Conductivity reconstruction from power density data in limited view

TL;DR

This paper investigates reconstructing electrical conductivity inside a body using limited boundary data in Acousto-Electric tomography, demonstrating the feasibility of the approach and adapting existing methods for partial boundary conditions.

Contribution

It extends the unique continuation principle and Runge approximation to limited boundary data, enabling conductivity reconstruction with fewer boundary conditions.

Findings

The Runge approximation property holds for limited boundary data.

Existence of boundary conditions ensuring non-vanishing gradients.

Numerical implementation shows potential and limitations of the method with two boundary conditions.

Abstract

In Acousto-Electric tomography, the objective is to extract information about the interior electrical conductivity in a physical body from knowledge of the interior power density data generated from prescribed boundary conditions for the governing elliptic partial differential equation. In this note, we consider the problem when the controlled boundary conditions are applied only on a small subset of the full boundary. We demonstrate using the unique continuation principle that the Runge approximation property is valid also for this special case of limited view data. As a consequence, we guarantee the existence of finitely many boundary conditions such that the corresponding solutions locally satisfy a non-vanishing gradient condition. This condition is essential for conductivity reconstruction from power density data. In addition, we adapt an existing reconstruction method intended for the full data situation to our setting. We implement the method numerically and investigate the opportunities and shortcomings when reconstructing from two fixed boundary conditions.