Overcoming Insufficient Microwave Scattering Data in Microwave Tomographic Imaging

One of the most challenging problems in microwave tomography is the reconstruction of dielectric properties inside parts of the human body. This is a crucial task for medical applications such as early cancer diagnostics and focused microwave thermotherapy. If not enough antennas are used, the amount of microwave scattering measurement data will be insufficient for satisfactory imaging with the conventional approach. This paper suggests a method for overcoming insufficient microwave scattering data using simultaneous reconstruction of the body shape and its interior dielectric properties, given the restriction that the shape must be regular and smooth. This restriction allows for reduced uncertainty in the reconstructed images. We developed a 2.5D electromagnetic solver and an iterative method for estimating body shape in the process of image reconstruction. This approach provides successful imaging even with insufficient scattering data. In addition, the simultaneous reconstruction leads to improved accuracy in both restorations.

SECTION I. Introduction

Microwave tomography (MT) for medical applications has attracted substantial interest over the past few decades [1]–​[3]. This interest arises primarily from the fact that the dielectric properties of abnormal tissue, e.g., cancer, differ significantly from those of normal tissue at microwave frequencies, and the microwave signals are non-ionizing, unlike x-ray computed tomography (CT), mammography, and positron emission tomography (PET). To date, the feasibility studies for medical applications have mostly been conducted using numerical simulations. The ultimate objective of any MT system is to serve as a cost-effective alternative to the existing imaging modalities for use in clinical settings. To meet this objective, an MT system should be developed beyond theory and simulations into a working prototype that can be used in clinical trials. However, there are many technical challenges that need to be solved before they can be used in clinical trials and commercial products.

One of the most challenging problems is the reconstruction of dielectric properties inside the human body, which is a non-linear electromagnetic inverse problem. The problem is inherently ill posed and nonlinear. Therefore, an iterative reconstruction algorithm is used, which involves a cost function that is either maximized or minimized. However, the nonlinearity of the problem often causes the algorithm to become trapped in a local minimum, thus leading to an incorrect reconstruction [4]. To address the non-uniqueness and the ill-posedness of nonlinear inverse problems to some extent, a large number of antennas must be used to collect a large microwave scattering data set, from which the dielectric properties are retrieved [5]. However, the number of antennas that can be used is limited by their finite size. Moreover, placing the antennas too close to each other may result in high mutual coupling between the antennas, thus introducing error in the measured scattering data.

To overcome this problem, it can be beneficial to use prior information about the object being imaged [6]–​[15]. The problem of getting trapped in a local minimum can often be avoided by a clever initial guess or by starting the reconstruction from an ideal model of the targets under reconstruction. For example, [6] takes prior dielectric data into account when reconstructing small objects. References [7] and [8] incorporate prior information about the boundaries between different tissues derived from magnetic resonance imaging (MRI). Recently, structural information derived from ultrasound rather than expensive MRI has been used as prior information for quantitative microwave imaging [9]–​[11]. As another approach, there are methods for direct shape reconstruction from microwave scattering data, such as linear sampling methods and level set methods [12], [13]. Although level set-based methods are relatively advanced compared to others, they require many iterations and take a very long time. Therefore, [14] proposed a fast variant of the level set method. In [15], a method for estimating the body’s boundary based on the antenna’s resonant frequency shift is explained.

In human body imaging applications, pure water is the safest bath liquid, and it is often used for cooling and matching with body tissue. However, the high dielectric constant of the water leads to small wavelengths relative to the size of the body part of interest at frequencies close to 1 GHz or higher. This necessitates the use of a relatively large number of receiving antennas in a circular array. The MT system we are interested in includes a cylindrical array with 16 Tx/Rx (transmitting/receiving) antennas (see Fig. 1). The system operates at a frequency of about 900 MHz, and it uses pure water as a matching and cooling liquid to fill a cylindrical tank (bath). The wavelength in water at this frequency is relatively small (about 40 mm), and it is necessary to use more than 20 antennas for successful imaging, as we will show in Sections II and III. Therefore, 16 antennas in the array are not enough to collect microwave scattering data. Another problem in MT is the very time-consuming nature of image reconstruction when using standard 3D EM forward solvers [16]. On the other hand, the application of common 2D EM solvers does not achieve high enough accuracy to provide acceptable image quality. To overcome these difficulties, we developed a new method for achieving acceptable image reconstruction despite these difficult circumstances.

FIGURE 1.

Schematic representation of our microwave tomographic imaging system. 16 antennas are arranged along a circular cavity wall filled with water. The object to be imaged is placed in the bath.

