Qualitative reconstruction methods for imaging interior Robin interfaces in EIT from Robin-to-Dirichlet data

TL;DR

This paper addresses inverse shape reconstruction in electrical impedance tomography where interior corrosion is modeled by Robin transmission conditions and data are given by the Robin-to-Dirichlet map difference $(M-M_0)$. It adapts two qualitative, non-iterative methods—the Linear Sampling Method and the Regularized Factorization Method—to the Robin-Dirichlet framework, deriving analytical characterizations that connect RtD data to the interior region $D$ and implementing them with regularization. A symmetric factorization-based analysis and a TTLS regularization extension yield robust imaging functionals that identify the location and boundary of $D$ from boundary measurements. Numerical results in the unit disk validate the approaches under noise, showing reliable localization and sharper delineation with RFM, and demonstrate practical potential for nondestructive testing where interior Robin interfaces occur.

Abstract

We consider an inverse shape problem arising in electrical impedance tomography (EIT) for nondestructive testing, in which interior defects are modeled through Robin transmission conditions. Unlike classical formulations, we impose Robin boundary conditions on both the exterior measurement surface and the interior interface, and use the Robin-to-Dirichlet (RtD) map as the available data. Within this setting, we develop qualitative (non-iterative) reconstruction methods based on the Linear Sampling Method (LSM) and the Regularized Factorization Method (RFM), and derive new analytical characterizations that enable these methods to identify interior regions. We further propose a numerical implementation that incorporates regularization strategies and demonstrate, through experiments, that the methods reliably reconstruct interior regions of interest.

Figures (6)

• Figure 1: Reconstruction of a circular region with $\rho=0.4$ marked with a dashed red line, via both LSM and RFM methods. In this case, there is no noise added in the matrix (i.e., $\delta=0$).
• Figure 2: Reconstruction of a circular region (radius $\rho = 0.4$) shown by the dashed red line, with conductivity $\sigma = 1$, using both the LSM and RFM methods. The data matrix includes $1\%$ additive noise ($\delta = 0.01$).
• Figure 3: Reconstruction of a circular region (radius $\rho = 0.4$) shown by the dashed red line, where the exterior conductivity is $\sigma_{\text{out}} = 1$ and the interior conductivity is $\sigma_{\text{in}} = 10$. The reconstruction is obtained using both the LSM and RFM methods, with $1\%$ additive noise included in the data matrix ($\delta = 0.01$).
• Figure 4: Reconstruction of a circular region (radius $\rho = 0.4$) shown by the dashed red line, where the exterior conductivity is $\sigma_{\text{out}} = 1$ and the interior conductivity is $\sigma_{\text{in}} = 10$. The reconstruction is obtained using both the LSM and RFM methods, with $1\%$ additive noise included in the data matrix ($\delta = 0.01$). The boundary data are sampled at $N = 64$ equiangular points.
• Figure 5: Reconstruction of a circular region (radius $\rho = 0.1$) shown by the dashed red line, with conductivity $\sigma = 1$, using both the LSM and RFM methods. The data matrix includes $1\%$ additive noise ($\delta = 0.01$).

Theorems & Definitions (23)

• Theorem 2.1
• Theorem 3.1
• proof
• Theorem 3.2
• proof
• Theorem 3.3
• proof
• Theorem 3.4
• proof
• Theorem 3.5