Real-time identification of small anomalies from scattering matrix without background information

Abstract

Several researches have confirmed the possibility of localizing small anomalies via Kirchhoff migration (KM); however, when the background information is unknown, small anomalies cannot be satisfactorily retrieved. This fact can be examined through the simulation results; however, related theoretical result to explain the reason of such phenomenon has not yet been investigated. In this contribution, we show that the imaging function of the KM can be expressed by an infinite series of the Bessel function of the first kind, material properties, and antenna arrangement, and applied alternative value of the background wavenumber. Based on the theoretical result, we explain why the exact location and shape of anomalies cannot be retrieved. The simulation results with synthetic data exhibited to support the theoretical result.

Keywords

Kirchhoff migration scattering matrix background information microwave imaging simulation results

1. Introduction

Retrieving shapes, locations, and material properties of small anomalies from measured scattering parameter data is an interesting problem in microwave imaging. In particular, the development of a real-time algorithm for retrieving small anomalies has been of great interest in various research areas, such as nondestructive evaluation, medical imaging, geophysics, and radar imaging. Related works can be found in [1–5] and references therein.

Kirchhoff migration (KM) is classified as a non-iterative technique for retrieving location and shape of unknown targets. Various studies have confirmed that the KM is a fast, effective, and robust method in both inverse scattering problem and microwave imaging (e.g., [6–8]). It is worth to emphasize that for a successful application of the KM, a priori information of the background should be known because the accurate value of the background wavenumber is essential to define an imaging function. In other words, if one of the exact values of background permittivity, conductivity, or permeability is unknown, the identified locations of anomalies are concentrated or far from the origin, the size of anomalies are smaller or larger than the true ones, or it will be extremely difficult to recognize the existence of anomalies because of the appearance of several artifacts. Until now, these phenomena have been examined through the simulation results. Motivated by this, mathematical theory for bifocuing method [10] and orthogonality sampling method [9] has been investigated. However, a reliable mathematical theory for KM to explain such phenomena has not yet been investigated.

In this study, we apply the KM without background information for retrieving the location and shape of small anomalies. To explain why the accurate location and shape of anomalies cannot be retrieved, we discover the mathematical structure of the imaging function of the KM. This is based on the fact that the scattered field S-parameter can be represented by an integral equation formula. Throughout the discovered structure, we can explain the theoretical reason of the appearance of an inaccurate location and shape of anomalies and some properties of the KM.

This manuscript is organized as follows. In Section 2, the scattering parameter in the presence of small anomalies and imaging function of the KM are introduced. In Section 3, the mathematical structure of the imaging function of the KM with an inaccurate wavenumber is discovered. In Section 4, the simulation results with synthetic data are presented. A short conclusion is provided in Section 5.

2. Scattering parameter and the imaging function

Let D_s, s = 1,2, …, S, be a small anomaly and D be the collection of D_s. Throughout this manuscript, we assume that every D_s is surrounded by a circular array of antennas A_n, n = 1,2, …, N (>2), with location a_n, refer to Fig. 1. We denote 𝛺 as a homogeneous region filled by matching liquid such that D ⊂ 𝛺 and assume that all materials D_s and 𝛺 are classified by their value of dielectric permittivity and electric conductivity at a given angular frequency of 𝜔 = 2𝜋f, that is, the value of magnetic permeability is constant for every x ∈ 𝛺 say, 𝜇(x) ≡𝜇_b = 1.257 × 10⁻⁶ H/m. We denote 𝜀_b and 𝜎_b as the background permittivity and conductivity, respectively. Analogously, let 𝜀_s and 𝜎_s be those of D_s. With this, we can define the following piecewise constant permittivity and conductivity, $\begin{eqnarray} \displaystyle {\varepsilon}(\mathbf{y})=\left\{ \begin{array}{@{} ll@{}}{\varepsilon}_{s} & \text{for }\quad \mathbf{y}\in D_{s}\\ {\varepsilon}_{b} & \text{for }\quad \mathbf{y}\in {\Omega}\setminus D \end{array} \right.\quad \text{and}\quad {\sigma}(\mathbf{y})=\left\{ \begin{array}{@{} ll@{}}{\sigma}_{s} & \text{for }\quad \mathbf{y}\in D_{s}\\ {\sigma}_{b} & \text{for }\quad \mathbf{y}\in {\Omega}\setminus D, \end{array} \right. & & \displaystyle \nonumber \end{eqnarray}$ respectively. Throughout this paper, we further assume that 𝜔𝜀_b ≫ 𝜎_b and $\sqrt{{\varepsilon}_{s}/{\varepsilon}_{b}}\times \text{diam}(D_{s})\lt {\lambda}$ , s = 1,2, …, S, where 𝜆 denotes the positive wavelength and diam(D_s) is the diameter of D_s.

