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

Abstract

We study the inverse problem in Optical Tomography of determining the optical properties of a medium Ω ⊂ R n , with n ≥ 3, under the so-called diffusion approximation. We consider the time-harmonic case where Ω is probed with an input field that is modulated with a fixed harmonic frequency ω = 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 Hölder stability of the derivatives of the absorption coefficient µa of any order at the boundary ∂Ω in terms of the measurements, in the case when the scattering coefficient µs 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.