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

Yu Liu , Hao Zhao, Rencheng Song , Xudong Chen , Chang Li and Xun Chen This
work was supported in part by the National Natural Science Foundation of China
(Grants 62176081, 61922075, and 41901350), the Provincial Natural Science
Foundation of Anhui (Grant 2108085MF201, 2008085QF285), the Anhui Key Project
of Research and Development Plan (Grants 202104d07020005), and the Fundamental
Research Funds for the Central Universities (Grants PA2021KCPY0051,
JZ2020HGPA0111 and JZ2021HGTB0078). (Corresponding author: Rencheng Song).Yu
Liu, Hao Zhao, Rencheng Song and Chang Li are with the Department of
Biomedical Engineering, and also with the Anhui Province Key Laboratory of
Measuring Theory and Precision Instrument, Hefei University of Technology,
Hefei 230009, China (e-mail<EMAIL_ADDRESS><EMAIL_ADDRESS><EMAIL_ADDRESS>changli@hfut.edu.cn).Xudong Chen is with the Department of
Electrical and Computer Engineering, National University of Singapore, 117583,
Singapore (e-mail: elechenx@nus.edu.sg).Xun Chen is with the Department of
Neurosurgery, The First Affiliated Hospital of USTC, Division of Life Sciences
and Medicine, and also with the Department of Electronic Engineering and
Information Science, University of Science and Technology of China, Hefei,
Anhui, 230001, China (e-mail: xunchen@ustc.edu.cn).

###### Abstract

In this paper, an unrolling algorithm of the iterative subspace-based
optimization method (SOM) is proposed for solving full-wave inverse scattering
problems (ISPs). The unrolling network, named SOM-Net, inherently embeds the
Lippmann-Schwinger physical model into the design of network structures. The
SOM-Net takes the deterministic induced current and the raw permittivity image
obtained from back-propagation (BP) as the input. It then updates the induced
current and the permittivity successively in sub-network blocks of the SOM-Net
by imitating iterations of the SOM. The final output of the SOM-Net is the
full predicted induced current, from which the scattered field and the
permittivity image can also be deduced analytically. The parameters of the
SOM-Net are optimized in a supervised manner with the total loss to
simultaneously ensure the consistency of the induced current, the scattered
field, and the permittivity in the governing equations. Numerical tests on
both synthetic and experimental data verify the superior performance of the
proposed SOM-Net over typical ones. The results on challenging examples like
scatterers with tough profiles or high permittivity demonstrate the good
generalization ability of the SOM-Net. With the use of deep unrolling
technology, this work builds a bridge between traditional iterative methods
and deep learning methods for solving ISPs.

###### Index Terms:

Inverse scattering problem, deep unrolling, subspace-based optimization.

##  I Introduction

Full-wave electromagnetic inverse scattering problems (ISPs) aim at
determining the shape, position, and constitutive parameters of unknown
scatterers from the measured scattering field. It is well known that the full-
wave ISP is usually highly nonlinear and ill-posed [1], which needs to be
solved by nonlinear iterative methods based on governing physical models.
Typical nonlinear iterative ISP methods include the distorted Born iterative
method (DBIM) [2], the contrast source inversion (CSI) method [3], the
contrast source extended Born (CS-EB) method [4], and the subspace-based
optimization method (SOM) [5, 6], etc. Those methods have achieved great
success in reconstructing scatterers in various ISP applications. However, one
main bottleneck of nonlinear iterative methods is the lack of real-time
reconstruction performance due to the expensive computational cost. In
addition, the reconstruction quality is also highly dependent on the
complicated regularization.

In recent years, researchers have successfully introduced deep learning (DL)
technology to solve ISPs [7, 8, 9]. Many studies demonstrate that DL-based
methods outperform conventional iterative ones for both reconstruction speed
and accuracy. Nevertheless, a major concern of DL-based methods is the
generalization ability of the learned models. Namely, the performance of DL-
based methods may degrade significantly, when the testing target is quite
different from those training ones on the aspects such as shape, size, and
parameter ranges. This is because DL models are typically data-driven, and the
constraints of the governing physical laws are only partially satisfied
compared to existing iterative full-wave methods.

Many efforts have been made to improve the model generalization ability of DL-
based inversion methods, which are often referred to as physics-inspired
models. In general, existing physics-inspired methods can be divided into two
types according to the different ways of model learning. The first type of
methods accelerate the conventional iterative methods using the DL technique,
which only adopt neural networks to replace some expensive components in the
traditional objective function method.

For example, Guo et al. [10] introduced a supervised descent method (SDM) to
learn the average descent directions from data. In [11], Guo et al. applied
the SDM to the pixel-based and model-based inversion of microwave data for
dielectric targets. This type of methods have a great advantage on
generalization ability, but the real-time performance is still a challenge.

