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Zbl 0992.78002
Dorn, Oliver
A transport-backtransport method for optical tomography. (English)
[J] Inverse Probl. 14, No.5, 1107-1130 (1998). ISSN 0266-5611

Summary: Optical tomography is modelled by an inverse problem for a time-dependent linear transport equation in $n$ spatial dimensions $(n=2,3)$. Based on measurements which are functionals of the outgoing density at the boundary $\partial\Omega$ for different sources $q_j$, $j=1,\cdots,p$, two coefficients of the equation, the absorption coefficient $\sigma_a(x)$ and the scattering coefficient $b(x)$, are reconstructed simultaneously inside $\Omega$. Starting from some initial guess $(\sigma_a,b)^{\top}$ 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 to the medium $\Omega$ by solving a corresponding adjoint transport problem. The correction $(h,k)^{\top T}_j$ to the guess $(\sigma_a,b)^{\top}$ is calculated from the densities of the direct and the adjoint problem inside the medium. Doing this for all source positions $q_j,j=1,\cdots,p$, one after the other yields one sweep of the algorithm. \par 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 an x-ray tomography with scattering). In the case of a 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.

MSC 2000:: *78A10 Physical optics
78Mxx Basic mathematical methods in optics
44A12 Radon transform
92C55 Tomography
78A46 Inverse scattering problems
78A55 Technical appl. of optics and electromagnetic theory

Keywords: optical tomography; inverse problem; time-dependent linear transport equation; reconstructed scattering coefficient

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Zentralblatt MATH Berlin [Germany]