In this paper, we suggest the following approaches to overcome the indicated difficulties:

1. Simultaneous reconstruction of shape and dielectric contrast with the restriction that the shape must be regular and smooth;

2. Consideration of an approximated fast 2.5D forward EM solver that considers only z-component electric fields instead of a rigorous 3D full-wave solver.

The first requirement means the use of additional prior information, which helps overcome the lack of antennas in the array. We apply a Gauss-Newton iteration method for body shape reconstruction, which is performed along with dielectric property image reconstruction. The restriction on the imaged shapes to be regular and smooth helps obtain satisfactory image quality even when there is insufficient microwave scattering data. In addition, simultaneous shape and dielectric image reconstruction improves the accuracy of both restorations.

SECTION II. Conventional Imaging Approach

In our MT system, as shown in Fig.1, the cylindrical body is immersed in a bath of matching liquid (pure water), and it is investigated by a cylindrical array of Tx/Rx waveguide-type antennas filled with a low loss dielectric. This system was modeled using the commercial 3D electromagnetic (EM) simulation software, CST Studio Suite, to gather MT signals. However, it is not practical to apply CST as a forward EM solver for image reconstruction because it requires time-consuming EM calculations. Therefore, it is necessary to apply EM solvers that use some kind of reasonable approximation [16].

The cylindrical symmetry of our MT system prompts us to use a 2D approximation for the forward EM solver. However, waveguide-type antennas naturally lead to 3D consideration of their EM field pattern. Consequently, in this work, we use a compromise solution that combines 2D and 3D approaches.

A. Calculation of Total Electric Field

Let us first consider a simplified model of the 2D scattering problem for transverse magnetic (TM) waves [17]. The circular array contains 16 line-current sources, which are oriented parallel to the z-axis. The MT system tests a body phantom, which is immersed in water with a complex permittivity $\varepsilon _{b}$ and wavenumber $k_{b}$ .

The 2D forward EM solver simulates both the total electric field $E_{z}^{tot}$ in the background and the phantom using a 2D variant of the volume integral equation (VIE) [17], [18]: $$\begin{equation*} \boldsymbol {E}_{z}^{tot}= \boldsymbol {E}_{z}^{inc}+k_{b}^{2}\int _{S} { \boldsymbol {G}_{0}\left ({\boldsymbol {\rho }- \boldsymbol {\rho }\mathrm {'} }\right)C(\boldsymbol {\rho }\mathrm {'})\cdot \boldsymbol {E}_{z}^{tot}\left ({\boldsymbol {\rho }\mathrm {'} }\right)} ds\mathrm {'}\tag{1}\end{equation*}$$ employing the 2D Green’s function $$\begin{equation*} G_{0}\left ({\boldsymbol {\rho }- \boldsymbol {\rho }' }\right)=\frac {j}{4}H_{0}^{\left ({2 }\right)}\left ({k_{b}\left |{ \boldsymbol {\rho }- \boldsymbol {\rho }' }\right | }\right)\tag{2}\end{equation*}$$

It should be noted that the same Green’s function can describe an incident electric field $E_{z}^{inc}$ , transmitted by line-current sources $$\begin{equation*} E_{z}^{inc}\left ({\boldsymbol {\rho }_{Tx}\vert \boldsymbol {\rho } }\right)=G_{0}\left ({\boldsymbol {\rho }_{Tx}- \boldsymbol {\rho } }\right)\tag{3}\end{equation*}$$

After the discretization of (1) with the grid period $\Delta$ , we can obtain the matrix form of 2D VIE [16]–​[18]: $$\begin{equation*} \boldsymbol {E}_{z}^{tot}= \boldsymbol {E}_{z}^{inc}-\left ({k_{b}^{2}\Delta ^{2} }\right)\cdot \boldsymbol {G}_{0}\cdot \boldsymbol {C}\cdot \boldsymbol {E}_{z}^{tot}\tag{4}\end{equation*}$$

Here, $\boldsymbol {G}_{0}$ is a matrix of the Green’s function, $\boldsymbol {C}\equiv diag\left ({\vec { \boldsymbol {C}} }\right)$ , and vector $\vec { \boldsymbol {C}}$ corresponds to the contrast $C\left ({\boldsymbol {\rho } }\right)\equiv \varepsilon \left ({\boldsymbol {\rho } }\right)/\varepsilon _{b}-1$ in the grid nodes.

