Stability of Electrical Impedance Tomography with Anisotropies and its Application to the Deep Calde\'on Method

In this work, we establish new conditional Lipschitz stability results for electrical impedance tomography (EIT) with anisotropies, of recovering the conductivity in a conformal class of a known anisotropic conductivity in both two- and multi-dimensional cases. Then we employ the stability theory to understand the property of the deep Calder\'on method, one deep learning-based technique for image reconstruction in EIT that has shown promising empirical results, but still lacks theoretical underpinnings. Specifically, we relate the stability theory to the robustness of the method with the proper choice of the training data, and present numerical results in two-dimension to complement the theoretical analysis.