# Kirchhoff Migration for Identifying Unknown Targets Surrounded by Random Scatterers
## 5 Further Result: Imaging of Extended Dielectric Targets in an Inhomogeneous Medium
So, motivated by [12], we consider the following normalized boundary measurement data \(\mathrm{B}_{\mathrm{meas}}(n,n^{\prime})\)  
\[\mathrm{B}_{\mathrm{meas}}(n,n^{\prime}):=\int_{\partial\Omega}\left(u^{(n)}( \mathbf{x})-u_{0}^{(n)}(\mathbf{x})\right)\frac{\partial v^{(n^{\prime})}( \mathbf{x})}{\partial\mathbf{\
u}(\mathbf{x})}dS(\mathbf{x})=\sum_{m=1}^{4} \int_{\partial\mathbf{A}_{m}}e^{-ik\theta_{n^{\prime}}\cdot\mathbf{y}}\phi^{( n)}(\mathbf{y})d\mathbf{y}+\mathcal{E}(\varepsilon_{0}), \tag{10}\]  
where \(v^{(n^{\prime})}(\mathbf{x})=e^{-ik\theta_{n^{\prime}}\cdot\mathbf{x}}\). Now, let us consider the following MSR matrix  
\[\mathbb{K}=\left[ \begin{array}{cccc}\mathrm{B}_{\mathrm{meas}}(1,1)& \mathrm{B}_{\mathrm{meas}}(1,2)&\cdots&\mathrm{B}_{\mathrm{meas}}(1,N)\\ \mathrm{B}_{\mathrm{meas}}(2,1)&\mathrm{B}_{\mathrm{meas}}(2,2)&\cdots& \mathrm{B}_{\mathrm{meas}}(2,N)\\ \vdots&\vdots&\ddots&\vdots\\ \mathrm{B}_{\mathrm{meas}}(N,1)&\mathrm{B}_{\mathrm{meas}}(N,2)&\cdots& \mathrm{B}_{\mathrm{meas}}(N,N)\end{array} \right].\]  
Then, \(\mathbb{K}\) can be decomposed as (see [14] for instance)  
\[\mathbb{K}\approx\sum_{m=1}^{4}\int_{\partial\mathbf{A}_{m}}\mathbb{E}( \mathbf{y})\mathbb{D}(\mathbf{y})d\mathbf{y},\]  
where \(\mathbb{E}(\mathbf{y})\) and \(\mathbb{D}(\mathbf{y})\) denote the illumination and resulting density vectors  
\[\mathbb{E}(\mathbf{y})=\left[e^{-ik\theta_{1}\cdot\mathbf{y}},e^{-ik\theta_{2 }\cdot\mathbf{y}},\cdots,e^{-ik\theta_{N}\cdot\mathbf{y}}\right]^{T}\quad \text{and}\quad\mathbb{D}(\mathbf{y})=\left[\phi^{(1)}(\mathbf{y}),\phi^{(2)} (\mathbf{y}),\cdots,\phi^{(N)}(\mathbf{y})\right], \tag{11}\]  
respectively. Based on [14], it should be noted that the range of \(\mathbb{K}\) is determined by the span of the \(\mathbb{E}(\mathbf{y})\) corresponding to \(\mathbf{A}_{m}\). This means that the signal subspace can be determined by selecting the singular vectors associated with the nonzero singular values of \(\mathbb{K}\). Thus, by taking the test vector \(\mathbf{W}(\mathbf{x})\) of (5) as  
\[\mathbf{W}(\mathbf{x})=\frac{1}{\sqrt{N}}\bigg{[}e^{-ik\theta_{1}\cdot \mathbf{x}},e^{-ik\theta_{2}\cdot\mathbf{x}},\cdots,e^{-ik\theta_{N}\cdot \mathbf{x}}\bigg{]}^{T},\]  
imaging functions \(\mathfrak{F}_{\mathrm{KIR}}(\mathbf{x})\) and \(\mathfrak{F}_{\mathrm{SUB}}(\mathbf{x})\) can be defined similarly with (4) and (7), respectively. **Example 5** (Imaging performances of Kirchhoff and subspace migrations). : _Figure 6 shows the imaging results via \(\mathfrak{F}_{\mathrm{KIR}}(\mathbf{x})\) and \(\mathfrak{F}_{\mathrm{SUB}}(\mathbf{x})\) when \(\lambda=0.3\). We also mention that 20 dB Gaussian random noise is added to the unperturbed data \(\mathrm{B}_{\mathrm{meas}}(n,n^{\prime})\), \(n,n^{\prime}=1,2,\cdots,N\). For performing subspace migration, first \(15\) singular values are selected. In contrast to the traditional results, it is very hard to identify the outline shapes of every inhomogeneity via \(\mathfrak{F}_{\mathrm{SUB}}(\mathbf{x})\) while it is possible to identify the outline shape of every inhomogeneity. Althoughthe size and permittivity of \(\mathbf{A}_{4}\) were both very small, the existence of \(\mathbf{A}_{4}\) was successfully recognized. On the basis of simulation results, we conclude that the result via Kirchhoff migration is better than the one via subspace migration for addressing problems of this type._  
**Example 6** (Influence of total number of incident fields). : _Based on the condition in Theorem 1, let us examine the effect of total number \(N\) of incident fields. Figure 7 exhibits the maps of \(\mathfrak{F}_{\mathrm{KIR}}(\mathbf{x})\) and \(\mathfrak{F}_{\mathrm{SUB}}(\mathbf{x})\) for \(N=32\) and \(64\). Similar to the traditional researches, it appears that small value of \(N\) might be a reason of poor result while large value of \(N\) will guarantee good imaging performance of Kirchhoff migration.