The mutual coupling between cells (pixels) of the grid with contrast $\vec { \boldsymbol {C}}$ is determined by the following solution of (4) in the imaging zone (IZ): $$\begin{equation*} \boldsymbol {E}_{z}^{tot}=\left ({\boldsymbol {I}+k_{b}^{2}\Delta ^{2} \boldsymbol {G}_{0}\cdot \boldsymbol {C} }\right)^{-1}\cdot \boldsymbol {E}_{z}^{inc}\equiv \boldsymbol {IGC}\cdot \boldsymbol {E}_{z}^{inc}\tag{5}\end{equation*}$$ where $\boldsymbol {I}$ is an identity matrix. Hence, equations (2)–​(5) allow us to calculate the transmitted electric fields $\boldsymbol {E}_{z}^{inc}$ and the total received signals $\boldsymbol {E}_{z}^{tot}$ .

To eliminate the problem of the singularity and improve the precision of the $\boldsymbol {IGC}$ calculation in (5), we approximate the Green’s function $G_{0}\left ({\boldsymbol {\rho } }\right)$ by using its average value $\hat {G}_{0}\left ({\boldsymbol {\rho } }\right)$ over the area $S_{m}$ of a mesh cell (pixel) following [19]: $$\begin{equation*} \hat {G}_{0}\left ({\boldsymbol {\rho }- \boldsymbol {\rho }_{m} }\right)\equiv \frac {1}{\Delta ^{2}}\int _{S_{m}} {G_{0}\left ({\boldsymbol {\rho }- \boldsymbol {\rho }' }\right)} ds\mathrm {'}\tag{6}\end{equation*}$$

Note that the transformation in (6) is especially perceptible for small distances between cells and that the differences between $\hat {G}_{0}\left ({\boldsymbol {\rho } }\right)$ and $G_{0}\left ({\boldsymbol {\rho } }\right)$ tend to be zero for large relative distances $k_{b}\left |{ \boldsymbol {\rho }_{m}- \boldsymbol {\rho }_{n} }\right |$ . Integration (6) has the following approximation for function $\hat {G}_{0}\left ({\boldsymbol {\rho } }\right)$ that follows from (13) in [19]: $$\begin{align*}&\hspace {-2pc}\hat {G}_{0}\left ({\boldsymbol {\rho }_{m}- \boldsymbol {\rho }_{n} }\right) \\\cong&\begin{cases} \displaystyle \frac {j}{\mathrm {2}k_{b}^{2}\mathrm {}\Delta ^{2}}\left [{ \pi k_{b}\mathrm {}a\mathrm { }H_{1}^{\left ({2 }\right)}\left ({k_{b}\mathrm {}a }\right)- 2j }\right], & m=n\\[8pt] \displaystyle \frac {j\mathrm {}\pi \mathrm {}a}{2k_{b}\mathrm {}\Delta ^{2}}J_{1}\left ({k_{b}\mathrm {}a }\right)H_{0}^{\left ({2 }\right)}\left ({k_{b}\left |{ \boldsymbol {\rho }_{m}- \boldsymbol {\rho }_{n} }\right | }\right), & m\ne n\\ \displaystyle \end{cases}\tag{7}\end{align*}$$ where $a$ is the equivalent radius of the fine mesh cell (pixel): $$\begin{equation*} \pi a^{2}=\Delta ^{2}\tag{8}\end{equation*}$$

B. 2.5D Forward Solver

As long as we use the array of waveguide-type antennas, we must apply a finer EM model for the image reconstruction program, which considers its field pattern. We developed a 2.5D EM solver that can calculate signals that are better matched to the data simulated in CST. For the 2.5D approximation, we considered it to be that case that there existed only z-components of the incident and scattered electric fields. To implement the 2.5D model, we utilized the EM field pattern of the transmitting antenna, which is also calculated in CST (see Fig. 2).

FIGURE 2.

Distribution of Ez field in vertical plane at 850 MHz. The electric field is normalized by the Ez field at the origin of the plane z = 0. (a) Empty bath with pure water. (b) The bath with cylindrical phantom (Fig. 3) submerged into the water.