The second type of physics-inspired methods are composed of two steps, i.e., a
rough permittivity image reconstruction by a fast imaging method, followed by
a permittivity image enhancement to get a high-resolution target [12, 13].
Among these methods, the back-propagation (BP) method is commonly used to
reconstruct the rough permittivity image from the scattered field. Then, the
resolution of rough input image is enhanced in an image-to-image translation
way using various networks, such as the U-Net [14], GANs [15, 16], cGANs [17],
etc. The performance of this kind of methods is dominated by both the quality
of input images and the way of conducting image resolution enhancement. Wei et
al. [18] proposed the dominant current scheme (DCS) instead of BP to generate
a better rough input for the U-Net. Zhang et al. [19] improved the input by
combining the rough images from both qualitative and quantitative methods. Yao
et al. [20] proposed to obtain the initial input image using a shallow
convolutional neural network (CNN) and then enhance the resolution using
U-Net. In addition to these efforts to improve the quality of input images,
Huang et al. [21] adopted the structural similarity index measure (SSIM) loss
and the mean squared error (MSE) loss to jointly constrained image features
and improve the reconstruction quality. Song et al. [22] defined a novel
perceptual adversarial network to improve the quality of image-to-image
translation. Wei et al. [23] introduced an induced current learning method
(ICLM) to solve the translation in the contrast source domain instead of the
contrast domain. Although these methods achieve better reconstructions, the
image-to-image translation in ISP is still lacking of physical insight,
thereby restricting the generalization ability of this kind of DL-based ISP
methods. Specifically, the image-to-image translation in existing ISP methods
only restricts the match of a single physical variable, which is usually the
permittivity. The ignoring of the consistency of other physical parameters may
also degenerate the model generalization ability.

Recently, the unrolling technique [24] has attracted extensive attention in
image processing and computational imaging research, which builds a bridge
between conventional iterative methods and DL-based methods. The deep
unrolling technique constructs neural networks with a few sub-network modules
to imitate iterations of conventional methods. The unknown target parameters
are successively updated in each sub-network module like doing iterations. It
effectively embeds physical knowledge into deep neural networks, which makes
the DL model more interpretable and efficient. The unrolling technique has
been verified to be efficient in solving various tasks like medical imaging
[25, 26, 27], and image restoration [28, 29], etc. It has also been introduced
to solve nonlinear ISPs very recently. Liu et al. [30] introduced the PM-Net
to simulate the ADMM iterative process, which defines four unconstrained sub-
problems to achieve the alternating optimization of contrast source and
contrast. Guo et al. [31] proposed an ISP SOLVER to simulate the iterative
gradient-based inversion method, where a pre-trained forward model is used
repeatedly to calculate the data fitness. The performance of the ISP SOLVER is
strongly affected by that of the pre-trained FWD SOLVER model. These methods
have been verified to obtain good generalization capability compared to direct
data-to-model networks. However, only partial physical variables are
explicitly enforced to satisfy the consistency in these methods. The embedding
of the governing physical law in learned networks can still be further
improved.

In this work, an unrolling of the subspace-based optimization, named SOM-Net,
is proposed to solve nonlinear ISPs. The input of the SOM-Net is taken as the
deterministic induced current obtained by the singular value decomposition
(SVD) and the raw permittivity image obtained from BP, while the output of
SOM-Net is the predicted full induced current. The induced current and the
permittivity image are updated alternatively by passing through the sub-
network blocks in SOM-Net, which is like performing successive iterations in
SOM. Particularly, the update of the induced current in the SOM-Net is
realized by the mapping of each sub-network, while the permittivity is still
updated analytically. Finally, the scattered field and the permittivity are
also deduced analytically from the output induced current. The whole SOM-Net
model is optimized with residuals defined simultaneously on the induced
current, the scattered field, and the permittivity image in a supervised
manner. The match of all these physical parameters which are governed by the
Lippmann-Schwinger equation ensures the generalization ability of the SOM-Net
model. Numerical tests on both synthetic and experimental data verify the
superior performance of the proposed SOM-Net over existing ones.

In summary, the contributions of this work are listed as follows:

  * •

The proposed SOM-Net inherently embeds physical knowledge by unrolling SOM,
which introduces an effective way to design physics-inspired deep learning
networks from the conventional iterative methods. Specifically, the update of
the induced current and the permittivity image within each sub-network of the
SOM-Net effectively reduces the nonlinearity of the mapping. Besides, the
permittivity is always updated analytically in SOM-Net which greatly
simplifies the network structure. Therefore, the SOM-Net is physically
interpretable and efficient.

  * •

The full constraints of physical variables, including the induced current, the
scattered field, and the permittivity image, make the output of the SOM-Net
physically consistent with the governing law. Guided by those constraints, the
prediction of the SOM-Net is more reliable and consistent, in terms of not
only the reconstructed images but also the induced current and the scattered
field.

  * •

The testing results on various challenging examples demonstrate the
effectiveness and good generalization ability of the SOM-Net. In particular,
the comparison results on scatterers with tough profiles or high permittivity
verify the advantages of the proposed method over other comparable ones in
terms of both accuracy and generalization ability.

This paper is organized as follows. In Section II, we introduce the
formulations of the proposed algorithm. Synthetic and experimental data are
then tested in Section III for the performance verification. In Section IV, we
discuss the benefit of SOM-net and its applications in other scenarios.
Finally, we conclude our work in Section V.

