# Tree Internal Defected Imaging Using Model-Driven Deep Learning Network
## 2 Method
### Scattering Problem  
The inverse scattering imaging model in this paper is shown in Figure 2, where the incident wave is mainly a two-dimensional plane wave, and the forward process of the electromagnetic wave includes the following two equations. The first is that the interaction form of the wave scatterer satisfies the Lippmann-Schwinger equation [21]:  
\[E^{t}(r)=E^{t}(r)+i\omega\mu_{0}\int_{D}g(r,r^{\prime})\left[-i\omega\varepsilon_ {0}(\varepsilon_{r}(r^{\prime})-1)E^{t}(r^{\prime})\right]dr^{\prime},r\in D \tag{1}\]  
The second is the equivalent current source radiation process of the scattered field:  
\[E^{s}(r)=i\omega\mu_{0}\int_{D}g(r,r^{\prime})\left[-i\omega\varepsilon_{0}( \varepsilon_{r}(r^{\prime})-1)E^{t}(r^{\prime})\right]dr^{\prime},r\in S \tag{2}\]  
Figure 1: The workflow chart of this paper.  
Figure 2: Inverse scattering imaging model.  
where \(\omega=2\pi f\) is the angular frequency, \(\mu_{0}\) is air-permeability, \(g(r,r^{\prime})\) is two-dimensional Green's function, \(\varepsilon_{0}\) is air-permittivity, and \(\varepsilon_{r}(r^{\prime})\) is relative permittivity in the electromagnetic detection model. In Equation (2), \(-i\omega\varepsilon_{0}(\varepsilon_{r}(r)-1)E^{t}(r)\) represents the induced contrast current density in a physical sense, where we define \(J(r)=(\varepsilon_{r}(r)-1)E^{t}(r)\) as the standardized contrast current density. \(k_{0}=\omega\sqrt{\mu_{0}\varepsilon_{0}}\) is the wavenumber of the homogeneous background medium, and \(\varepsilon_{r}(r)-1\) reflects the contrast between xylem and defects \(\chi(r)\).  
\[k_{0}^{2}\int_{D}g\big{(}r,r^{\prime}\big{)}J\big{(}r^{\prime}\big{)}dr^{\prime }=\left\{\begin{array}{l}|G_{S}(J),r\in S\\ |G_{D}(J),r\in D\end{array}\right. \tag{3}\]  
Then the governing equation can be written in two incompatible forms. The first type is called field equations, where the electric field involves two equations:  
\[E^{t}(r)=E^{i}(r)+G_{D}\big{(}\chi E^{t}\big{)},r\in D \tag{4}\]  
\[E^{s}(r)=G_{S}\big{(}\chi E^{t}\big{)},r\in S \tag{5}\]  
The second type is called the source equation, where the current source involves two equations:  
\[J(r)=\chi(r)\Big{[}E^{i}(r)+G_{D}(J)\Big{]},r\in D \tag{6}\]  
\[E^{s}(r)=G_{S}(J),r\in S \tag{7}\]  
The core of the CSI is to process the source integral Equations (6) and (7). The inverse problem is interpreted as the minimization of the objective function, which is a linear combination of the normalized data equation and the normalized state equation. The objective function formula is as follows:  
\[F\big{(}J_{j},\chi,\beta\big{)}=\beta\frac{\sum_{j}\|\chi E_{j}^{i}-J_{j}+\chi G _{D}J_{j}\|_{D}^{2}}{\sum_{j}\|\chi E_{j}^{i}\|_{D}^{2}}+\frac{\sum_{j}\|E_{j }^{s}-G_{S}J_{j}\|_{S}^{2}}{\sum_{j}\|E_{j}^{s}\|_{S}^{2}} \tag{8}\]  
where \(\|\chi E_{j}^{i}-J_{j}+\chi G_{D}J_{j}\|_{D}^{2}\) is the \(\|\cdot\|_{D}\) measure of the right side of the equation minus the left end of the equation in Equation (7), and \(\|E_{j}^{s}-G_{S}J_{j}\|_{S}^{2}\) is the \(\|\cdot\|_{S}\) measure of the right side of the equation minus the left residual of the Equation (6).  
The CSI uses the conjugate gradient descent method to control the iterative process. The contrast source \(J_{j}\) and the contrast \(\chi\) are updated alternately, and the output value of the objective function is gradually reduced and minimized to reach the exact solution.