Let us use cylindrical coordinates $\left ({\boldsymbol {\rho },z }\right)$ with the origin at the antenna’s aperture center; then, we express the fields, $\boldsymbol {E}_{Rx}^{inc}\left ({\boldsymbol {\rho },z }\right)$ , incident at the Rx antenna, so the total, $\boldsymbol {E}_{Tx}^{tot}\left ({\boldsymbol {\rho },z }\right)$ , for Tx antenna takes the following form: $$\begin{align*} \boldsymbol {E}_{Rx}^{inc}\left ({\boldsymbol {r} }\right)\cong&\hat { \boldsymbol {z}}\,\mathrm { }E_{z,Rx}^{inc}\left ({\boldsymbol {\rho },0 }\right)\chi _{Rx}\left ({\boldsymbol {\rho },z }\right) \\ \boldsymbol {E}_{Tx}^{tot}\left ({\boldsymbol {r} }\right)\cong&\hat { \boldsymbol {z}}\,\mathrm { }E_{z,Tx}^{tot}\left ({\boldsymbol {\rho },0 }\right)\chi _{Tx}\left ({\boldsymbol {\rho },z }\right)\tag{9}\end{align*}$$ where $\chi _{Rx}$ and $\chi _{Tx}$ are certain scalar real-valued functions. It should be noted that $\chi _{Tx}$ and $\chi _{Rx}$ are almost the same for both the total field and for the incident field, as demonstrated in Fig. 2.

To consider the pattern of the field, which is transmitted by a real antenna, we can apply the reciprocity theorem [18] and use (9) to obtain the following equation: $$\begin{align*}&\hspace {-0.5pc}V_{Rx}^{scat}=\frac {j\omega \varepsilon _{0}\mathrm {}\varepsilon _{b}}{I_{0}}\int _{S} E_{z,Rx}^{inc}\left ({\boldsymbol {\rho }\mathrm {'}\mathrm {,0} }\right)C\left ({\boldsymbol {\rho }\mathrm {'}\mathrm {,0} }\right) \\&\qquad\qquad\qquad\qquad\qquad\quad\times \,E_{z\mathrm {, }Tx}^{tot}\left ({\boldsymbol {\rho }\mathrm {'}\mathrm {,0} }\right)h\left ({\boldsymbol {\rho }{\mathrm {'}} }\right) ds\mathrm {'}\tag{10}\end{align*}$$

Here, $V_{Rx}^{scat}$ is signal of scattered waves received by Rx antenna, $S$ is an area in the section of IZ by plane $z=0$ , and $h\left ({\boldsymbol {\rho } }\right)$ is an effective height function of the cylindrical phantom: $$\begin{equation*} h\left ({\boldsymbol {\rho } }\right)\equiv \int _{Zmin}^{Zmax} {\chi _{Rx}\left ({\boldsymbol {\rho },z }\right)\chi _{Tx}\left ({\boldsymbol {\rho },z }\right)} dz\tag{11}\end{equation*}$$

After the discretization of (10) with period $\Delta$ and using (5), we obtain $$\begin{equation*} V_{Rx}^{scat}\cong -\frac {j\omega \varepsilon _{0}\varepsilon _{b}}{I_{0}}\Delta ^{2}\left ({\boldsymbol {E}_{Rx} }\right)^{T}\cdot \boldsymbol {CH}\cdot \boldsymbol {IGC}\cdot \boldsymbol {E}_{Tx}\tag{12}\end{equation*}$$ where $\boldsymbol {CH}\equiv diag\left ({\vec { \boldsymbol {C}}.\ast \vec { \boldsymbol {h}} }\right)$ and vector $\vec { \boldsymbol {h}}$ corresponds to values of $h\left ({\boldsymbol {\rho } }\right)$ (10) in the grid nodes, and $\boldsymbol {E}_{Rx}$ and $\boldsymbol {E}_{Tx}$ are vector representations of $E_{z,\mathrm { }Rx}^{inc}$ and $E_{z,Tx}^{inc}$ in the grid nodes, respectively (see (5) and (10)).

Our 2.5D model can be improved if it can modify $\boldsymbol {IGC}$ matrices (5), where 2D Green’s functions $G_{0}\left ({\boldsymbol {\rho }- \boldsymbol {\rho }' }\right)$ still infinitely spread and remain constant along the z-direction. In addition, coupling between pixels in 2D is handled by (4), and (5) corresponds to coupling between tin cylinders in 3D, which are also infinite in the z-direction. To provide finer modeling, we must restrict the area of field distribution and mutual coupling to an effective width of Tx antennas beams in the z-direction (Fig. 2). To this end, let us define the following function of effective beam height: $$\begin{equation*} h_{Tx}\left ({\boldsymbol {\rho } }\right)\equiv \int _{Zmin}^{Zmax} {\chi _{Tx}\left ({\boldsymbol {\rho },z }\right)} dz\tag{13}\end{equation*}$$