Figure 1: The configuration of a 2-D ISP under TM illuminations.

##  II Method

For clarity, the notations are declared first. We use $\bar{\bar{X}}$ and
$\bar{X}$ to denote the matrix and vector of the discretized operator or
parameter $X$, respectively. The superscripts $H$ and $*$ respectively
indicate the conjugate transpose and the complex conjugate of a matrix or
vector.

###  II-A Formulation of the Problem

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.

####  II-A1 Forward Scattering

The forward scattering aims to calculate scattering field with known
permittivity of scatterers. The method of moments (MoM) [33] is taken to
calculate the scattered field. The domain $D$ is discretized into $M$
subunits, and the centers of subunits are located at $r_{m}$ with
$m=1,2,...M$. Accordingly, taking into account of all subunits, the state
equation is discretized as

$\bar{{E}}^{t}=\bar{{E}}^{i}+\bar{\bar{{G}}}_{D}\cdot\bar{J},$ (1)

where $\bar{{E}}^{i}$ and $\bar{{E}}^{t}$ represent the incident and total
electric field in domain $D$, respectively. The matrix $\bar{\bar{{G}}}_{D}$
with $M\times M$ dimensions maps the induced current to the scattered field in
domain $D$. In the forward problem, (1) describes the wave-scattering
interaction in domain $D$.

The induced current $\bar{J}$ is defined as
$\bar{J}=\bar{\bar{{\chi}}}\cdot\bar{{E}}^{t}$. It is well known that the
contrast $\bar{\bar{\chi}}$ and the relative permittivity
$\bar{\bar{\epsilon}}_{r}$ have a relationship of
$\bar{\bar{{\chi}}}=-i\left(k_{0}/\eta_{0}\right)\left(\bar{\bar{\epsilon}}_{r}-1\right)$,
where $i=\sqrt{-1}$, $k_{0}$ and $\eta_{0}$ respectively represent the
wavenumber and the impedance of homogeneous medium background.

The discretized data equation is expressed as

$\bar{{E}}^{s}=\bar{\bar{{G}}}_{S}\cdot\bar{J},$ (2)

where $\bar{{E}}^{s}$ is the scattered field measured by all receivers. The
matrix $\bar{\bar{{G}}}_{S}$ with $N_{r}\times M$ dimensions maps the induced
current from the domain $D$ to the space of scattered field in domain $S$. In
the forward problem, (2) describes the scattered field as a reradiation of the
induced current.

####  II-A2 SOM Algorithm

The purpose of the inverse problem is to retrieve the relative permittivity
$\bar{\bar{\epsilon}}_{r}$ of the scatterer from the measured scattering field
$\bar{{E}}^{s}$. For solving nonlinear ISPs, the induced current cannot be
obtained directly from (2). Suppose the singular value decomposition (SVD) of
$\bar{\bar{{G}}}_{S}$ is
$\bar{\bar{{G}}}_{S}={\textstyle\sum_{n}}\bar{u}_{n}\sigma_{n}\bar{\nu}_{n}^{\small
H}$ with $\sigma_{1}\geq\sigma_{2}\geq,...,\sigma_{M}=0$, where
$\bar{\mu}_{n}$, $\bar{\nu}_{n}$ and $\sigma_{n}$ are the orthogonal left,
right singular vectors, and singular values of $\bar{\bar{{G}}}_{S}$,
respectively. The singular values are sorted in the descending order.

In SOM, the full induced current $\bar{J}$ is decoupled into the deterministic
induced current $\bar{J}^{+}$ and the ambiguous induced current $\bar{J}^{-}$
as

$\bar{J}=\bar{J}^{+}+\bar{J}^{-}=\bar{J}^{+}+\bar{\bar{V}}^{-}\cdot\bar{\alpha}^{-},$
(3)

where $\bar{\bar{V}}^{-}$ is composed of the last $M-L$ columns of right
singular vectors, $\bar{\alpha}^{-}$ is the unknown coefficient vector with
the length of $M-L$, and $\bar{{J}}^{+}$ satisfies

$\bar{{J}}^{+}=\sum_{j=1}^{L}\frac{\bar{\mu}_{j}^{\scriptsize
H}\cdot\bar{E}^{s}}{\sigma_{j}}\bar{\nu}_{j}.$ (4)

The initializations of SOM are taken as the deterministic induced current
$\bar{{J}}^{+}$ and the contrast $\bar{\bar{\chi}}^{bp}$ obtained by back-
propagation in (5),

$\bar{\bar{\chi}}^{bp}(r)=\frac{{\textstyle\sum_{l=1}^{N_{i}}\bar{J}_{l}(r)\cdot\left[\bar{E}_{l}^{t}(r)\right]^{*}}}{{\textstyle\sum_{l=1}^{N_{i}}\left|\bar{E}_{l}^{t}(r)\right|^{2}}}.$
(5)

