Inverse diffusion problems with redundant internal information

TL;DR

This paper develops explicit methods for reconstructing a diffusion coefficient in elliptic PDEs using internal measurements, applicable to medical imaging techniques like UMEIT, UMOT, and MREIT.

Contribution

It introduces two novel reconstruction procedures for the diffusion coefficient from internal functionals, generalizing previous methods to higher dimensions and various parameter values.

Findings

Reconstruction via ODE systems for specific internal functionals.

Unique solvability of a linear elliptic system in certain cases.

Applicability to ultrasound and MRI-based imaging modalities.

Abstract

This paper concerns the reconstruction of a scalar diffusion coefficient $\sigma(x)$ from redundant functionals of the form $H_i(x)=\sigma^{2\alpha}(x)|\nabla u_i|^2(x)$ where $\alpha\in\Rm$ and $u_i$ is a solution of the elliptic problem $\nabla\cdot \sigma \nabla u_i=0$ for $1\leq i\leq I$. The case $\alpha=\frac12$ is used to model measurements obtained from modulating a domain of interest by ultrasound and finds applications in ultrasound modulated electrical impedance tomography (UMEIT) as well as ultrasound modulated optical tomography (UMOT). The case $\alpha=1$ finds applications in Magnetic Resonance Electrical Impedance Tomography (MREIT). We present two explicit reconstruction procedures of $\sigma$ for appropriate choices of $I$ and of traces of $u_i$ at the boundary of a domain of interest. The first procedure involves the solution of an over-determined system of ordinary differential equations and generalizes to the multi-dimensional case and to (almost) arbitrary values of $\alpha$ the results obtained in two and three dimensions in \cite{CFGK} and \cite{BBMT}, respectively, in the case $\alpha=\frac12$. The second procedure consists of solving a system of linear elliptic equations, which we can prove admits a unique solution in specific situations.