## Appendix A D-Bar Algorithm  
The D-Bar method uses complex geometrical optics (CGO) solutions to solve the conductivity \(\sigma\) inside \(\Omega\), using the DN-map as an input. As a first step, the Laplace Equation (1) is transformed to the time-independent Schrodinger equation, which is achieved through the change of variables. With \(q(z)\;=\;\sigma^{-1/2}\Delta\sigma^{1/2}\) and \(v=\sigma^{1/2}u\), Equation (1) becomes  
\[(-\Delta\;+\;q(z))v\;=\;0. \tag{1}\]  
Using the assumption that \(\sigma\) is constant near the boundary \(\partial\Omega\), the now more abstract potential \(q(z)\) is extended from the domain \(\Omega\) to the whole complex plane with \(q(z)=0\) outside \(\Omega\). This enables the integration along the boundary to be simpler. The CGO solutions can now be used to solve \(q(z)\). After introducing a parameter \(k\), we define the solution by \(\Psi(k,z)\)--both \(\mathbf{k}\) and \(\mathbf{\
abla}\) are complex-valued--which satisfies  
\[(-\Delta\;+\;q(z))\Psi(k,z)\;=\;0. \tag{2}\]  
These CGO solutions were used by [PERSON] in the context of quantum scattering [38]. Subsequently, [PERSON] and [PERSON] used CGO solutions for their solution of the Calderon problem (to determine the conductivity given the DN map) [27]. In order to solve for \(\mathbf{\
abla}\) of Equation (2), we define a function \(\mu(k,z)\) by  
\[\mu(k,z)\;:=\;e^{-jkz}\Psi(k,z). \tag{3}\]  
It is defined this way because \(\mathbf{\
abla}\) is asymptotic to \(e^{ikz}\) and it also satisfies the \(\bar{\mathbf{\delta}}\) (D-Bar) equation, which gave the algorithm its name. It is given in integral form as [28]  
\[\mu(k,z)=1+\frac{1}{(2\pi)^{2}}\int_{\mathbb{R}^{2}}\frac{\mathbf{t}(k)}{(s-k)\bar {k}}e_{-z}\overline{\mu(k,z)}dk_{1}dk_{2}. \tag{4}\]  
Here, \(e_{-z}=e^{-j(kz+\bar{k}z)}\) and \(\mathbf{t}(k)\) is the so-called scattering transform. We recommend reading [28] for further details on the mathematical background. The scattering transform can be regarded as a nonlinear Fourier transformation, i.e., it transforms inputs to a complex frequency-like domain with nonlinear waves. It is defined as  
\[\mathbf{t}(k)\;=\;\int_{\partial\Omega}e^{\bar{k}\bar{z}}(\Lambda_{\sigma}\;-\; \Lambda_{1})\Psi(k,z)ds, \tag{5}\]where \(\Lambda_{1}\) is the DN map for the case of conductivity 1 inside the domain \(\Omega\) and \(ds\) is the arc length of the domain boundary. When the D-Bar equation is solved for \(\mu(k,z)\), one can calculate the conductivity inside the domain by  
\[\lim_{k\to 0}\,\mu(k,z)^{2}\,=\,\sigma(z),\ z\in\Omega \tag{10}\]  
Note that (10) is dependent on the \(z\)-position. Therefore, the D-Bar algorithm can solve the conductivity for specific points inside the domain. The density of reconstruction points can be selected to balance the computation speed and the potential spatial resolution. Another option is to just reconstruct points in a certain region of interest. We recommend reading [28] for a thorough explanation of the mathematical background.  
The steps described above have a disadvantage in the case of real-world measurements. When the data are noisy, the reconstruction results in distortions. One can imagine the space of the scattering transform as a complex frequency-like domain, where \(|k|\) can be interpreted as a kind of frequency. High-frequency (larger \(|k|\)) components provide greater detail in terms of spatial resolution, for example, for sharp edges, but are simultaneously more prone to noise. Thus, the scattering transformation \(t(k)\) is not evaluated in the whole complex plane, but, rather, on a truncated disk of radius \(|k|<=R_{1}\) around the origin. The scattering transformation outside \(R_{1}<|k|\) is set to 0.