On an inverse problem for anisotropic conductivity in the plane

TL;DR

This paper presents an explicit method to reconstruct a domain, isotropic conductivity, and a boundary quasiconformal map from the Dirichlet-to-Neumann operator in an anisotropic conductivity problem in the plane.

Contribution

It introduces a novel explicit procedure to recover isotropic conductivity and domain geometry from boundary measurements in anisotropic inverse problems.

Findings

Unique reconstruction of domain and conductivity from boundary data

Explicit algorithm for transforming anisotropic to isotropic conductivity

Establishment of a quasiconformal map linking original and reconstructed domains

Abstract

Let $\hat \Omega \subset \mathbb R^2$ be a bounded domain with smooth boundary and $\hat \sigma$ a smooth anisotropic conductivity on $\hat \Omega$. Starting from the Dirichlet-to-Neumann operator $\Lambda_{\hat \sigma}$ on $\partial \hat \Omega$, we give an explicit procedure to find a unique domain $\Omega$, an isotropic conductivity $\sigma$ on $\Omega$ and the boundary values of a quasiconformal diffeomorphism $F:\hat \Omega \to \Omega$ which transforms $\hat \sigma$ into $\sigma$.