Then, we can rewrite equation (4) in the following form $$\begin{align*} \vec { \boldsymbol {h}}_{Tx}.\ast \boldsymbol {E}_{z}^{tot}+k_{b}^{2}\Delta ^{2} \boldsymbol {G}_{0}\cdot \boldsymbol {C}\cdot \left ({\vec { \boldsymbol {h}}_{Tx}.\ast \boldsymbol {E}_{z}^{tot} }\right)=\vec { \boldsymbol {h}}_{Tx}.\ast \boldsymbol {E}_{z}^{inc} \\\tag{14}\end{align*}$$ where vector $\vec { \boldsymbol {h}}_{Tx}$ represents $h_{Tx}\left ({\boldsymbol {\rho } }\right)$ in the grid nodes and operator “$.\ast$ ” represents element-wise multiplication. Thus, the $\boldsymbol {IGC}$ term in (5) can be replaced with $$\begin{equation*} \boldsymbol {IG} \boldsymbol {C}_{2.5}=\left ({\boldsymbol {I}+k_{b}^{2}\Delta ^{2} \boldsymbol {H}_{Tx}^{-1}\cdot \boldsymbol {G}_{0}\cdot \boldsymbol {H}_{Tx}\cdot \boldsymbol {C} }\right)^{-1}\tag{15}\end{equation*}$$ where $\boldsymbol {H}_{Tx}\equiv diag\left ({\vec { \boldsymbol {h}}_{Tx} }\right)$ .

C. Solving of Inverse Problem

We reconstructed the phantom’s image using a Gauss-Newton algorithm that iteratively updates the contrast vector $\boldsymbol {C}$ : $$\begin{equation*} \boldsymbol {C}_{n+1}= \boldsymbol {C}_{n}+\Delta \boldsymbol {C}_{n}\tag{16}\end{equation*}$$

At each iteration, the increment $\Delta \boldsymbol {C}_{n}$ of the contrast vector is defined by the following linearized equation: $$\begin{align*} \boldsymbol {J}\cdot \Delta \boldsymbol {C}_{n}=&\Delta \boldsymbol {\gamma }_{n} =\left ({\boldsymbol {\gamma }_{n}- \boldsymbol {\gamma }_{meas} }\right) \equiv \left ({\boldsymbol {V}_{n}- \boldsymbol {V}_{meas} }\right). / \boldsymbol {V}_{0} \\\tag{17}\end{align*}$$ where $\boldsymbol {J}$ is a Jacobian matrix; $\boldsymbol {V}_{n}$ and $\boldsymbol {\gamma }_{n}$ are vectors for the voltage and normalized voltage of all received signals calculated for contrast $\boldsymbol {C}_{n}$ , respectively; $\boldsymbol {V}_{0}$ is the voltage vector of all signals received in an empty bath; and $\boldsymbol {\gamma }_{meas}$ is a normalized vector for all measured (or simulated) signals caused by the object being imaged. Note that for the 2D forward solver, we must apply $\boldsymbol {V}_{n}= \boldsymbol {E}_{z}^{tot}\left ({\boldsymbol {\rho }_{Tx}\vert \boldsymbol {\rho }_{Rx} }\right)$ and $\boldsymbol {V}_{0}= \boldsymbol {E}_{z}^{inc}\left ({\boldsymbol {\rho }_{Tx}\vert \boldsymbol {\rho }_{Rx} }\right)$ .

The Jacobian is typically an ill-conditioned matrix, and the approximate solution of (17) can be determined with Tikhonov regularization: $$\begin{equation*} \Delta \boldsymbol {C}_{n}=-\left ({\boldsymbol {J}^{H}\cdot \boldsymbol {J}+\alpha ^{2} \boldsymbol {I} }\right)^{-1}\cdot \boldsymbol {J}^{H}\cdot \Delta \boldsymbol {\gamma }_{n}\tag{18}\end{equation*}$$ where $\alpha$ is the parameter of Tikhonov regularization [17]. The superscript “$H$ ” denotes a Hermitian conjugate. Equation (4) allows us to derive the following elements of the Jacobian matrix: $$\begin{align*} \boldsymbol {J}=-k_{b}^{2}\Delta ^{2}\left ({\boldsymbol {G}_{hg}\left ({\boldsymbol {r}_{Rx}\vert \boldsymbol {r}_{i} }\right).\ast \boldsymbol {G}_{hg}\left ({\boldsymbol {r}_{Tx}\vert \boldsymbol {r}_{i} }\right) }\right)./ \boldsymbol {E}_{z}^{inc}\left ({\boldsymbol {r}_{Tx}\vert \boldsymbol {r}_{Rx} }\right) \\\tag{19}\end{align*}$$ where $\boldsymbol {r}_{Tx}$ and $\boldsymbol {r}_{Rx}$ are the positions of the $Tx$ and $Rx$ antennas, respectively, and $\boldsymbol {r}_{i}$ are points in the IZ, while operator “./” is an element-wise division. Here, $\boldsymbol {G}_{hg}$ denotes Green’s functions matrix for heterogeneous media with a body, and it can be expressed as $$\begin{equation*} \boldsymbol {G}_{hg}\equiv \boldsymbol {G}_{0}\cdot \boldsymbol {IG} \boldsymbol {C}_{2.5}^{T}\tag{20}\end{equation*}$$