In SOM, the unknown parameters, $\bar{\alpha}^{-}$ and $\bar{\bar{\chi}}$ are
updated alternatively. For the $m$th subunits, the induced current
$(\bar{J}_{l})_{m}$ is updated by (3) , the total field
$(\bar{E}^{t}_{l})_{m}$ is updated by (1) and the contrast
$(\bar{\bar{\chi}})_{m}$ is straightforwardly updated by (6) [1],

$(\bar{\bar{\chi}})_{m}=\left[\sum_{l=1}^{N_{i}}\frac{(\bar{E}_{l}^{t})_{m}^{*}}{\left\|\bar{{J}}_{l}^{+}\right\|}\cdot\frac{(\bar{J}_{l})_{m}}{\left\|\bar{{J}}_{l}^{+}\right\|}\right]/\left[\sum_{l=1}^{N_{i}}\left|\frac{(\bar{E}_{l}^{t})_{m}}{\left\|\bar{J}_{l}^{+}\right\|}\right|^{2}\right],$
(6)

where ${N_{i}}$ is the total number of transmitters, which is set as 16.

###  II-B The SOM-Net Algorithm

Figure 2: The structure of the proposed SOM-Net. The input of the SOM-Net is a
combined variable with 3 input channels, which contains the deterministic
induced current and the raw permittivity image obtained from BP. The output of
the SOM-Net is the predicted full induced current $\bar{\bar{J}}^{pre}$ with 2
output channels, which is further used to calculate the predicted scattered
field $\bar{E}^{s,pre}$ and the predicted permittivity image
$\bar{\bar{\epsilon}}_{r}^{pre}$ through (2) and (6), respectively. The
superscript $pre$ indicates the variable is obtained from prediction. The
symbol $\oplus$ denotes the concatenation operation for the input channels.
The orange arrows denote the variables are updated analytically.

####  II-B1 Structure of SOM-Net

The flowchart of the proposed SOM-Net is shown in Fig. 2. The SOM-Net is
designed by unrolling the iterations in SOM, which not only incorporates
physical knowledge into the model, but also alleviates the nonlinear coupling
of the induced current and the permittivity.

As seen, there are $K$ sub-networks in SOM-Net to imitate the SOM iterations.
Suppose $\bar{\bar{\epsilon}}_{r}^{bp}$ denotes the permittivity image
reconstructed by the BP method with all scattered field. The
$\bar{\bar{J}}^{+}$ represents the deterministic induced current matrix
reformulated from the vector $\bar{{J}}^{+}$. For the $l$th incidence, the
deterministic induced current $\bar{\bar{J}}^{+}_{l}$ is concatenated with the
permittivity image $\bar{\bar{\epsilon}}_{r}^{bp}$ as a single input sample of
the SOM-Net. Accordingly, the input of the SOM-Net model is denoted as
$\bar{\bar{J}}^{+}\oplus\bar{\bar{\epsilon}}_{r}^{bp}$ with the number of
input channels as 3, in which the first two channels are occupied by the real
and imaginary part of the induced current and the third channel is occupied by
the permittivity image, and the symbol $\oplus$ indicates the concatenation
operation of two variables on the real and imaginary channels. For the same
reconstruction, all the $N_{i}$ resulting samples from the total $N_{i}$
incidences will be simultaneously employed to train the SOM-Net. Namely, the
batch size is set to $N_{i}$.

Similar to SOM, the SOM-Net updates the induced current $\bar{\bar{J}}_{(k)}$
and the permittivity variable $\bar{\bar{\epsilon}}_{r,(k)}$ alternatively by
passing variables across the sub-network $S_{\theta,k},k=1,2,\ldots,K$, where
$\theta$ indicates model parameters. Finally, the predicted induced current
$\bar{\bar{J}}^{pre}$ with the channel number as 2 at all incidence is
directly obtained by network reconstruction. Then, the predicted scattered
field $\bar{E}^{s,pre}$ is calculated analytically by (2) and the permittivity
image $\bar{\bar{\epsilon}}_{r}^{pre}$ can also be reconstructed directly from
the contrast $\bar{\bar{\chi}}^{pre}$ by (6). In SOM-Net, the analytical
update of physical variables is indicated by orange arrows in Fig. 2. It
significantly simplifies the network complexity and embeds physical knowledge
into the model.

Overall, the data flow in SOM-Net is summarized in Algorithm 1, where $F$
represents the analytical update of permittivity from the contrast
$\bar{\bar{\chi}}$ by (6). Specifically, four sub-networks (i.e., $K=4$) are
experimentally taken to design SOM-Net in this paper. It is worth noting that
$K$ can be adjusted according to the complexity of the target ISP.

