含各向异性的电阻抗断层成像的稳定性及其在深度卡尔德隆方法中的应用
Stability of Electrical Impedance Tomography with Anisotropies and its Application to the Deep Caldeón Method
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中文总结 AI 辅助
本研究建立含各向异性的电阻抗断层成像（EIT）的条件利普希茨稳定性结果，将其应用于分析深度卡尔德隆方法的鲁棒性，辅以二维数值结果补充理论分析，为该深度学习EIT重建技术提供理论支撑。
中文摘要 AI 辅助
在本研究中，我们针对含各向异性的电阻抗断层成像（EIT）建立了新的条件利普希茨稳定性结果，可在二维和多维情况下，于已知各向异性电导率的共形类中重建电导率。随后，我们运用该稳定性理论分析深度卡尔德隆方法的性质，该方法是一种基于深度学习的EIT图像重建技术，虽已取得良好的实证结果，但仍缺乏理论支撑。具体而言，我们将稳定性理论与该方法在训练数据恰当选择下的鲁棒性相关联，并给出二维数值结果以补充理论分析。
英文摘要
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 Calderón 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.
