[論文レビュー] Qualitative reconstruction methods for imaging interior Robin interfaces in EIT from Robin-to-Dirichlet data

Rafael Ceja Ayala, Malena I. Español+1 more|arXiv (Cornell University)|Jan 15, 2026

Electrical and Bioimpedance Tomography被引用数 0

ひとこと要約

This paper develops two qualitative, non-iterative reconstruction methods (LSM and RFM) for locating interior Robin interfaces in EIT using Robin-to-Dirichlet data, with theoretical guarantees and numerical validation.

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.

研究の動機と目的

• Motivate nondestructive testing applications where interior defects are modeled by Robin transmission conditions.
• Formulate the inverse shape problem using the Robin-to-Dirichlet map on both the exterior surface and interior interface.
• Develop non-iterative, qualitative reconstruction methods that identify interior regions from boundary data.
• Provide analytical results proving unique determination of interior regions and implement robust numerical schemes.

提案手法

• Formulate the direct and inverse problems with Robin boundary conditions on both the exterior boundary and interior interface.
• Define the data operator as the difference (M - M0) between Robin-to-Dirichlet maps with and without the interior region.
• Derive a first factorization (M - M0) = G S to enable Linear Sampling Method (LSM) analysis.
• Derive a more detailed, symmetric factorization (M - M0) = S* T S to enable the Regularized Factorization Method (RFM).
• Introduce and analyze the Robin Green’s function G(., z) to test sampling points.
• Propose imaging functionals for LSM and RFM using regularization schemes (Spectral cutoff, Tikhonov, TTLS) and demonstrate binary behavior (inside vs outside D).

実験結果

リサーチクエスチョン

• RQ1 Can the Robin-to-Dirichlet data determine the location and shape of an interior Robin interface uniquely?
• RQ2 How can LSM and RFM be adapted to the Robin-Dirichlet framework to reconstruct interior regions from boundary measurements?
• RQ3 What are the analytical properties (injectivity, dense range, compactness) of the associated operators that enable qualitative reconstruction?
• RQ4 How can regularization be employed to robustly implement LSM and RFM in this setting?
• RQ5 Do numerical experiments in 2D validate the proposed methods for interior interface reconstruction?

主な発見

• The RtD map difference (M - M0) uniquely determines the location and shape of the interior region under Robin transmission conditions.
• LSM and RFM can be adapted to the Robin-Dirichlet setting via operator factorizations of (M - M0) and yield imaging functionals with binary behavior.
• A first factorization (M - M0) = G S connects boundary data to the interior jump across ∂D, enabling LSM testing of sampling points.
• A symmetric factorization (M - M0) = S* T S enables the Regularized Factorization Method to provide a more robust reconstruction of D.
• The paper proves injectivity of S and of G, and shows dense range for S, supporting the theoretical basis of the methods.
• Numerical experiments in the unit disk validate the theoretical imaging functionals and demonstrate reliable reconstruction of interior regions.