A transport-backtransport method for optical tomography

Abstract

Optical tomography is modelled by the inverse problem of the time-dependent linear transport equation in n spatial dimensions (n = 2, 3). Based on the measurements which consist of some functionals of the outgoing density at the boundary ∂ for different sources q j , j = 1, . . . , p, two coefficients of the equation, the absorption coefficient σ a (x) and the scattering coefficient b(x), are reconstructed simultaneously inside . Starting out from some initial guess (σ a , b) T for these coefficients, the transport-backtransport (TBT) algorithm calculates the difference between the computed and the physically given measurements for a fixed source q j by solving a 'direct' transport problem, and then transports these residuals back into the medium by solving a corresponding adjoint transport problem. The correction (h, k) T j to the guess (σ a , b) T is calculated from the densities of the direct and the adjoint problem inside the medium. Doing this for all source positions q j , j = 1, . . . , p, one after the other yields one sweep of the algorithm. Numerical experiments are presented for the case when n = 2. They show that the TBT-method is able to reconstruct and to distinguish between scattering and absorbing objects in the case of large mean free path (which corresponds to x-ray tomography with scattering). In the case of very small mean free path (which corresponds to optical tomography), scattering and absorbing objects are located during the early sweeps, but phantoms are built up in the reconstructed scattering coefficient at positions where an absorber is situated and vice versa.

Level Set Techniques For Structural Inversion In Medical Imaging

“…Notice that the velocity function (32) needs to be recalculated at each step of the artificial shape evolution (33) for each point of the current shape boundary x ∈ ∂D. This means that a Radon transform needs to be applied to the current attenuation distribution µ(x; t) at time step t, and then the backprojection operator T * needs to be applied to the difference in the data T µ(x; t) − g in order to calculate (32).…”

“…(See, for example, [33,34] for possible choices.) The parameter γ in (70) is the mean cosine of the scattering function.…”

Investigation on the reflection at the boundaries for reconstruction laser-based imaging

“…For authors' best knowledge, effects of boundary conditions on the quality of imaging have not been investigated. Some studies have been done by considering boundaries, [7,8,10], while others not, [13,14],inthe case of inverse problem analysis and numerous studies have investigated boundary conditions for the diffusion approximation, see a comprehensive development in [15], but the effect on imaging has not been elucidated. For inverse reconstruction problems, it is worth noting that reflective boundary conditions are a way to enforce photons to stay longer within the sample and should have the same effects as long-term filtering while being applicable to steady-state [16] or frequency-domain techniques [17], where time filtering is not available.…”

Laguerre Runge--Kutta--Fehlberg Method for Simulating Laser Pulse Propagation in Biological Tissue

“…In the past, Dorn [15] has used the inverse problem of the time-dependent transport equation for modeling optical tomography. Larsen et al [16] has carried out asymptotic analysis of radiative transfer problems, and Tarvainen et al [17] have worked on finite element modeling of the coupled radiative transfer equation and diffusion approximation considering the time dependency.…”