Algorithm 1 Data Flow in SOM-Net

1: Calculate $\bar{{E}}^{i}$,$\bar{\bar{{G}}}_{D}$,$\bar{\bar{{G}}}_{S}$;

2: Obtain network input $\bar{\bar{J}}_{(0)}\leftarrow\bar{\bar{J}}^{+}$,
$\bar{\bar{\epsilon}}_{r,(0)}\leftarrow\bar{\bar{\epsilon}}_{r}^{bp}$;

3: for each $k\in[2,K]$ do

4:
$\bar{\bar{J}}_{(k-1)}=S_{\theta,k-1}\left(\bar{\bar{J}}_{(k-2)}\oplus\bar{\bar{\epsilon}}_{r,(k-2)}\right)$;

5: $\bar{\bar{\epsilon}}_{r,(k-1)}=F\left(\bar{\bar{J}}_{(k-1)}\right)$;

6: end for

7: $\bar{\bar{J}}^{pre}=\bar{\bar{J}}_{(K)}$;

8: $\bar{E}^{s,pre}=\bar{\bar{{G}}}_{S}\cdot\bar{\bar{J}}^{pre}$;

9: $\bar{\bar{\epsilon}}_{r}^{pre}=\bar{\bar{\epsilon}}_{r,(K)}$;

####  II-B2 Loss Functions

In this paper, a comprehensive loss function is defined to train the SOM-Net
$S_{\theta}$. The full loss function $L_{S}(\theta)$ of SOM-Net is defined as

$L_{S}(\theta)=L_{J}(\theta)+L_{E}(\theta)+\lambda_{1}L_{SSIM}(\theta)+\lambda_{2}L_{MSE}(\theta),$
(7)

where $L_{J}(\theta)$ denotes the induced current loss, $L_{E}(\theta)$ is the
scattered field loss, $L_{SSIM}(\theta)$ indicates the structural similarity
loss of two permittivity images, and $L_{MSE}(\theta)$ is the pixel-wise MSE
loss. Here, $\lambda_{i}(i=1,2)$ are hyperparameters.

Specifically, each term of $L_{S}(\theta)$ is defined as

$L_{J}=\frac{1}{N_{i}}\sum_{l=1}^{N_{i}}\left|\bar{\bar{J}}^{pre}_{l}-\bar{\bar{J}}^{\mathrm{MoM}}_{l}\right|^{2},$
(8)
$L_{E}=\frac{1}{N_{r}}\sum_{q=1}^{N_{r}}\left|\bar{E}^{s,pre}_{q}-\bar{E}^{s}_{q}\right|,$
(9)
$L_{SSIM}=1-\mathrm{SSIM}(\bar{\bar{\epsilon}}_{r}^{pre},\bar{\bar{\epsilon}}_{r}^{t}),$
(10)
$L_{MSE}=\left|\bar{\bar{\epsilon}}_{r}^{pre}-\bar{\bar{\epsilon}}_{r}^{t}\right|^{2},$
(11)

where $\bar{\bar{J}}^{pre}$, $\bar{E}^{s,pre}$ and
$\bar{\bar{\epsilon}}_{r}^{pre}$ are the predicted induced current, the
predicted scattered field and the permittivity image, respectively, while
$\bar{\bar{J}}^{\mathrm{MoM}}$, $\bar{E}^{s}$ and
$\bar{\bar{\epsilon}}_{r}^{t}$ are the corresponding reference variables,
respectively.

Figure 3: Reconstruction results of Test#1 to Test#4 from the MNIST data set with 10% white Gaussian noise. TABLE I: Error metrics of reconstruction results for 1500 testing samples from MNIST data set by SOM, U-Net, and SOM-Net in Fig.3. Method | Test#1 | Test#2 | Test#3 | Test#4 | 1500 MNIST
---|---|---|---|---|---
SSIM | RMSE | SSIM | RMSE | SSIM | RMSE | SSIM | RMSE | SSIM | RMSE
SOM | 0.84 | 0.09 | 0.84 | 0.11 | 0.77 | 0.18 | 0.83 | 0.12 | 0.80 | 0.13
U-Net | 0.89 | 0.09 | 0.88 | 0.11 | 0.84 | 0.20 | 0.87 | 0.10 | 0.88 | 0.13
SOM-Net | 0.92 | 0.07 | 0.94 | 0.07 | 0.88 | 0.14 | 0.95 | 0.07 | 0.91 | 0.10

It can be seen that the total loss $L_{S}(\theta)$ enforces the consistency of
the induced current, the scattered fiel, and the permittivity. The use of SSIM
and MSE constraints on the permittivity image further ensures the structural
and pixel-wise match with the reference one. Hence, from the physical point of
view, the total loss $L_{S}(\theta)$ guarantees the consistency of all
physical variables in the Lippmann-Schwinger governing equation, thereby
improving the prediction accuracy of the SOM-Net model. The proper design of
network and loss functions enables the model to learn physical laws from the
data, thereby making the model have good generalization ability.

Figure 4: The fitting of induced currents by the first transmitter for Test#1
to Test#4: (a) Test#1, (b) Test#2, (c) Test#3, (d) Test#4. For each test, the
left, middle, and right columns are the deterministic induced current, the
output induced current of SOM-Net, and the target induced current,
respectively. And the first row is the real part (Re), and the second row is
the imaginary part (Im).
Figure 5: The fitting of all scattered fields for Test#1 to Test#4: (a)
Test#1, (b) Test#2, (c) Test#3, (d) Test#4. For each test, the predicted
scattered field by SOM-Net is represented by the orange dots, and the
reference one is indicated by the blue line.