Fig. 1.

Illustration of the problem configuration.

Let S (n, m) be the scattering parameter (or S-parameter) which is defined as the ratio of the output voltage at the A_n antenna and the input voltage at the A_m. We denote S_inc(n, m) and S_tot(n, m) as the incident and total field S-parameters, respectively, because of the absence and presence of D. Correspondingly, we let S_scat(n, m) = S_tot(n, m) − S_inc(n, m) be the scattered field S-parameter. Then, based on [11,12], S_scat(n, m) can be represented as the following integral equation formula $\begin{eqnarray}\displaystyle S_{\text{scat}}(n,m)=-\frac{ik_{0}^{2}}{4{\omega}{\mu}_{b}}\int _{{\Omega}}\left(\frac{{\varepsilon}(\mathbf{y})-{\varepsilon}_{b}}{{\varepsilon}_{b}}+i\frac{{\sigma}(\mathbf{y})-{\sigma}_{b}}{{\omega}{\varepsilon}_{b}}\right)\mathbf{E}_{\text{inc}}(\mathbf{y},\mathbf{a}_{m})\mathbf{E}_{\text{tot}}(\mathbf{a}_{n},\mathbf{y})\text{d}\mathbf{y}, & & \displaystyle\end{eqnarray}$ (1) where $k_{0}={\omega}\sqrt{{\varepsilon}_{b}{\mu}_{b}}$ denotes the lossless background wavenumber, E_inc(y, a_m) denotes the incident electric field in a homogeneous medium due to the point current density at A_m, and E_tot(a_n, y) is the corresponding total field in the presence of D measured at A_n. Here, time-harmonic dependence e^−i𝜔t is assumed. At this moment, it is very difficult to design an imaging function of KM with (1). In order to derive an alternative expression of (1), let us mention that the simulation configuration is same as the previous studies [12,13] thus, only the z-component of the incident and total fields can be handled. Correspondingly, S_scat(n, m) of (1) can be written as $\begin{eqnarray}\displaystyle S_{\text{scat}}(n,m)=-\frac{ik_{0}^{2}}{4{\omega}{\mu}_{b}}\int _{{\Omega}}\left(\frac{{\varepsilon}(\mathbf{y})-{\varepsilon}_{b}}{{\varepsilon}_{b}}+i\frac{{\sigma}(\mathbf{y})-{\sigma}_{b}}{{\omega}{\varepsilon}_{b}}\right)\text{E}_{\text{inc}}^{(z)}(\mathbf{y},\mathbf{a}_{m})\text{E}_{\text{tot}}^{(z)}(\mathbf{a}_{n},\mathbf{y})\text{d}\mathbf{y}, & & \displaystyle\end{eqnarray}$ (2) where $\text{E}_{\text{inc}}^{(z)}$ and $\text{E}_{\text{tot}}^{(z)}$ denote the z-component of E_inc and E_tot, respectively.

Notice that the incident field $\text{E}_{\text{inc}}^{(z)}(\mathbf{y},\mathbf{a}_{m})$ can be expressed as $\begin{eqnarray}\displaystyle \text{E}_{\text{inc}}^{(z)}(\mathbf{y},\mathbf{a}_{m})=-\frac{i}{4}H_{0}^{(1)}(k_{b}|\mathbf{a}_{m}-\mathbf{y}|), & & \displaystyle \nonumber\end{eqnarray}$ where $H_{0}^{(1)}$ denotes the Hankel function of order zero of the first kind and k_b is the background wavenumber that satisfies $k_{b}^{2}={\omega}^{2}{\mu}_{b}({\varepsilon}_{b}-i{\sigma}_{b}/{\omega})$ . However, $\text{E}_{\text{tot}}^{(z)}(\mathbf{a}_{n},\mathbf{y})$ cannot be expressed without complete information of D. Previously, we assumed that $\sqrt{{\varepsilon}_{s}/{\varepsilon}_{b}}\times \text{diam}(D_{s})<{\lambda}$ for all s so every D_s can be regarded as a small target. Then, based on [15], it is possible to apply the Born approximation $\text{E}_{\text{tot}}^{(z)}(\mathbf{a}_{n},\mathbf{y})\approx \text{E}_{\text{inc}}^{(z)}(\mathbf{a}_{n},\mathbf{y})$ and correspondingly, (2) can be written as follows $\begin{eqnarray}\displaystyle S_{\text{scat}}(n,m)\approx \frac{ik_{0}^{2}}{64{\omega}{\mu}_{b}}\int _{D}\left(\frac{{\varepsilon}(\mathbf{y})-{\varepsilon}_{b}}{{\varepsilon}_{b}}+i\frac{{\sigma}(\mathbf{y})-{\sigma}_{b}}{{\omega}{\varepsilon}_{b}}\right)H_{0}^{(1)}(k_{b}|\mathbf{a}_{m}-\mathbf{y}|)H_{0}^{(1)}(k_{b}|\mathbf{a}_{n}-\mathbf{y}|)\text{d}\mathbf{y}. & & \displaystyle\end{eqnarray}$ (3)

