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

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 σ(t, x, θ)f (x) (with σ 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 σ satisfies a certain positivity condition. We show that for generic term σ 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.

Ultrasound Modulated Bioluminescence Tomography and Controllability of the Radiative Transport Equation

“…Note that other control theory results for the time-dependent RTE have had applications to inverse problems as well. See for example [1,2].…”

Global Lipschitz stability for a fractional inverse transport problem by Carleman estimates

“…The remainder of this paper is organized as follows. In §3, we derive a first-order equation in time by multiplying ∂ 1/2 t by the fractional radiative transport equation in (1). In §4, we establish our key Carleman estimate.…”

“…For the usual radiative transport equation with ∂ t , the Lipschitz stability was shown for −T < t < T [18], for the purely absorbing case of σ s ≡ 0 [11], and for 0 < t < T [23]. The recovery of σ t was also considered in [1]. The exact controllability was proved [19] and the case that σ t depends on x, v, t was considered in [25].…”