###  II-C Computational Complexity

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 $O\left(M\right)$.

Figure 6: Reconstruction results of Test#5 to Test#12 for scatterers with complex profiles.  TABLE II: Error metrics of reconstruction results for Test#5-Test#12. Method | Test#5 | Test#6 | Test#7 | Test#8 | Test#9 | Test#10 | Test#11 | Test#12
---|---|---|---|---|---|---|---|---
SSIM | RMSE | SSIM | RMSE | SSIM | RMSE | SSIM | RMSE | SSIM | RMSE | SSIM | RMSE | SSIM | RMSE | SSIM | RMSE
SOM | 0.77 | 0.07 | 0.84 | 0.07 | 0.80 | 0.13 | 0.78 | 0.13 | 0.91 | 0.07 | 0.83 | 0.10 | 0.82 | 0.11 | 0.85 | 0.12
U-Net | 0.84 | 0.09 | 0.85 | 0.07 | 0.76 | 0.20 | 0.69 | 0.17 | 0.94 | 0.08 | 0.92 | 0.10 | 0.84 | 0.17 | 0.90 | 0.12
SOM-Net | 0.93 | 0.05 | 0.91 | 0.05 | 0.83 | 0.12 | 0.86 | 0.10 | 0.97 | 0.04 | 0.94 | 0.08 | 0.92 | 0.09 | 0.95 | 0.07
TABLE III: Error metrics of reconstruction results for Test#13-Test#16. Method | Test#13 | Test#14 | Test#15 | Test#16
---|---|---|---|---
SSIM | RMSE | SSIM | RMSE | SSIM | RMSE | SSIM | RMSE
SOM | 0.69 | 0.15 | 0.68 | 0.15 | 0.67 | 0.16 | 0.66 | 0.16
U-Net | 0.80 | 0.16 | 0.77 | 0.17 | 0.76 | 0.18 | 0.77 | 0.19
SOM-Net | 0.90 | 0.11 | 0.88 | 0.11 | 0.88 | 0.12 | 0.86 | 0.13

##  III Numerical and experimental results

In this section, we conduct verifications of SOM-Net with both synthetic and
experimental examples. The performance of the SOM-Net is compared with the
classical SOM method and U-Net [18]. The comparison with SOM indicates the
benefit of the improvement on reconstruction quality and speed of SOM-Net,
while the comparison with U-Net denotes the superior performance of SOM-Net
over existing physics-inspired deep learning methods.

###  III-A Configuration of the Scattering System

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.

###  III-B Training Details

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.

Figure 7: Reconstruction results of the “Austria” profile with the relative
permittivity of 2.0 under different noise levels. From Test#13 to Test#16,
10%, 20%, 25% and 30% white Gaussian noise are added to the scattered field,
respectively.

###  III-C Within-Database Test: Synthetic Data

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.

Figure 8: The fitting of induced currents by the first transmitter for Test#15
and Test#16: (a) Test#15, (b) Test#16. For each test, the left, middle, and
right columns are the deterministic induced current, the output induced
current of SOM-Net, and the target induced current, respectively. And the
first row is the real part (Re), and the second row is the imaginary part Im).
Figure 9: The fitting of all scattered fields for Test#15 and Test#16: (a)
Test#15, (b) Test#16. For each test, the predicted scattered field by SOM-Net
is represented by the orange dots, and the reference one is indicated by the
blue line.

###  III-D Cross-Database Test: Synthetic Data

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.

Figure 10: Reconstruction results of the star profile with the relative permittivity of 1.5, 2.0, 2.5, and 3.0, respectively.  TABLE IV: Error metrics of reconstruction results for Test#17-Test#20. Method | Test#17 | Test#18 | Test#19 | Test#20
---|---|---|---|---
SSIM | RMSE | SSIM | RMSE | SSIM | RMSE | SSIM | RMSE
SOM | 0.92 | 0.05 | 0.91 | 0.09 | 0.92 | 0.11 | 0.78 | 0.18
U-Net | 0.93 | 0.06 | 0.94 | 0.08 | 0.83 | 0.23 | 0.83 | 0.24
SOM-Net | 0.94 | 0.05 | 0.94 | 0.08 | 0.93 | 0.11 | 0.92 | 0.16

####  III-D1 Complex Profiles

We test Test#5-Test#12 where scatterers have complex shapes different from the
MNIST training data. All the scattered field are contaminated with $10\%$
white Gaussian noise. The reconstruction results of Test#5-Test#12 are shown
in Fig. 6, where the quality metrics are summarized in Table II. The results
shows that the SOM-Net outperforms the comparison methods. Particularly, in
Test#11 and Test#12, although there are three different scatterers and
scatterers have diverse permittivity values, the SOM-Net still accurately
reconstructs all examples. The results effectively verify the good
generalization ability of the SOM-Net, which is attributed to the unrolling
model design and the physical loss constraints for model training.

