Optical tomography in two dimensions

We consider in two dimensions, the inverse boundary problem of reconstructing the absorption and scattering coefficient of an inhomogeneous medium by probing it with diffuse light. The problem is modeled as an inverse boundary problem for the stationary linear Boltzmann equation. The information is encoded in the albedo operator. We show that we can recover the absorption and the scattering kernel from this information provided that the latter is small in an appropriate topology. We also give stability estimates and propose an approximate reconstruction procedure. Optical Tomography concerns with the determination of spatially-varying optical ab-sorption and scattering properties of a medium by measuring the response of the medium to transmitted near-infrared light. This has been proposed recently as a possible diagnostic tool in medicine (see [A] for a review). A probabilistic approach to optical tomography is the so-called diffused tomography [SGKZ]. Other areas of applications are atmospheric remote sensing [Bi], nuclear physics (see [MC] for a review), etc. We describe in more detail the mathematical problem below. Let X ⊂ R2 be an open convex set with smooth boundary and consider the 2D stationary transport equation −v(θ) · ∇xf − σ(x, θ)f + S1 k(x, θ′, θ)f(x, θ′) dθ ′ = 0, (1) where x ∈ X, θ, θ ′ ∈ S1 and throughout this paper we will use the notation v(θ) = (cos θ, sin θ)