Time-harmonic diffuse optical tomography: H\"older stability of the derivatives of the optical properties of a medium at the boundary

TL;DR

This paper establishes Hölder stability estimates for derivatives of optical properties at the boundary in time-harmonic diffuse optical tomography, assuming known scattering and using singular solutions to the forward problem.

Contribution

It proves Hölder stability for derivatives of the absorption coefficient at the boundary in a time-harmonic setting, extending previous elliptic results to the optical tomography inverse problem.

Findings

Hölder stability of derivatives at the boundary under certain conditions

Construction of singular solutions for the forward elliptic system

Extension of elliptic PDE techniques to optical tomography

Abstract

We study the inverse problem in Optical Tomography of determining the optical properties of a medium $\Omega\subset\mathbb{R}^n$, with $n\geq 3$, under the so-called diffusion approximation. We consider the time-harmonic case where $\Omega$ is probed with an input field that is modulated with a fixed harmonic frequency $\omega=\frac{k}{c}$, where $c$ is the speed of light and $k$ is the wave number. Under suitable conditions that include a range of variability for $k$, we prove a result of H\"older stability of the derivatives of the absorption coefficient $\mu_a$ of any order at the boundary $\partial\Omega$ in terms of the measurements, in the case when the scattering coefficient $\mu_s$ is assumed to be known. The stability estimates rely on the construction of singular solutions of the underlying forward elliptic system, which extend results obtained in J. Differential Equations 84 (2): 252-272 for the single elliptic equation.