To introduce the imaging function of the KM, let us generate the scattering matrix $\mathbb{K}$ $\begin{eqnarray}\displaystyle \mathbb{K}=\left[\begin{array}{@{}cccc@{}}0 & S_{\text{scat}}(1,2) & \cdots ∼ & S_{\text{scat}}(1,N)\\ S_{\text{scat}}(2,1) & 0 & \cdots ∼ & S_{\text{scat}}(2,N)\\ \vdots & \vdots & \ddots & \vdots \\ S_{\text{scat}}(N,1) & S_{\text{scat}}(N,2) & \cdots ∼ & 0\end{array}\right]. & & \displaystyle\end{eqnarray}$ (4) Then, on the basis of (3) and (4), we introduce a unit vector: for each x ∈ 𝛺, $\begin{eqnarray} \displaystyle \hspace{-5.0pt}\mathbf{W}(k_{b},\mathbf{x})=\frac{\mathbf{F}(k_{b},\mathbf{x})}{|\mathbf{F}(k_{b},\mathbf{x})|},\quad \mathbf{F}(k_{b},\mathbf{x})=\Big[H_{0}^{(1)}(k_{b}|\mathbf{a}_{1}-\mathbf{x}|),H_{0}^{(1)}(k_{b}|\mathbf{a}_{2}-\mathbf{x}|),\ldots ,H_{0}^{(1)}(k_{b}|\mathbf{a}_{N}-\mathbf{x}|)\Big]^{T}. & & \displaystyle \end{eqnarray}$ (5) Then, the imaging function of the KM is given by the following: $\begin{eqnarray}\displaystyle \mathfrak{F}_{\text{KIR}}(k_{b},\mathbf{x})=|\overline{\mathbf{W}(k_{b},\mathbf{x})}^{T}\mathbb{K}\overline{\mathbf{W}(k_{b},\mathbf{x})}|. & & \displaystyle\end{eqnarray}$ (6) Then, the anomaly D can be identified through the map of $\mathfrak{F}_{\text{KIR}}(k_{b},\mathbf{x})$ , refer to [7,8].

3. Mathematical structure of the imaging function

As we have noted, the exact value of k_b, that is, the values of 𝜀_b and 𝜎_b, should be known to generate W(k_b, x) of (5). Because we assumed that the exact value of k_b is unknown, we apply another wavenumber k_a instead of k_b and generate unit vector W(k_a, x) from (5). Then, based on the imaging results via $\mathfrak{F}_{\text{KIR}}(k_{a},\mathbf{x})$ of (6), it will be possible to recognize the existence of D_s; however, the exact location and shape of D_s cannot be retrieved unless k_a = k_b. Notice that for some values of k_a, the retrieved locations are shifted in a specific direction and the sizes are smaller or larger than the true ones. To explain this phenomenon, we discover the mathematical structure of the imaging function.

TheoremA .
Let 𝜃_n = a_n∕|a_n| = (cos𝜃_n, sin𝜃_n) and k_ax − k_by = |k_ax − k_by|(cos𝜙, sin𝜙). If a_n satisfies $|\mathbf{a}_{n}-\mathbf{x}|\gg \{1/4|k_{a}|,1/4|k_{b}|\}$ for all n = 1,2, …, N, $\mathfrak{F}_{\text{KIR}}(\mathbf{x},k_{a})$ can be represented as follows: $\begin{eqnarray}\displaystyle \mathfrak{F}_{\text{KIR}}(k_{a},\mathbf{x}) & {\approx} & \displaystyle \frac{N{\omega}{\varepsilon}_{b}}{32Rk_{b}{\pi}}\left|\int _{D}{\mathcal{O}}(\mathbf{y})\left(J_{0}(|k_{a}\mathbf{x}-k_{b}\mathbf{y}|)+\frac{{\mathcal{E}}(k_{b},k_{a},\mathbf{x})}{N}\right)^{2}\text{d}\mathbf{y}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -\left.\frac{1}{N}\int _{D}{\mathcal{O}}(\mathbf{y})\left(J_{0}(2|k_{a}\mathbf{x}-k_{b}\mathbf{y}|)+\frac{{\mathcal{E}}(2k_{b},2k_{a},\mathbf{x})}{N}\right)\text{d}\mathbf{y}\right|,\end{eqnarray}$ (7) where J_p denotes the Bessel function of order p, $\begin{eqnarray}\displaystyle {\mathcal{O}}(\mathbf{y})=\frac{{\varepsilon}(\mathbf{y})-{\varepsilon}_{b}}{{\varepsilon}_{b}}+i\frac{{\sigma}(\mathbf{y})-{\sigma}_{b}}{{\omega}{\varepsilon}_{b}},\quad \text{and}\quad {\mathcal{E}}(k_{b},k_{a},\mathbf{x})=\mathop{\sum }_{n=1}^{N}\mathop{\sum }_{p=-\infty ,p\neq 0}^{\infty }i^{p}J_{p}(|k_{a}\mathbf{x}-k_{b}\mathbf{y}|)e^{ip({\theta}_{n}-{\phi})}. & & \displaystyle \nonumber\end{eqnarray}$

