## 2 Direct Scattering Problem and Subspace Migration
### Subspace Migration Imaging Algorithm  
Here, we apply the (1) to introduce the subspace migration algorithm for a direct imaging of \(\Gamma_{m}\) from collected far-field pattern data  
\[\mathbb{F}=\Big{\{}u_{\infty}(\boldsymbol{\theta}_{j},\boldsymbol{\theta}_{l} ):j,l=1,2,\cdots,N,\;\boldsymbol{\theta}_{j}=-\boldsymbol{\theta}_{j}\in \mathbb{S}^{1}_{\text{SUB}}\Big{\}},\]  
where \(\mathbb{S}^{1}_{\text{SUB}}\) denotes a connected, proper subset of \(\mathbb{S}^{1}\) (see Figure 1). With this, we generate the following MSR matrix \(\mathbb{M}\in\mathbb{C}^{N\times N}\) such that  
\[\mathbb{M}= \begin{bmatrix}\Delta S_{1,1}&\Delta S_{1,2}&\cdots&\Delta S_{1,N }\\ \Delta S_{2,1}&\Delta S_{2,2}&\cdots&\Delta S_{2,N}\\ \vdots&\vdots&\ddots&\vdots\\ \Delta S_{N,1}&\Delta S_{N,2}&\cdots&\Delta S_{N,N}\end{bmatrix} ,\quad\Delta S _{j,l}=u_{\infty}(\boldsymbol{\theta}_{j},\boldsymbol{\theta}_{l}).\]  
Based on the (1), each element \(\Delta S_{j,l}\) can be written as  
\[\Delta S_{j,l}\approx\frac{k^{2}\pi(1+i)}{4\sqrt{k\pi}}\sum_{m=1}^{M}\alpha_{ m}^{2}\bigg{(}\frac{\varepsilon_{m}-\varepsilon_{\text{b}}}{\sqrt{ \varepsilon_{\text{b}}\mu_{\text{b}}}}\bigg{)}e^{ik(\boldsymbol{\theta}_{j}+ \boldsymbol{\theta}_{l})\cdot\mathbf{y}_{m}}.\]  
Based on the above representation, we introduce the following unit vector: for each \(\mathbf{x}\in\Omega\),  
\[\mathbf{W}(\mathbf{x})\approx\frac{1}{\sqrt{N}}\bigg{(}e^{ik\boldsymbol{\theta }_{1}\cdot\mathbf{x}},e^{ik\boldsymbol{\theta}_{2}\cdot\mathbf{x}},\cdots,e^{ ik\boldsymbol{\theta}_{N}\cdot\mathbf{x}}\bigg{)}^{T}.\]  
Then, \(\mathbb{M}\) can be decomposed as follows:  
\[\mathbb{M}=\frac{k^{2}\pi(1+i)}{4\sqrt{k\pi\varepsilon_{\text{b}}\mu_{\text{b }}}}\sum_{m=1}^{M}\mathbf{W}(\mathbf{x})\begin{bmatrix}\alpha_{1}^{2}( \varepsilon_{1}-\varepsilon_{\text{b}})&0&\cdots&0\\ 0&\alpha_{2}^{2}(\varepsilon_{2}-\varepsilon_{\text{b}})&\cdots&0\\ \vdots&\vdots&\ddots&\vdots\\ 0&0&\cdots&\alpha_{M}^{2}(\varepsilon_{M}-\varepsilon_{\text{b}})where \(\sigma_{n}\) denotes the \(n\)th singular value, \(\mathbf{U}_{n}\) and \(\mathbf{V}_{n}\) are the \(n\)th left- and right-singular vectors of \(\mathbb{M}\), respectively. Since the following relation already be examined in [48]  
\[\langle\mathbf{U}_{n},\mathbf{W}(\mathbf{x})\rangle\approx 1,\quad \langle\overline{\mathbf{\
abla}}_{n},\mathbf{W}(\mathbf{x})\rangle\approx 1 \quad\text{if}\quad\mathbf{x}=\mathbf{y}_{m},\] \[0<\langle\mathbf{U}_{n},\mathbf{W}(\mathbf{x})\rangle<1,\quad 0 <\langle\overline{\mathbf{\
abla}}_{n},\mathbf{W}(\mathbf{x})\rangle<1\quad \text{if}\quad\mathbf{x}\
eq\mathbf{y}_{m},\]  
we can introduce the following imaging function: for each \(\mathbf{x}\in\Omega\)  
\[\mathfrak{F}_{\text{SUB}}(\mathbf{x})=\bigg{|}\sum_{n=1}^{M}\langle\mathbf{U} _{n},\mathbf{W}(\mathbf{x})\rangle\langle\overline{\mathbf{\
abla}}_{n}, \mathbf{W}(\mathbf{x})\rangle\bigg{|}.\]  
Here, \(\langle\mathbf{U},\mathbf{V}\rangle=\mathbf{U}\cdot\overline{\mathbf{V}}\). Then, \(\mathfrak{F}_{\text{SUB}}(\mathbf{x})\approx 1\) when \(\mathbf{x}=\mathbf{y}_{m}\in\Gamma_{m}\) and \(\mathfrak{F}_{\text{SUB}}(\mathbf{x})<1\) when \(\mathbf{x}\
eq\Omega\backslash\Gamma\) so that the location \(\mathbf{y}_{m}\) can be identified through the map of \(\mathfrak{F}_{\text{SUB}}(\mathbf{x})\). It is worth mentioning that subspace migration can be applied to the full- and limited-view inverse scattering problems. However, one cannot retrieve good results in the limited-view problem when total number \(N\) is small, while good results can be guaranteed in a full-view problem. A theoretical reason is as follows. Refer to [43]:  
**Theorem 1**. : _Let \(\mathbf{x}-\mathbf{y}_{m}=|\mathbf{x}-\mathbf{y}_{m}|(\cos\phi_{m},\sin\phi_{ m})\) and \(\boldsymbol{\theta}_{n}=(\cos\theta_{n},\sin\theta_{n})\).