D. Results of Image Reconstruction

Let us demonstrate that the 16 antennas in our cylindrical array inside the matching liquid (water) are not sufficient to produce a satisfactory image using the conventional reconstruction method. This follows from the theoretical analysis in [20], which estimates the minimum number of antennas needed, $N_{ant\,min}$ , to provide an accurate representation of a scattered EM field. In this context, $N_{ant\,min}$ is determined by the phantom’s outline length, $L_{phantom}$ , and the wavelength $\lambda _{b}$ in the background medium: $$\begin{equation*} N_{ant\,min}\cong L_{phantom}/\left ({\lambda _{b}/2 }\right)\tag{21}\end{equation*}$$

For example, we tested two 2D phantoms under the following identical conditions:

1. frequency 850 MHz

2. bath liquid: pure water ($\varepsilon =78$ , $\sigma =0.2$ S/m)

3. array 180 mm diameter, 16 Tx / Rx antennas

4. 70 iterations

Both phantoms are immersed into pure water and $\lambda _{b}\mathrm {/2\cong }~20$ mm at 850 MHz. We can show in both cases that the measured data from the array with 16 antennas is not sufficient for acceptable imaging.

The first phantom has an elliptical shape, is covered by skin, and contains a 2D tumor model (Fig. 3). This phantom is described by the following parameters:

• inner homogeneous material: $\varepsilon =40$ , $\sigma =0.7$ S/m

• tumor: $\varepsilon =56.3$ , $\sigma =0.99$ S/m

• skin: $\varepsilon =46.0$ , $\sigma =0.85$ S/m, thickness 2 mm.

FIGURE 3.

Result of conventional imaging approach using the equations from Sec. II for elliptical shape phantom with skin and tumor. Using 16 Tx / Rx array in pure water at 850 MHz. (a) Phantom permittivity and (b) conductivity. (c) Reconstructed permittivity and (d) conductivity.

Fig. 3 demonstrates the failure in image reconstruction for this phantom. The direct Gauss-Newton iteration method with Tikhonov regularization was used for image reconstruction [17]. This result can be explained by the fact that the scattering measurement data was insufficient for successful image reconstruction because $N_{ant\,min}=26$ according to (21).

The second phantom was a knee with skin, and it contained a tumor where $L_{phantom}=405$ mm (Fig. 4); it was created using a 3D whole-body model as described in [21]. Fig. 4 also illustrates the failure of image reconstruction, because 16 antennas are not sufficient to obtain a satisfactory reconstructed image, since (21) gives $N_{ant\,min}=20$ .

FIGURE 4.

Results of conventional imaging approach using equations from Sec. II for a phantom of a knee with skin and a tumor. Using 16 Tx / Rx array in pure water at 850 MHz. (a) Phantom permittivity and (b) conductivity. (c) Reconstructed permittivity and (d) conductivity.

SECTION III. Proposed Imaging Approach

To overcome the problem of not enough antennas, we suggest applying a special method of image reconstruction that is based on restricting the object’s image to be regular and smooth. This condition eliminates irregularly-shaped solutions like those seen in Fig. 3 and Fig. 4, and it provides prior smooth reconstruction images. We found that such an approach is very helpful for solving the problem of an insufficient number of antennas.

There are several methods for shape reconstruction, such as the linear sampling methods and level set methods [12], [13]. However, these do not guarantee that the reconstructed image will be regular and smooth. In the present work, we developed an alternative Gauss-Newton method for shape reconstruction, where the algorithm searches only regular and smooth shapes and excludes irregular solutions. It demonstrated an advantage and improved stability over traditional reconstruction methods in the case in which there is not enough data for successful imaging. In addition, the use of simultaneous shape and contrast reconstruction helps improve the accuracy of both.

A. Solving of Inverse Problem

Let us consider Fig. 5, which presents sketches illustrating the geometrical details that are important for the derivation of our shape reconstruction algorithm. We define the body’s outline curve $D$ in polar coordinates and describe it using a smooth function $\rho =R(\varphi)$ (see Fig. 5(b)). Let us discretize this curve by an array of phases $\boldsymbol {\varphi }=\left [{ \Delta \varphi,2\Delta \varphi,\ldots,2\pi }\right]$ with a phase period $\Delta \varphi$ . Thus, the vector $\boldsymbol {R}=\left [{ R\left ({\Delta \varphi }\right),\ldots,R\left ({2\pi }\right) }\right]$ fully describes the body’s regular shape $\Omega$ .

