A transport-backtransport method for optical tomography

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 partial derivative Omega for different sources q(j), j = l,..., p, two coefficients of the equation, the absorption coefficient sigma(a)(x) and the scattering coefficient b(x), are reconstructed simultaneously inside Omega. Starting out from some initial guess (sigma(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 qj by solving a 'direct' transport problem, and then transports these residuals back into the medium Omega by solving a corresponding adjoint transport problem. The correction (h, k)(j)(T) to the guess (sigma(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 = l,..., 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.