Learning-Enhanced Variational Regularization for Electrical Impedance Tomography via Calder\'on's Method

Abstract

This paper aims to numerically solve the two-dimensional electrical impedance tomography (EIT) with Cauchy data. This inverse problem is highly challenging due to its severe ill-posed nature and strong nonlinearity, which necessitates appropriate regularization strategies. Choosing a regularization approach that effectively incorporates the \textit{a priori} information of the conductivity distribution (or its contrast) is therefore essential. In this work, we propose a deep learning-based method to capture the \textit{a priori} information about the shape and location of the unknown contrast using Calder\'on's method. The learned \textit{a priori} information is then used to construct the regularization functional of the variational regularization method for solving the inverse problem. The resulting regularized variational problem for EIT reconstruction is then solved using the Gauss-Newton method. Extensive numerical experiments demonstrate that the proposed inversion algorithm achieves accurate reconstruction results, even in high-contrast cases, and exhibits strong generalization capabilities. Additionally, some stability and convergence analysis of the variational regularization method underscores the importance of incorporating \textit{a priori} information about the support of the unknown contrast.