TheoremB .
Since $|\mathbf{a}_{n}-\mathbf{x}|\gg \{1/4|k_{a}|,1/4|k_{b}|\}$ for all n = 1,2, …, N, the following asymptotic form of the Hankel function holds (e.g., [4, Theorem 2.5]): $\begin{eqnarray}\displaystyle H_{0}^{(1)}(k_{b}|\mathbf{x}-\mathbf{y}|)\approx \frac{(1-i)e^{ik_{b}|\mathbf{x}|}}{\sqrt{k_{b}{\pi}|\mathbf{a}_{n}|}}e^{-ik_{b}\boldsymbol{{\theta}}_{n}\cdot \mathbf{y}}\quad \text{and}\quad H_{0}^{(1)}(k_{a}|\mathbf{x}-\mathbf{y}|)\approx \frac{(1-i)e^{ik_{a}|\mathbf{x}|}}{\sqrt{k_{a}{\pi}|\mathbf{a}_{n}|}}e^{-ik_{a}\boldsymbol{{\theta}}_{n}\cdot \mathbf{y}}. & & \displaystyle\end{eqnarray}$ (8) Then, the test vector and scattering matrix can be represented as $\begin{eqnarray}\displaystyle \mathbf{W}(k_{a},\mathbf{x})=\frac{1}{\sqrt{N}}[e^{-ik_{a}\boldsymbol{{\theta}}_{1}\cdot \mathbf{x}},e^{-ik_{a}\boldsymbol{{\theta}}_{2}\cdot \mathbf{x}},\ldots ,e^{-ik_{a}\boldsymbol{{\theta}}_{N}\cdot \mathbf{x}}]^{T} & & \displaystyle \nonumber\end{eqnarray}$ and $\begin{eqnarray}\displaystyle \mathbb{K}=\frac{e^{2iRk_{b}}{\omega}{\varepsilon}_{b}}{32Rk_{b}{\pi}}\left[\begin{array}{@{}cccc@{}}0 & \displaystyle \int _{D}{\mathcal{O}}(\mathbf{y})e^{-ik_{b}(\boldsymbol{{\theta}}_{1}+\boldsymbol{{\theta}}_{2})\cdot \mathbf{y}}\text{d}\mathbf{y} & \cdots ∼ & \displaystyle \int _{D}{\mathcal{O}}(\mathbf{y})e^{-ik_{b}(\boldsymbol{{\theta}}_{1}+\boldsymbol{{\theta}}_{N})\cdot \mathbf{y}}\text{d}\mathbf{y}\\[12.0pt] \displaystyle \int _{D}{\mathcal{O}}(\mathbf{y})e^{-ik_{b}(\boldsymbol{{\theta}}_{2}+\boldsymbol{{\theta}}_{1})\cdot \mathbf{y}}\text{d}\mathbf{y} & 0 & \cdots ∼ & \displaystyle \int _{D}{\mathcal{O}}(\mathbf{y})e^{-ik_{b}(\boldsymbol{{\theta}}_{2}+\boldsymbol{{\theta}}_{N})\cdot \mathbf{y}}\text{d}\mathbf{y}\\ \vdots & \vdots & \ddots & \vdots \\ \displaystyle \int _{D}{\mathcal{O}}(\mathbf{y})e^{-ik_{b}(\boldsymbol{{\theta}}_{N}+\boldsymbol{{\theta}}_{1})\cdot \mathbf{y}}\text{d}\mathbf{y} & \displaystyle \int _{D}{\mathcal{O}}(\mathbf{y})e^{-ik_{b}(\boldsymbol{{\theta}}_{N}+\boldsymbol{{\theta}}_{2})\cdot \mathbf{y}}\text{d}\mathbf{y} & \cdots ∼ & 0\\ \end{array}\right], & & \displaystyle \nonumber\end{eqnarray}$ respectively. Based on [14], the following relation holds for n = 1,2, …, N, $\begin{eqnarray}\displaystyle \mathop{\sum }_{n=1}^{N}e^{i\boldsymbol{{\theta}}_{n}\cdot (k_{a}\mathbf{x}-k_{b}\mathbf{y})}=\mathop{\sum }_{n=1}^{N}e^{i|k_{a}\mathbf{x}-k_{b}\mathbf{y}|\cos ({\theta}_{n}-{\phi}_{s})}=NJ_{0}(|k_{a}\mathbf{x}-k_{b}\mathbf{y}|)+{\mathcal{E}}(k_{b},k_{a},\mathbf{x}), & & \displaystyle\end{eqnarray}$ (9) we have $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \overline{\mathbf{W}(k_{b},\mathbf{x})}^{T}\mathbb{K}\overline{\mathbf{W}(k_{b},\mathbf{x})}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =\frac{e^{2iRk_{b}}{\omega}{\varepsilon}_{b}}{32NRk_{b}{\pi}}\left[\begin{array}{@{}c@{}}\displaystyle \int _{D}{\mathcal{O}}(\mathbf{y})e^{-ik_{b}\boldsymbol{{\theta}}_{1}\cdot \mathbf{y}}\left(\mathop{\sum }_{n=1}^{N}e^{i\boldsymbol{{\theta}}_{n}\cdot (k_{a}\mathbf{x}-k_{b}\mathbf{y})}-e^{i\boldsymbol{{\theta}}_{1}\cdot (k_{a}\mathbf{x}-k_{b}\mathbf{y})}\right)\text{d}\mathbf{y}\\[15.0pt] \displaystyle \int _{D}{\mathcal{O}}(\mathbf{y})e^{-ik_{b}\boldsymbol{{\theta}}_{2}\cdot \mathbf{y}}\left(\mathop{\sum }_{n=1}^{N}e^{i\boldsymbol{{\theta}}_{n}\cdot (k_{a}\mathbf{x}-k_{b}\mathbf{y})}-e^{i\boldsymbol{{\theta}}_{2}\cdot (k_{a}\mathbf{x}-k_{b}\mathbf{y})}\right)\text{d}\mathbf{y}\\ \vdots \\ \displaystyle \int _{D}{\mathcal{O}}(\mathbf{y})e^{-ik_{b}\boldsymbol{{\theta}}_{N}\cdot \mathbf{y}}\left(\mathop{\sum }_{n=1}^{N}e^{i\boldsymbol{{\theta}}_{n}\cdot (k_{a}\mathbf{x}-k_{b}\mathbf{y})}-e^{i\boldsymbol{{\theta}}_{N}\cdot (k_{a}\mathbf{x}-k_{b}\mathbf{y})}\right)\text{d}\mathbf{y}\end{array}\right]∼\cdot ∼\left[\begin{array}{@{}c@{}}e^{ik_{a}\boldsymbol{{\theta}}_{1}\cdot \mathbf{x}}\\ e^{ik_{a}\boldsymbol{{\theta}}_{2}\cdot \mathbf{x}}\\ \vdots \\ e^{ik_{a}\boldsymbol{{\theta}}_{N}\cdot \mathbf{x}}\end{array}\right]\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =\frac{e^{2iRk_{b}}{\omega}{\varepsilon}_{b}}{32NRk_{b}{\pi}}\left[\begin{array}{@{}c@{}}\displaystyle \int _{D}{\mathcal{O}}(\mathbf{y})e^{-ik_{b}\boldsymbol{{\theta}}_{1}\cdot \mathbf{y}}(NJ_{0}(|k_{a}\mathbf{x}-k_{b}\mathbf{y}|)+{\mathcal{E}}(k_{b},k_{a},\mathbf{x}))\text{d}\mathbf{y}\\[12.0pt] \displaystyle \int _{D}{\mathcal{O}}(\mathbf{y})e^{-ik_{b}\boldsymbol{{\theta}}_{2}\cdot \mathbf{y}}(NJ_{0}(|k_{a}\mathbf{x}-k_{b}\mathbf{y}|)+{\mathcal{E}}(k_{b},k_{a},\mathbf{x}))\text{d}\mathbf{y}\\ \vdots \\ \displaystyle \int _{D}{\mathcal{O}}(\mathbf{y})e^{-ik_{b}\boldsymbol{{\theta}}_{N}\cdot \mathbf{y}}(NJ_{0}(|k_{a}\mathbf{x}-k_{b}\mathbf{y}|)+{\mathcal{E}}(k_{b},k_{a},\mathbf{x}))\text{d}\mathbf{y}\end{array}\right]∼\cdot ∼\left[\begin{array}{@{}c@{}}e^{ik_{a}\boldsymbol{{\theta}}_{1}\cdot \mathbf{x}}\\ e^{ik_{a}\boldsymbol{{\theta}}_{2}\cdot \mathbf{x}}\\ \vdots \\ e^{ik_{a}\boldsymbol{{\theta}}_{N}\cdot \mathbf{x}}\end{array}\right].\nonumber\end{eqnarray}$ Now, applying (9) again, we can evaluate that $\begin{eqnarray}\displaystyle \text{} & \text{} & \displaystyle \mathop{\sum }_{n=1}^{N}\int _{D}{\mathcal{O}}(\mathbf{y})e^{i\boldsymbol{{\theta}}_{n}\cdot (k_{a}\mathbf{x}-k_{b}\mathbf{y})}(NJ_{0}(|k_{a}\mathbf{x}-k_{b}\mathbf{y}|)+{\mathcal{E}}(k_{b},k_{a},\mathbf{x})-e^{i\boldsymbol{{\theta}}_{n}\cdot (k_{a}\mathbf{x}-k_{b}\mathbf{y})})\text{d}\mathbf{y}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =\int _{D}{\mathcal{O}}(\mathbf{y})\left[(NJ_{0}(|k_{a}\mathbf{x}-k_{b}\mathbf{y}|)+{\mathcal{E}}(k_{b},k_{a},\mathbf{x}))^{2}\text{d}\mathbf{y}-\int _{D}\mathop{\sum }_{n=1}^{N}e^{2i\boldsymbol{{\theta}}_{n}\cdot (k_{a}\mathbf{x}-k_{b}\mathbf{y})}\right]\text{d}\mathbf{y}\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle \quad =\int _{D}{\mathcal{O}}(\mathbf{y})[(NJ_{0}(|k_{a}\mathbf{x}-k_{b}\mathbf{y}|)+{\mathcal{E}}(k_{b},k_{a},\mathbf{x}))^{2}-(NJ_{0}(|2k_{a}\mathbf{x}-k_{b}\mathbf{y}|)+{\mathcal{E}}(2k_{b},2k_{a},\mathbf{x}))]\text{d}\mathbf{y}.\nonumber\end{eqnarray}$ Hence, $\begin{eqnarray}\displaystyle \overline{\mathbf{W}(k_{b},\mathbf{x})}^{T}\mathbb{K}\overline{\mathbf{W}(k_{b},\mathbf{x})}\text{} & =\text{} & \displaystyle \frac{Ne^{2iRk_{b}}{\omega}{\varepsilon}_{b}}{32Rk_{b}{\pi}}\left[\int _{D}{\mathcal{O}}(\mathbf{y})\left(J_{0}(|k_{a}\mathbf{x}-k_{b}\mathbf{y}|)+\frac{{\mathcal{E}}(k_{b},k_{a},\mathbf{x})}{N}\right)^{2}\text{d}\mathbf{y}\right.\nonumber\\ \displaystyle \text{} & \text{} & \displaystyle -\left.\frac{1}{N}\int _{D}{\mathcal{O}}(\mathbf{y})\left(J_{0}(2|k_{a}\mathbf{x}-k_{b}\mathbf{y}|)+\frac{{\mathcal{E}}(2k_{b},2k_{a},\mathbf{x})}{N}\right)\text{d}\mathbf{y}\right].\nonumber\end{eqnarray}$ With this, we can obtain (7). □

