{bp}$ by BP and get the deterministic induced current $\bar{{J}}^{+}$. The computational complexity of BP is $O\left(N_{i}M\log{M}\right)$ [18] if the Fast Fourier transform (FFT) is applied in the matrix-vector multiplication, and $N_{i}$ is the number of incidences. The computational cost of obtaining $\bar{{J}}^{+}$ is $O\left(N_{r}^{2}M\right)$ [34], which lies in the thin SVD decomposition of $\bar{\bar{{G}}}_{S}$.

For the feed-forward of SOM-Net, the computational cost includes several basic operations like convolutions, activation function, and max pooling, where the complexity is dominated by convolutions. Especially, assume that when performing convolution operation, the number of input feature maps is $Q_{i}$ and the number of output feature maps is $Q_{o}$. Thus, the complexity is calculated as $O\left(Q_{i}Q_{o}M_{1}M_{2}K_{f}^{2}\right)$ [18], where the feature map size and the convolution kernel size are $M_{1}\times M_{2}$ and $K_{f}\times K_{f}$ ($K_{f}=3$ in this paper), respectively.

To calculate the scattered field, the computational cost of the matrix-vector multiplication in (2) is $O\left(N_{r}M\right)$. To calculate the permittivity, the computational cost of the vector-vector multiplication in (6) is $O\left(M\right)$.

The configuration of synthetic ISPs is defined as follows. 16 line sources and 32 line receivers are equally placed on a circle with a radius of 3 m centered at the origin. The operating frequency is 400 MHz. For each transmitting antenna, all receivers measure the scattered field. By default, each scatterer is assumed to be lossless if not mentioned. And the inversion of lossy scatterers will be described in the discussions. The domain $D$ has the size of 2 m $\times$ 2 m in a free space background. It is discretized into $100\times 100$ grids to simulate the measured scattered field by MoM, while the discretization is changed to $64\times 64$ grids in inversion to avoid the inverse crime. Suppose that the measured scattered fields of all receivers from all transmitters are recorded in a matrix $\bar{\bar{{E}}}^{s}$ with the size of $N_{r}\times N_{i}$. $\bar{\bar{{E}}}^{s}$ is noise-free in the training stage, while Gaussian noise is added to $\bar{\bar{{E}}}^{s}$ in the testing stage to validate the robustness of SOM-Net against noise. The noise level is defined as ($\left\|\bar{\bar{n}}\right\|_{F}/\left\|\bar{\bar{E}}^{s}\right\|_{F}$), where $\left\|\cdot\right\|_{F}$ represents Frobenius norm of a matrix.

The training details of the SOM-Net are as follows. The MNIST data set [35] is used to generate the training data. Considering the complexity of the ISP, we randomly choose 5000 handwritten digits in the MNIST database for training and another 2500 digits for validation. In order to increase the richness of data and enhance the generalization ability of the model, each digit is randomly rotated with an angle of $[-170^{\circ},170^{\circ}]$ degrees. Besides, a circle with a random radius between 0.1 m to 0.5 m is also added in $D$. The relative permittivity of all training scatterers is set between 1.5 and 2.5. Here ${L}=15$ is used in (4) to generate deterministic induced current following the criterion of [18].

The hyperparameters for training the SOM-Net are set as follows. Both the $\lambda_{1}$ and $\lambda_{2}$ are experimentally set to 2.0. The Adam optimizer is taken to optimize the SOM-Net with $\beta_{1}=0.9$ and $\beta_{2}=0.999$. The batch size is set to 16, and the training epochs are set to 40. The first 20 epochs are running with the same learning rate of 0.0002, while the learning rate decays linearly to 0 from the 21st to the 40th epochs. The code is implemented using PyTorch on a server with Intel(R) Core(TM) i9-10900X CPU @3.70GHz and GeForce RTX 3090 GPU. The training process of the SOM-Net took about 7.26 hours under the above configurations.

In the first example, we randomly select another 1500 images from MNIST database as the testing set. All the scattered fields of testing samples are added with $10\%$ White Gaussian noise. The relative permittivity of all testing scatterers is between 1.5 and 2.5. The reconstruction results of four randomly selected examples, i.e., Test#1 to Test#4, are shown in Fig. 3. We take the structural similarity measure (SSIM) and the root-mean-square error (RMSE) of the relative permittivity of scatterers as the quality metrics [22] to evaluate the performance of all methods. The results are summarized in Table I. It can be seen that the SOM-Net achieves superior performance compared to other two methods.

Since the SOM-Net is trained by enforcing the consistency of the induced current, the scattered field, and the permittivity images, Fig. 4 shows the deterministic induced current, the predicted induced current distribution, and the corresponding target one for Test#1 to Test#4. The induced currents of other transmitters are similar and are omitted here. Meanwhile, the predicted scattered field and its reference with respect to all incidences are plotted in Fig. 5, where the predicted field is represented by the orange dots and the reference one is indicated by the blue line. It is observed that the physical constraints of SOM-Net on the induced current and the scattered field make all those variables consistent with their references. Therefore, the SOM-Net learns the governing physical law and achieves much better reconstruction results compared to U-Net.

Considering the simulation time, the SOM-Net only takes about 0.53 seconds to reconstruct a single result with given inputs. Specifically, it takes about 0.03 seconds to obtain the rough contrast by BP, and another 1.4 seconds to obtain the deterministic induced current $\bar{J}^{+}$. So, the whole simulation time of SOM-Net to reconstruct one example is about 1.96 seconds. In comparison, the SOM method needs about 27.5 seconds for 50 iterations of reconstruction. The simulation time has been significantly reduced by 92% with the SOM-Net.

