$\gamma(z)$ can be recovered from the CGO solutions at $k=0$ by first reconstructing $Q$ $$\label{eq:MtoQ}
Q_{12}(z)=\frac{\operatorname{\operatorname{\partial}_z}\left[M_{11}(z,0)+M_{12}(z,0)\right]}{M_{22}(z,0)+M_{21}(z,0)},\qquad \text{and}\qquad Q_{21}(z)=\frac{\operatorname{{\bar{\partial}}_z}\left[M_{22}(z,0)+M_{21}(z,0)\right]}{M_{11}(z,0)+M_{12}(z,0)},$$ and then undoing the change of variables by computing $$\label{eq:QtoGamma}
\gamma(z)=\exp\left\{-\frac{2}{\pi \bar{z}}\ast Q_{12}(z)\right\} \quad \text{or equivalently }\quad \gamma(z)=\exp\left\{-\frac{2}{\pi {z}}\ast Q_{21}(z)\right\},$$ where $\ast$ denotes convolution in $z$ over ${{\mathbb R}}^2$.

The two formulations of this matrix-based D-bar method differ in their connections of the scattering data to the DN map $\Lambda_\gamma$. In the first approach \[[@Hamilton2012]\], traces of the CGO solutions $\Psi_{12}(z,k)$ and $\Psi_{21}(z,k)$ are computed for $z\in\operatorname{\partial\Omega}$ via $$\label{eq:Fran_Psibndry}
\begin{array}{lcl}
\Psi_{12}(z,k)&=& \int_{\operatorname{\partial\Omega}} \frac{e^{i\bar{k}(k-\zeta)}}{4\pi(z-\zeta)}\left(\Lambda_\gamma-\Lambda_1\right)u_2(\zeta,k)ds(\zeta)\\
\Psi_{21}(z,k)&=& \int_{\operatorname{\partial\Omega}} \overline{\left[\frac{e^{i{k}(k-\zeta)}}{4\pi(z-\zeta)}\right]}\left(\Lambda_\gamma-\Lambda_1\right)u_1(\zeta,k)ds(\zeta).
\end{array}$$ Replacing $u_1$ and $u_2$ by their asymptotic behaviors ($u_1\sim \frac{e^{ikz}}{ik}$ and $u_2\sim \frac{e^{-ik\bar{z}}}{-ik}$) yields the [*Born approximations*]{} as in \[[@Hamilton2017_PhysMeas]\]: $$\label{eq:Fran_PsiExp}
\begin{array}{lcl}
\operatorname{\Psi_{12}^{\mbox{\tiny{\textbf{exp}}}}}(z,k)&=& \int_{\operatorname{\partial\Omega}} \frac{e^{i\bar{k}(k-\zeta)}}{4\pi(z-\zeta)}\left(\Lambda_\gamma-\Lambda_1\right)\left(\frac{e^{-ik\bar{\zeta}}}{-ik}\right)ds(\zeta)\\
\operatorname{\Psi_{21}^{\mbox{\tiny{\textbf{exp}}}}}(z,k)&=& \int_{\operatorname{\partial\Omega}} \overline{\left[\frac{e^{i{k}(k-\zeta)}}{4\pi(z-\zeta)}\right]}\left(\Lambda_\gamma-\Lambda_1\right)\left(\frac{e^{ik\zeta}}{ik}\right)ds(\zeta),
\end{array}$$ and corresponding scattering data $$\label{eq:Fran_scatPsiExp}
\begin{array}{lcl}
S^{\mbox{\tiny{$\Psi$exp}}}_{12}(k)&=&\frac{i}{2\pi}\int_{\operatorname{\partial\Omega}}e^{-i\bar{k}{z}}\operatorname{\Psi_{12}^{\mbox{\tiny{\textbf{exp}}}}}(z,k){\nu(z)}ds(z)\\
S^{\mbox{\tiny{$\Psi$exp}}}_{21}(k)&=&\frac{i}{2\pi}\int_{\operatorname{\partial\Omega}}e^{i\bar{k}\bar{z}}\operatorname{\Psi_{21}^{\mbox{\tiny{\textbf{exp}}}}}(z,k)\bar{\nu}(z)ds(z).
\end{array}$$

The reconstruction process is then:

1.  Given $[\Lambda_\gamma,\Lambda_1]$, evaluate the approximate CGOs $\left\{\operatorname{\Psi_{12}^{\mbox{\tiny{\textbf{exp}}}}},\; \operatorname{\Psi_{21}^{\mbox{\tiny{\textbf{exp}}}}}\right\}$ from and compute the corresponding approximate scattering data for $\left\{S^{\mbox{\tiny{$\Psi$exp}}}_{12}(k),\;S^{\mbox{\tiny{$\Psi$exp}}}_{21}(k)\right\}$ from , for $0<|k|\leq R$ for some chosen radius $R$.

2.  Solve the $\operatorname{{\bar{\partial}}_k}$ equation , using $S^{\mbox{\tiny{$\Psi$exp}}}(k)$, for each $z\in\Omega$.

3.  Recover the admittivity $\gamma^{\mbox{\tiny{$\Psi$exp}}}$ using the matrix potential $Q^{\mbox{\tiny{$\Psi$exp}}}$ .

The second approach is to re-write the equations for the scattering 