FIGURE 5.

Position of body $\Omega$ in the imaging zone (IZ) $A$ . $C_{0}\left ({r }\right)$ is a contrast distribution over the whole IZ, $C_{D}\left ({r }\right)$ is the contrast distribution in the body’s outline $D$ , $\chi \left ({r }\right)$ is the body’s characteristic function (22), $\Delta \chi \left ({r }\right)$ is an increment of the body’s characteristic function. (b) Outline of the body shape in polar coordinates $\left ({\rho,\varphi }\right)$ with period of discretization $\Delta \varphi$ in the angle coordinate.

Let us introduce a characteristic function $\chi \left ({\boldsymbol {r} }\right)$ for the shape specification (see Fig. 5(a)) $$\begin{align*} \chi \left ({\boldsymbol {r} }\right)=\begin{cases} 1, & inside\,\, of\,\,\ \Omega \\ 0, & outside \,\,of \,\,\Omega \\ \end{cases}\tag{22}\end{align*}$$ and represent contrast $C\left ({\boldsymbol {r} }\right)$ inside the body in the form of $$\begin{equation*} C\left ({\boldsymbol {r} }\right)=C_{0}\left ({\boldsymbol {r} }\right)\chi \left ({\boldsymbol {r} }\right)\tag{23}\end{equation*}$$ where $C_{0}\left ({\boldsymbol {r} }\right)$ is a contrast distribution over the whole IZ.

The total contrast deviation $\Delta C^{tot}$ includes two components: the first relates to the deviation $\Delta C_{0}\left ({\boldsymbol {r} }\right)$ in the shape $\chi \left ({\boldsymbol {r} }\right)$ while the second relates to the shape deviation $\Delta \chi \left ({\boldsymbol {r} }\right)$ : $$\begin{equation*} \Delta C^{tot}\left ({\boldsymbol {r} }\right)=\Delta C_{0}\left ({\boldsymbol {r} }\right)\chi \left ({\boldsymbol {r} }\right)+C_{D}\left ({\boldsymbol {r} }\right)\Delta \chi \left ({\boldsymbol {r} }\right)\tag{24}\end{equation*}$$ where $C_{D}$ is the contrast in the body’s boundary, as shown in Fig. 5(a). After discretization, we obtain the following equation in vector form: $$\begin{equation*} \Delta \boldsymbol {C}^{tot}=\Delta \boldsymbol {C}_{0}.\ast \boldsymbol {\chi }+ \boldsymbol {C}_{D}.\ast \Delta \boldsymbol {\chi }\equiv \Delta \boldsymbol {C}+ \boldsymbol {C}_{D}.\ast \Delta \boldsymbol {\chi }\tag{25}\end{equation*}$$

Note that we define vector $\Delta \boldsymbol {\chi }$ on the basis of areas $\Delta S$ bounded by intervals $\Delta R$ , $\Delta \varphi$ , and it can be approximated by $$\begin{equation*} \Delta \boldsymbol {\chi }\cong \Delta \boldsymbol {S}/\Delta ^{2},\quad where \Delta \boldsymbol {S}\cong \boldsymbol {R}.\ast \Delta \boldsymbol {R}.\ast \hat { \boldsymbol {n}}_{s}\Delta \varphi\tag{26}\end{equation*}$$ where $\hat { \boldsymbol {n}}_{s}$ is normal to the outline curve. Hence $$\begin{equation*} \Delta \boldsymbol {C}^{tot}\cong \Delta \boldsymbol {C}+\frac {1}{\Delta ^{2}}\left ({\boldsymbol {C}_{D}.\ast \boldsymbol {R}.\ast \Delta \boldsymbol {R}.\ast \hat { \boldsymbol {n}}_{s}\Delta \varphi }\right)\tag{27}\end{equation*}$$

Although our shape updating program utilizes a slightly more accurate equation for $\Delta \boldsymbol {S}$ , it is equivalent to (26) for small $\Delta \boldsymbol {S}$ .

