Time-harmonic diffuse optical tomography: Hölder stability of the derivatives of the optical properties of a medium at the boundary

Jason Curran; Romina Gaburro; Clifford Nolan; Erkki Somersalo

doi:10.3934/ipi.2022044

Article Contents

2023, Volume 17, Issue 2: 338-361. Doi: 10.3934/ipi.2022044

This issue Previous Article Variational image motion estimation by preconditioned dual optimization Next Article Multi-target detection with rotations

Time-harmonic diffuse optical tomography: Hölder stability of the derivatives of the optical properties of a medium at the boundary

1.
Department of Mathematics and Statistics, CONFIRM-Science Foundation Ireland, University of Limerick, Ireland
2.
Department of Mathematics and Statistics, Health Research Institute (HRI), University of Limerick, Ireland
3.
Department of Mathematics, Applied Mathematics and Statistics, Case Western University, Cleveland, OH, USA

^*Corresponding author: Romina Gaburro
^*Corresponding author: Romina Gaburro

Received: May 2022

Revised: August 2022

Early access: September 2022

Published: April 2023

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Abstract

We address the inverse problem in Optical Tomography of stably determining the optical properties of an anisotropic medium $ \Omega\subset\mathbb{R}^n $, with $ n\geq 3 $, under the so-called diffusion approximation. Assuming that the scattering coefficient $ \mu_s $ is known, we prove Hölder stability of the derivatives of any order of the absorption coefficient $ \mu_a $ at the boundary $ \partial\Omega $ in terms of the measurements, in the time-harmonic case, where the anisotropic medium $ \Omega $ is interrogated 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. The stability estimates are established under suitable conditions that include a range of variability for $ k $ and they 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 and those obtained in Applicable Analysis DOI:10.1080/00036811.2020.1758314, where a Lipschitz type stability estimate of $ \mu_a $ on $ \partial\Omega $ was established in terms of the measurements.

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Mathematics Subject Classification: Primary: 35J25, 35J47, 35R30; Secondary: 92C50.

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References

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