# Through-the-Wall Microwave Imaging: Forward and Inverse Scattering Modeling

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## Abstract

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## 1. Introduction

## 2. Theoretical Approach to the Through-Wall Imaging Problem

#### 2.1. Forward-Scattering Problem Formulation

#### 2.2. Inverse-Scattering Problem Formulation

- Set the outer iteration index to $n=0$ and initialize the contrast function at the first outer step with ${c}_{0}=0$.
- Linearize the scattering problem by computing the Fréchet derivative ${\mathcal{T}}_{n}^{\prime}$ of the operator $\mathcal{T}$ around the current solution ${c}_{n}$. A linear problem ${\mathcal{T}}_{n}^{\prime}{\xi}_{n}\left(x,y\right)={E}_{scatt}\left(x,y\right)-\mathcal{T}\left({c}_{n}\right)\left(x,y\right)$ is then obtained. It is worth remarking that, similarly to the corresponding procedures in free space [31,33], the computation of the right-hand side of the linear problem and of the Fréchet derivative ${\mathcal{T}}_{n}^{\prime}$ requires the solution of a set of forward problems. To this end, a forward solver based on the MoM is adopted.
- Solve the obtained linear problem in a regularized sense by means of the Lebesgue-space procedure detailed in [31,33]. Specifically, the solution of the linear problem obtained in step 2, i.e., ${\xi}_{n}$, is computed by means of the following Landweber-type iterations:$${\xi}_{n,l+1}={\mathcal{J}}_{q}\left({\mathcal{J}}_{p}\left({\xi}_{n,l}\right)-\beta {\mathcal{T}}_{n}^{\u2019*}{\mathcal{J}}_{p}\left({\mathcal{T}}_{n}^{\u2019}{\xi}_{n,l}-{E}_{scatt}\left(x,y\right)+\mathcal{T}\left({c}_{n}\right)\right)\right)$$
- Update the contrast function by adding the solution of the linear problem ${\xi}_{n}$ found at step 3 to the current value, i.e., ${c}_{n+1}={c}_{n}+{\xi}_{n}$
- Iterate from step 2 until a proper stopping criterion is satisfied.

## 3. Numerical Results

#### 3.1. Validation of the Forward Methods

#### 3.2. Inversion Scheme

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

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**Figure 2.**Actual configuration. (

**a**) Single dielectric cylinder and (

**b**) two separate dielectric cylinders.

**Figure 3.**(

**a**) Amplitude and (

**b**) phase of the scattered fields in some of the considered view computed by the analytical forward solver based on the cylindrical wave approach (CWA) and by the integral equation formulation used in the inverse scattering procedure. Single dielectric cylinder.

**Figure 4.**(

**a**) Amplitude and (

**b**) phase of the scattered fields in some of the considered view computed by the analytical forward solver based on the CWA and by the integral equation formulation used in the inverse scattering procedure. Two separate dielectric cylinders.

**Figure 5.**Reconstructed distribution of the relative dielectric permittivity inside the through-wall (TW) investigation domain. Single dielectric cylinder. (

**a**) Optimal value of the norm parameter (${p}_{opt}=1.3$) and (

**b**) standard Hilbert-space approach ($p=2$ ).

**Figure 6.**Reconstructed distribution of the relative dielectric permittivity inside the TW investigation domain. Two dielectric cylinders. (

**a**) Optimal value of the norm parameter (${p}_{opt}=1.3$) and (

**b**) standard Hilbert-space approach ($p=2$ ).

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**MDPI and ACS Style**

Fedeli, A.; Pastorino, M.; Ponti, C.; Randazzo, A.; Schettini, G. Through-the-Wall Microwave Imaging: Forward and Inverse Scattering Modeling. *Sensors* **2020**, *20*, 2865.
https://doi.org/10.3390/s20102865

**AMA Style**

Fedeli A, Pastorino M, Ponti C, Randazzo A, Schettini G. Through-the-Wall Microwave Imaging: Forward and Inverse Scattering Modeling. *Sensors*. 2020; 20(10):2865.
https://doi.org/10.3390/s20102865

**Chicago/Turabian Style**

Fedeli, Alessandro, Matteo Pastorino, Cristina Ponti, Andrea Randazzo, and Giuseppe Schettini. 2020. "Through-the-Wall Microwave Imaging: Forward and Inverse Scattering Modeling" *Sensors* 20, no. 10: 2865.
https://doi.org/10.3390/s20102865