####  III-D2 High-level Noise Interference

We also test the noise robustness of the SOM-Net model under different noise
levels. In this testing, we choose the “Austria” profile as the scatterer with
the $\varepsilon_{r}$ as 2.0. And $10\%,20\%,25\%,$ and $30\%$ white Gaussian
noise are respectively added to the scattered field. The reconstruction
results of Test#13-Test#16 with different noise levels are shown in Fig. 7.
The corresponding error metrics are summarized in Table III. It demonstrates
that the SOM-Net is still robust against noise, which achieves better
reconstruction results compared to the other methods. It is worth noting that
compared with the “Austria” test in existing unrolling ISP methods [30] and
[31], the SOM-Net obtains better imaging results, especially showing clear
gaps between two small rings and large rings. It can be considered that the
high-quality restoration of induced current plays an important role in the
reconstruction of permittivity images.

The fitting of induced current and the scattered field are shown in Fig. 8 and
Fig. 9, respectively. The induced current is still from the first transmitter
and the one for other transmitters are similar. Fig. 8 and Fig. 9 show that
the SOM-Net can accurately predict the induced current and the scattered field
for “Austria” profile with 25% and 30% high-level noise. These results verify
the noise robustness of the SOM-Net.

####  III-D3 Scatterers With Different Permittivities

To further check the generalization ability of the SOM-Net, we also test the
reconstruction performance of a star profile with varying permittivities. The
results of Test#17-Test#20 are shown in Fig. 10, where the relative
permittivity ranges from 1.5 to 3.0. And $10\%$ white Gaussian noise is added
to all testing cases. The corresponding error metrics are summarized in Table
IV. It is worth noting that the SOM-Net has shown superior performance even
for the case of the relative permittivity of 3.0, where the shape and the
acute angles of the scatterer can still be recovered.

Although the SOM-Net successfully retrieves the high-permittivity scatterer,
it is still very challenging to reconstruct scatterers with a quite high
permittivity. One potential way to overcome this issue is to unroll the
existing model that can handle the ISP with high permittivity. For example,
the work in [36] introduces an approximate model for solving high-permittivity
ISP, and it is meaningful to unroll such iterative methods into neural
networks to retrieves high-permittivity scatterers.

Figure 11: The “FoamDielExt” profile from Fresnel experimental data.
Figure 12: Reconstruction results of the “FoamDielExt” profile at 3GHz and 4GHz, respectively. TABLE V: Error metrics of reconstruction results in Fig.12 for experimental data at 3GHz and 4GHz, respectively. f/GHz | 3.0 | 4.0
---|---|---
Method | SSIM | RMSE | SSIM | RMSE
SOM | 0.91 | 0.09 | 0.88 | 0.12
U-Net | 0.91 | 0.07 | 0.89 | 0.15
SOM-Net | 0.94 | 0.07 | 0.92 | 0.13

###  III-E Cross-Database Test: Experimental Data

We finally validate the proposed method with the experimental data provided by
Fresnel Institute [37]. 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.

Figure 13: The fitting of induced currents by the first transmitter for
“FoamDielExt” profile at 3 GHz and 4 GHz, respectively: (a) 3 GHz, (b) 4 GHz.
For each test, the left, middle, and right columns are the deterministic
induced current, the output induced current of SOM-Net, and the target induced
current, respectively. And the first row is the real part (Re), and the second
row is the imaginary part (Im).
Figure 14: The fitting of all scattered fields for the “FoamDielExt” profile
at 3GHz and 4GHz, respectively: (a) 3 GHz, (b) 4 GHz. For each test, the
predicted scattered field by SOM-Net is represented by the orange dots every 5
points, and the reference one is indicated by the blue line.

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.

Figure 15:  The Fresnel experimental data: (a) the “FoamDielInt” profile and
(b) the “FoamTwinDiel” profile.
Figure 16: Reconstruction results of the experimental data: (a) the “FoamDielExt” profile, (b) the “FoamDielInt” profile, and (c) the “FoamTwinDiel” profile at 4GHz, respectively. TABLE VI: Error metrics of reconstruction results of SOM-Net in Fig. 16 for experimental data at 4GHz. Test | SSIM | RMSE
---|---|---
FoamDielExt | 0.95 | 0.09
FoamDielInt | 0.95 | 0.08
FoamTwinDiel | 0.90 | 0.16

##  IV Discussion

In this section, we further discuss some aspects of the SOM-Net. It includes
the comparison of SOM-Net with existing unrolling methods [30, 31], and the
extension of SOM-Net for other cases like limited aperture and lossy
scatterers.

###  IV-A Comparison with Existing Unrolling Methods

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.