Equations (26) and (27) allow us to modify equation (17) for updating contrast: $$\begin{equation*} \boldsymbol {J}\cdot \Delta \boldsymbol {C}+ \boldsymbol {J}_{R}\cdot \Delta \boldsymbol {R}=\Delta \boldsymbol {\gamma }\tag{28}\end{equation*}$$

Here, $\boldsymbol {J}$ is a Jacobian (19) and $\boldsymbol {J}_{R}$ is a Jacobian for shape reconstruction: $$\begin{equation*} \boldsymbol {J}_{R}=\frac {\Delta \varphi }{\Delta ^{2}} \boldsymbol {J}_{D}\cdot diag\left ({\boldsymbol {R}.\ast \hat { \boldsymbol {n}}_{s} }\right)\tag{29}\end{equation*}$$

$\boldsymbol {J}_{D}$ is a Jacobian $\boldsymbol {J}$ (19), which is calculated for points on the shape outline (see Fig. 5(a)). This means that we use vector contrast $\boldsymbol {C}_{D}$ in (15) rather than a vector contrast for the inner points $\boldsymbol {C}$ . Based on (28), we obtain the following equations for simultaneously updating the contrast $\Delta \boldsymbol {C}$ and shape, while using constant preselected weighting coefficients $w_{1},w_{2}$ and Tikhonov regularization as in (18): $$\begin{align*} \Delta \boldsymbol {C}=&-w_{1}\left ({\boldsymbol {J}^{H}\cdot \boldsymbol {J}+\alpha ^{2} \boldsymbol {I} }\right)^{-1}\cdot \boldsymbol {J}^{H}\cdot \Delta \boldsymbol {\gamma } \tag{30}\\ \Delta \boldsymbol {R}=&-w_{2}\left ({\boldsymbol {J}_{R}^{H}\cdot \boldsymbol {J}_{R}+\alpha ^{2} \boldsymbol {I} }\right)^{-1}\cdot \boldsymbol {J}_{R}^{H}\cdot \Delta \boldsymbol {\gamma }\tag{31}\end{align*}$$

Note that the value at $w_{1}=1$ in equation (30) is the same as that in (18), which is natural.

B. Results of Image Reconstruction

In this section, we present the results of simultaneous shape and dielectric contrast image reconstruction based on equations (30), (31) for the two phantoms that are described in Sec. II, D. In contrast to the poor image quality produced by the conventional method, the new approach is considered to be successful.

For both these phantoms, we applied the same conditions as above in Sec. II, D. The best results were obtained when the reconstruction process was carried out in the following manner:

1. During the first 40 iterations, shape reconstruction was mainly being performed: the weighting coefficients were $w_{1}=0.2$ for the contrast reconstruction and $w_{2}=0.5$ for the shape reconstruction;

2. During the last 30 iterations, contrast reconstruction was mainly being performed: the weighting coefficients were $w_{1}=0.5$ for the contrast reconstruction and $w_{2}=0.2$ for the shape reconstruction.

Fig. 6 shows the successful results of the image reconstruction for the body’s phantom with elliptical shape (Fig. 3(a), (b)), which were obtained using simultaneous reconstruction of the shape and contrast. As shown in Fig. 6, the reconstructed shape is well delineated, which is very different in the image quality in comparison to Fig. 3.

FIGURE 6.

Results of simultaneous imaging approach of an elliptical shape body phantom with skin and tumor (Fig. 2(a), (b)). We utilized a 2.5D VIE forward solver (Sec. II, B) and 3D CST simulated field data over 70 iterations. (a) Reconstructed permittivity of the phantom. (b) Reconstructed conductivity of the phantom. (c) Misfit between original phantom shape and reconstructed shape.

The results of imaging the knee phantom (Fig. 4(a), (b)) using (30), (31) are presented in Fig. 7. This figure shows satisfactory image reconstructions, in contrast with the irregular images presented in Fig. 4(c), (d), despite the fact that we used the same parameters as the conventional approach (see Sec. II, D).

FIGURE 7.

Results of simultaneous imaging approach of knee phantom with skin and tumor (Fig. 3(a), (b)). We applied a 2.5D VIE forward solver (Sec. II, B) and 3D CST simulated field data over 70 iterations. (a) Reconstructed permittivity of the phantom. (b) Reconstructed conductivity of the phantom. (c) Misfit between original phantom shape and reconstructed shape.

SECTION IV. Conclusion

We proposed a method for overcoming the problem of insufficient microwave scattering data by using simultaneous reconstruction of body tomographic image and its shape under the restriction that the shape must be regular and smooth. While seeking a balance between computation time and accuracy, we developed an original 2.5D forward EM solver for image reconstruction. Through numerical simulations using human body models, we demonstrated that this approach can produce satisfactory images of permittivity and conductivity despite insufficient microwave scattering data. The suggested method is simple, fast, and not computationally expensive, and it provides good accuracy.