To verify the generalization ability of SOM-Net, we also test the SOM-Net with more challenging examples, including scatterers with complex shapes, high-level noise interference, and scatterers with high relative permittivity. We still use the same testing model as the MNIST examples.

We finally validate the proposed method with the experimental data provided by Fresnel Institute [39]. As shown in Fig. 11, the “FoamDielExt” profile is considered in this section. It is composed of two cylinders, a foam cylinder with a diameter of 80 mm and $\varepsilon_{r}=1.45\pm 0.15$, and a plastic cylinder with a diameter of 31 mm and $\varepsilon_{r}=3.0\pm 0.3$. Different from the previous synthetic examples, there are 8 linear transmitters and 241 receivers to measure experimental data, which are located on a circle with a radius of $1.67m$ from the center of the target.

The operating frequency is 3 GHz and 4 GHz, respectively, instead of the 400 MHz used in the synthetic examples. Accordingly, the size of DOI is also changed from 2 m $\times$ 2 m to 0.2 m $\times$ 0.2 m. In this experimental example, we use the same MNIST profiles with a random circle to generate the training data set, and the relative permittivity range of all scatterers is between 1.1 and 3.0.is In Fig. 12, we present the reconstruction profiles of SOM-Net, SOM, and U-Net, respectively. The corresponding error metrics are summarized in Table V. Specifically, the best SSIM is 0.94 at 3GHz, and 0.92 at 4GHz, respectively. The comparison results of the induced current and the scattered field are demonstrated in Fig. 13 and Fig. 14, respectively. The preferable reconstruction results once again verify the strong generalization ability of the SOM-Net.

Here we take a further comparison of SOM-Net with the existing unrolling work [30, 31] using the Fresnel’s experimental data in Fig. 11 and Fig. 15. In order to take a fair comparison, we employ the same configurations of generating training data as [31], where each training sample consists of 2 or 3 random circles. More details can be seen in Table II in [31]. Specially, the predicted results of the SOM-Net are presented in Fig. 16, where both the reconstructed relative permittivity values and the shape are of high precision. Meanwhile, the corresponding SSIM and RMSE metrics are summarized in Table VI. Considering the complexity to implement the methods in [30, 31], we directly compare the SOM-Net results with those ones in the two papers. In comparison, the reconstruction results of SOM-Net are visually superior to the comparison algorithms. It should be noted that there is no quality metric given in [30], and only the metric of data misfit ($rms_{D}$) on the scattered field is defined in [31]. Accordingly, the $rms_{D}$ of SOM-Net results for the “FoamDielExt” and “FoamDielInt” examples in Fig. 16 (a) and (b) reaches 0.08 and 0.07, respectively, while that in [31] is 0.12 and 0.08, respectively. It shows that the predicted scattered field of the proposed method also matches better with the observed field.

Measurement within a limited aperture is a common ISP configuration in many real applications. Here, we conduct the limited-aperture experiment using the same setup as the above full-aperture ones for synthetic examples, except that only measurements from transmitters and receivers within the observable aperture are used. Since available measurements are limited if the aperture is too small, we only consider an experiment with a $270^{\circ}$ aperture, where 12 line sources and 24 line receivers are available.

The reconstruction results with the $270^{\circ}$ aperture are demonstrated in Fig. 17, and the corresponding quality metrics are summarized in Table VII, where we use (L) and (F) to represent the results of limited and full apertures, respectively. As observed, the proposed SOM-Net can still achieve better reconstruction performance for challenging cases than the SOM and U-Net. Meanwhile, the fitting results of the induced current by the first transmitter and the corresponding scattered fields for Test#21 are shown in Fig. 18, which indicates a good consistency with the references.

Meanwhile, it is also found that the reconstruction performance of SOM-Net degrades in the limited-aperture case compared to the full-aperture ones. As shown in the comparison results of SOM-Net(L) and SOM-Net(F) in Fig. 17 and Table VII, it is observed that the SOM-Net with full-aperture data achieves better reconstruction results compared to that of the SOM-Net(L). For example, there is a noticeable deterioration in the image quality for Test#21 in the limited-aperture case. Meanwhile, the SSIM decreases from 0.91 to 0.88, and the RMSE changes from 0.13 to 0.11. Similar quality dropping has also been observed in other examples. This suggests that extra regularization is required to improve the reconstruction of SOM-Net for dealing with such a challenging case, which needs further research but is beyond the scope of the current work.

Finally, we verify the performance of the SOM-Net for the reconstruction of lossy profiles. Accordingly, we still conduct the lossy experiments using the same configurations as the synthetic lossless ones. The only difference is that each scatterer here has an imaginary part randomly defined between 0 and 0.9. In addition, another channel representing the imaginary part of the complex relative permittivity image for lossy profiles also needs to be added to the SOM-Net. The reconstruction results of the real and imaginary parts for some typical samples are shown in Fig. 19 and Fig. 20, respectively. The corresponding quality metrics are listed in Table VIII.

According to the reconstruction results, it can be seen that the average results of 1500 lossy samples in the MNIST data set verify the superior performance of SOM-Net over the compared ones. In particular, as observed in Fig. 19 and Fig. 20, the proposed SOM-Net can still reconstruct challenging lossy profiles in high quality. In Fig. 21, the fitting results of the induced current and all scattered fields for the “Austria” profile are drawn to further demonstrate the effective learning of physical knowledge. Note that in the presentation of concrete results, the first row is the real part (Re) and the second row is the imaginary part (Im) for the induced current and the scattered field.

It should be mentioned that, although we only describe the SOM-Net with a TM case, considering the similarity of the iterative SOM algorithms for the TM [5] and transverse electric (TE) [40] cases, it is also easy to extend the SOM-Net to the TE case. However, the computational complexity will increase quickly compared to the TM case.