Recovery of the absorption coefficient in radiative transport from a single measurement

TL;DR

This paper develops a method to recover the absorption coefficient in radiative transport using a single boundary measurement, combining linear and nonlinear inverse problem techniques with stability analysis.

Contribution

It introduces a two-step approach for reconstructing optical properties, including a linear framework for initial recovery and a nonlinear step for the absorption coefficient, with stability guarantees.

Findings

Unique solvability for generic conditions

Stability estimates for the inverse problem

Effective reconstruction of optical properties

Abstract

In this paper, we investigate the recovery of the absorption coefficient from boundary data assuming that the region of interest is illuminated at an initial time. We consider a sufficiently strong and isotropic, but otherwise unknown initial state of radiation. This work is part of an effort to reconstruct optical properties using unknown illumination embedded in the unknown medium. We break the problem into two steps. First, in a linear framework, we seek the simultaneous recovery of a forcing term of the form $\sigma(t,x,\theta) f(x)$ (with $\sigma$ known) and an isotropic initial condition $u_{0}(x)$ using the single measurement induced by these data. Based on exact boundary controllability, we derive a system of equations for the unknown terms $f$ and $u_{0}$. The system is shown to be Fredholm if $\sigma$ satisfies a certain positivity condition. We show that for generic term $\sigma$ and weakly absorbing media, this linear inverse problem is uniquely solvable with a stability estimate. In the second step, we use the stability results from the linear problem to address the nonlinearity in the recovery of a weak absorbing coefficient. We obtain a locally Lipschitz stability estimate.