According to the result in Theorem 3.1, we can give some physical explanations.

TheoremC . (Appearance of Inaccurate Results).

Since J₀(|k_ax − k_by|) = 1 and ${\mathcal{E}}(k_{b},k_{a},\mathbf{x})=∼0$ when x = (k_b∕k_a)y, y ∈ D, the inaccurate location and size of D will be retrieved through the map of $\mathfrak{F}_{\text{KIR}}(k_{a},\mathbf{x})$ because J₀(0) = 1 and J_p(0) = 0 for nonzero p. Thus, the factor ${\mathcal{E}}(k_{b},k_{a},\mathbf{x})$ is disturbing the imaging performance. This is the theoretical reason why the accurate location and size of D cannot be retrieved without the background information. Notice that although the exact location and size of D cannot be identified, the outline shape of D can be recognized.

TheoremD . (Imaging Quality Against to the Applied Background Permittivity).

Assume that only the exact value of 𝜀_b is unknown, i.e., 𝜎_a = 𝜎_b. Then, since the condition 𝜔𝜀_b ≫ 𝜎_b holds, if 𝜔𝜀_a ≫ 𝜎_b then the identified location becomes $\begin{eqnarray}\displaystyle \mathbf{x}=\left(\frac{k_{b}}{k_{a}}\right)\mathbf{y}=\sqrt{\frac{{\omega}{\varepsilon}_{b}-i{\sigma}_{b}}{{\omega}{\varepsilon}_{a}-i{\sigma}_{b}}}\approx \sqrt{\frac{{\varepsilon}_{b}}{{\varepsilon}_{a}}}\mathbf{y},\quad \mathbf{y}\in D. & & \displaystyle \nonumber\end{eqnarray}$ Then, the locations of retrieved objects will be concentrated at the origin and their sizes will be smaller than the true ones if 𝜀_a > 𝜀_b. If 𝜀_a < 𝜀_b then opposite to the previous examination, retrieved locations of retrieved objects will be far from the origin and their sizes are larger then then the true ones.

TheoremE . (Imaging Quality Against to the Applied Background Conductivity).

Assume that only the exact value of 𝜎_b is unknown, i.e., 𝜀_a = 𝜀_b. Then, if the conditions 𝜔𝜀_b ≫ 𝜎_a and 𝜔𝜀_b ≫ 𝜎_b hold, the identified location becomes $\begin{eqnarray}\displaystyle \mathbf{x}=\left(\frac{k_{b}}{k_{a}}\right)\mathbf{y}=\sqrt{\frac{{\omega}{\varepsilon}_{b}-i{\sigma}_{b}}{{\omega}{\varepsilon}_{b}-i{\sigma}_{a}}}\approx \sqrt{\frac{{\omega}{\varepsilon}_{b}}{{\omega}{\varepsilon}_{b}}}\mathbf{y}=\mathbf{y},\quad \mathbf{y}\in D. & & \displaystyle \nonumber\end{eqnarray}$ Hence, although exact value of 𝜎_b is unknown, it will be possible to retrieve the objects by selecting a sufficiently small value of 𝜎_a. However, if the 𝜔𝜀_b ≫ 𝜎_a condition no longer holds, it will be very difficult to retrieve the objects.

TheoremF . (Imaging Quality Against to the Antenna Arrangement).

For a more general case, one can assume that only partial or the whole most outside boundary of an object is surrounded by the antennas. Which is regarded as the limited-aperture measurement configuration case. In this case, total number of antennas N becomes small so that the imaging quality becomes poor because the influence of disturbing factor ${\mathcal{E}}(k_{b},k_{a},\mathbf{x})/N$ is getting grow. We refer to [16] for related a study about subspace migration imaging technique.

4. Simulation results and discussions

In this section, we exhibit the simulation results to support the result in Theorem 3.1. To this end, we select N = 16 antennas A_n uniformly arranged on the circle of radius 0.09 m. We set 𝛺 as a square region (−0.1 m, 0.1 m)² with material properties (𝜀_b, 𝜎_b) = (20, 0. 2 S∕m) at f = 1.2 GHz and select two anomalies, D₁ and D₂ as circles with the same radii 0.01 m. The center and material properties of D₁ are set to y₁ = (0. 01 m, 0. 01 m) and (𝜀₁, 𝜎₁) = (55, 1. 2 S∕m), respectively, and those of D₂ are set to y₂ = (−0. 04 m, −0. 02 m) and (𝜀₂, 𝜎₂) = (45, 1. 0 S∕m), respectively. With this, the measurement data S_scat(n, m) are generated by the CST STUDIO SUITE.

Figure 2 shows the maps of $\mathfrak{F}_{\text{KIR}}(k_{a},\mathbf{x})$ when the exact value of 𝜀_b is unknown, that is, $k_a=\omega\sqrt{\mu_b(\varepsilon_a-i\sigma_b/\omega)}$ for some 𝜀_a. Notice that if 𝜀_a > 𝜀_b then the retrieved anomalies are concentrated at the origin, and their sizes are smaller than the true ones. In contrast, the identified anomalies are located far from the origin, and their retrieved sizes are larger than the true ones when 𝜀_a < 𝜀_b. However, it is extremely difficult to recognize the existence of D₁ ∪ D₂ because of the appearance of several artifacts when 𝜀_a ≫ 𝜀_b and the identified locations are too far away from the origin when 𝜀_a ≪ 𝜀_b. Therefore, although almost every information of the background is known, it is extremely difficult to retrieve the accurate location and shape of anomalies unless the exact value of 𝜀_b is known.

Fig. 2.

Maps of $\mathfrak{F}_{\text{KIR}}(k_{a},\mathbf{x})$ at f =1.2 GHz. Violet-colored circles describe the boundary of anomalies.

