Cauchy problem for Ultrasound Modulated EIT

TL;DR

This paper investigates the mathematical properties of a nonlinear hyperbolic equation derived from ultrasound modulated electrical impedance tomography, establishing conditions for existence, uniqueness, and stability of solutions, and exploring partial and global reconstructions.

Contribution

It introduces a novel analysis of the Cauchy problem for the 0-Laplacian in the context of hybrid imaging, providing new theoretical results for inverse problems with boundary data.

Findings

Existence and uniqueness of solutions are established.

Stable reconstruction of the diffusion coefficient is possible on a domain of influence.

Global reconstruction methods are analyzed for specific geometries.

Abstract

Ultrasound modulation of electrical or optical properties of materials offers the possibility to devise hybrid imaging techniques that combine the high electrical or optical contrast observed in many settings of interest with the high resolution of ultrasound. Mathematically, these modalities require that we reconstruct a diffusion coefficient $\sigma(x)$ for $x\in X$, a bounded domain in $\Rm^n$, from knowledge of $\sigma(x)|\nabla u|^2(x)$ for $x\in X$, where $u$ is the solution to the elliptic equation $-\nabla\cdot\sigma\nabla u=0$ in $X$ with $u=f$ on $\partial X$. This inverse problem may be recast as a nonlinear equation, which formally takes the form of a 0-Laplacian. Whereas $p-$Laplacians with $p>1$ are well-studied variational elliptic non-linear equations, $p=1$ is a limiting case with a convex but not strictly convex functional, and the case $p<1$ admits a variational formulation with a functional that is not convex. In this paper, we augment the equation for the 0-Laplacian with full Cauchy data at the domain's boundary, which results in a, formally overdetermined, nonlinear hyperbolic equation. The paper presents existence, uniqueness, and stability results for the Cauchy problem of the 0-Laplacian. In general, the diffusion coefficient $\sigma(x)$ can be stably reconstructed only on a subset of $X$ described as the domain of influence of the space-like part of the boundary $\partial X$ for an appropriate Lorentzian metric. Global reconstructions for specific geometries or based on the construction of appropriate complex geometric optics solutions are also analyzed.