Figure 17: Reconstruction results of Test#21 to Test#28 for scatterers with 10% white Gaussian noise. TABLE VII: Error metrics of reconstruction results for Test#21-Test#28 and 1500 MNIST tests. Method | Test#21 | Test#22 | Tset#23 | Test#24 | Test#25 | Test#26 | Test#27 | Test#28 | 1500 MNIST
---|---|---|---|---|---|---|---|---|---
SSIM | RMSE | SSIM | RMSE | SSIM | RMSE | SSIM | RMSE | SSIM | RMSE | SSIM | RMSE | SSIM | RMSE | SSIM | RMSE | SSIM | RMSE
SOM(L) | 0.69 | 0.15 | 0.91 | 0.07 | 0.83 | 0.10 | 0.81 | 0.12 | 0.78 | 0.16 | 0.82 | 0.11 | 0.92 | 0.11 | 0.76 | 0.18 | 0.78 | 0.14
U-Net(L) | 0.79 | 0.15 | 0.93 | 0.07 | 0.91 | 0.10 | 0.88 | 0.10 | 0.84 | 0.18 | 0.82 | 0.15 | 0.89 | 0.15 | 0.85 | 0.24 | 0.86 | 0.13
SOM-Net(L) | 0.86 | 0.13 | 0.96 | 0.04 | 0.93 | 0.08 | 0.91 | 0.09 | 0.87 | 0.13 | 0.91 | 0.15 | 0.92 | 0.11 | 0.90 | 0.15 | 0.88 | 0.12
SOM-Net(F) | 0.90 | 0.11 | 0.97 | 0.04 | 0.94 | 0.08 | 0.95 | 0.07 | 0.91 | 0.11 | 0.92 | 0.09 | 0.93 | 0.11 | 0.92 | 0.16 | 0.91 | 0.10

Figure 18: Fitting results of induced currents and scattered field in Test#21:
(a) the fitting of induced currents by the first transmitter, where the left,
middle, and right columns are the input, the output, and the target induced
current, respectively. (b) the fitting of all scattered fields, where the
predicted scattered field by SOM-Net is represented by the orange dots, and
the reference one is indicated by the blue line.

###  IV-B Limited Aperture

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. 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.

###  IV-C Lossy Profiles

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) [38] 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.

Figure 19: Reconstruction results of the real parts for lossy scatterers in
Test#29 to Test#36 with 10% white Gaussian noise.
Figure 20:  Reconstruction results of the imaginary parts for lossy scatterers in Test#29 to Test#36 with 10% white Gaussian noise. TABLE VIII: Error metrics of reconstruction results for Test#29-Test#36 and 1500 MNIST tests. Test | Test#29 | Test#30 | Tset#31 | Tset#32 | Tset#33 | Tset#34 | Tset#35 | Tset#36 | 1500 MNIST
---|---|---|---|---|---|---|---|---|---
SSIM | Re | Im | Re | Im | Re | Im | Re | Im | Re | Im | Re | Im | Re | Im | Re | Im | Re | Im
SOM | 0.64 | 0.13 | 0.67 | 0.21 | 0.68 | 0.29 | 0.77 | 0.19 | 0.76 | 0.23 | 0.74 | 0.23 | 0.86 | 0.21 | 0.61 | 0.10 | 0.69 | 0.12
U-Net | 0.87 | 0.75 | 0.89 | 0.81 | 0.75 | 0.67 | 0.95 | 0.90 | 0.94 | 0.89 | 0.89 | 0.82 | 0.92 | 0.79 | 0.87 | 0.79 | 0.89 | 0.59
SOM-Net | 0.87 | 0.84 | 0.92 | 0.86 | 0.89 | 0.80 | 0.96 | 0.94 | 0.95 | 0.92 | 0.94 | 0.91 | 0.91 | 0.88 | 0.91 | 0.88 | 0.90 | 0.70

Figure 21: Fitting results of induced currents and scattered field in Test#29:
(a) the fitting of induced currents by the first transmitter, where the left,
middle, and right columns are the input, the output, and the target induced
current, respectively. (b) the fitting of all scattered fields, where the
predicted scattered field by SOM-Net is represented by the orange dots, and
the reference one is indicated by the blue line.

##  V Conclusion

In this work, we have proposed the SOM-Net for solving the full-wave ISPs by
unrolling the iterative SOM. The SOM-Net has been designed with several sub-
networks to imitate the iterations of the SOM. The deterministic induced
current and the rough permittivity image got by BP are used as the input,
while the direct output of the SOM-Net is the full induced current. The
induced current and the permittivity variables have been updated alternately
in the cascade sub-networks of SOM-Net, which greatly reduces the nonlinearity
of the ISP. Joint physical constraints of the induced current, the scattered
field, and the permittivity image have been employed to guide the training of
the SOM-Net model. The generalization ability of the resulting SOM-Net model
has been greatly enhanced, considering the inherent embedding of physical law
on the network structure and the consistent matches of all physical variables
of the governing Lippmann-Schwinger equations. All numerical tests have
validated that the proposed SOM-Net has superior performance than SOM and
U-Net. Overall, the SOM-Net has been verified to be a learning-based fast
inversion method with strong generalization ability and good physical
interpretability, which provides a new idea for the application of deep
unrolling technology to solve ISPs.

Despite the good results achieved, the proposed SOM-Net still needs to be
further improved and extended in many aspects, such as the inversion under a
limited aperture, and the extension to the 3D case.

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