Figure 3 shows maps of $\mathfrak{F}_{\text{KIR}}(k_{a},\mathbf{x})$ when exact value of 𝜎_b is unknown, that is, $k_a=\omega\sqrt{\mu_b(\varepsilon_b-i\sigma_a/\omega)}$ for some 𝜎_a. Opposite to the results in Fig. 2, the location and shape of D₁ ∪ D₂ were successfully retrieved if 𝜎_a ≪ 𝜎_b, that is, when the value of 𝜎_a is too small; however, it is extremely difficult to recognize D₁ ∪ D₂ because of the appearance of a huge artifacts with a large magnitude if 𝜎_a ≫ 𝜎_b. Hence, we can say that it will be possible to retrieve the almost accurate shape and location of anomalies by selecting an extremely small (close to zero) value of 𝜎_a when only the exact value of 𝜎_b is unknown.

Fig. 3.

Maps of $\mathfrak{F}_{\text{KIR}}(k_{a},\mathbf{x})$ at f = 1.2 GHz. Violet-colored circles describe the boundary of anomalies.

5. Conclusion

Based on the integral representation formula for the scattering parameter in the presence of small anomaly, we designed an imaging function of the KM from the scattering matrix. By a careful analysis, we show that the designed imaging function can be expressed by an infinite series of the Bessel functions of the first kind, material properties, and antenna arrangement when the complete information of the background is unknown. Based on the theoretical result, we successfully explained why the accurate location and shape of anomalies cannot be retrieved when the background information is unknown. Correspondingly, an investigation of an algorithm for retrieving exact value of background wavenumber is needed for a successful retrieving, and this will be the forthcoming research. Finally, this paper concerned the two-dimensional microwave imaging. We believe that the analysis and numerical simulations can be extended to the three-dimensional microwave imaging and this will be an interesting research topic.

Footnotes

Acknowledgements

The author would like to acknowledge anonymous reviewers for their comments that help to increase the quality of the paper. This research was supported by the National Research Foundation of Korea (NRF) grant funded by the Korea government (MSIT) (NRF-2020R1A2C1A01005221).

References

Ammari

and Kang

, Reconstruction of Small Inhomogeneities from Boundary Measurements, Lecture Notes in Mathematics, Vol. 1846, Springer-Verlag, Berlin, 2004.

Aster

R.C.

Borchers

and Thurber

C.H.

, Parameter Estimation and Inverse Problems, 2nd edn, Elsevier, 2013.

Bleistein

Cohen

and Stockwell Jr

J.S.

, Mathematics of Multidimensional Seismic Imaging, Migration, and Inversion, Interdisciplinary Applied Mathematics, Vol. 13, Springer, New York, 2001.

Colton

and Kress

, Inverse Acoustic and Electromagnetic Scattering Problems, Mathematics and Applications Series, Vol. 93, Springer, New York, 1998.

Zhdanov

M.S.

, Geophysical Inverse Theory and Regularization Problems, Elsevier, Amsterdam, The Netherlands, 2002.

Ahn

C.Y.

and Park

W.-K.

, Kirchhoff migration for identifying unknown targets surrounded by random scatterers, Appl. Sci.9 (2019), 4446.

Ammari

Garnier

Kang

Park

W.-K.

and Sølna

, Imaging schemes for perfectly conducting cracks, SIAM J. Appl. Math.71 (2011), 68–91.

Lee

K.-J.

Son

S.-H.

and Park

W.-K.

, A real-time microwave imaging of unknown anomaly with and without diagonal elements of scattering matrix, Results Phys.17 (2020), 103104.

Park

W.-K.

, A novel study on the orthogonality sampling method in microwave imaging without background information, J. Appl. Math. Lett.145 (2023), 108766.

10.

Son

S.-H.

and Park

W.-K.

, Application of the bifocusing method in microwave imaging without background information, J. Korean Soc. Ind. Appl. Math.27 (2023), 109–122.

11.

Haynes

Stang

and Moghaddam

, Real-time microwave imaging of differential temperature for thermal therapy monitoring, IEEE Trans. Biomed. Eng.61 (2014), 1787–1797.

12.

Park

W.-K.

, Real-time microwave imaging of unknown anomalies via scattering matrix, Mech. Syst. Signal Proc.118 (2019), 658–674.

13.

Park

W.-K.

, Application of MUSIC algorithm in real-world microwave imaging of unknown anomalies from scattering matrix, Mech. Syst. Signal Proc.153 (2021), 107501.

14.

Park

W.-K.

Kim

H.P.

Lee

K.-J.

and Son

S.-H.

, MUSIC algorithm for location searching of dielectric anomalies from S-parameters using microwave imaging, J. Comput. Phys.348 (2017), 259–270.

15.

Slaney

Kak

A.C.

and Larsen

L.E.

, Limitations of imaging with first-order diffraction tomography, IEEE Trans. Microwave Theory Tech.32 (1984), 860–874.

16.

Park

W.-K.

, Real-time detection of small anomaly from limited-aperture measurements in real-world microwave imaging, Mech. Syst. Signal Proc.171 (2022), 108937.