3D Electrical Impedance Tomography reconstructions from simulated electrode data using direct inversion \begin{document}$ \mathbf{t}^{\rm{{\textbf{exp}}}} $\end{document} and Calderón methods

The first numerical implementation of a \begin{document}$ \mathbf{t}^{\rm{{\textbf{exp}}}} $\end{document} method in 3D using simulated electrode data is presented. Results are compared to Calderon's method as well as more common TV and smoothness regularization-based methods. The \begin{document}$ \mathbf{t}^{\rm{{\textbf{exp}}}} $\end{document} method for EIT is based on tailor-made non-linear Fourier transforms involving the measured current and voltage data. Low-pass filtering in the non-linear Fourier domain is used to stabilize the reconstruction process. In 2D, \begin{document}$ \mathbf{t}^{\rm{{\textbf{exp}}}} $\end{document} methods have shown great promise for providing robust real-time absolute and time-difference conductivity reconstructions but have yet to be used on practical electrode data in 3D, until now. Results are presented for simulated data for conductivity and permittivity with disjoint non-radially symmetric targets on spherical domains and noisy voltage data. The 3D \begin{document}$ \mathbf{t}^{\rm{{\textbf{exp}}}} $\end{document} and Calderon methods are demonstrated to provide comparable quality to their 2D counterparts and hold promise for real-time reconstructions due to their fast, non-optimized, computational cost.