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Numerical Methods for Optical Diffusion Tomography

Investigator: Matthias Schlottbom -- Advisors: Herbert Egger, Wolfgang Marquardt


Optical Diffusion Tomography is a medical imaging technique based on absorption and scattering of near infrared light in biological tissue. The propagation of light in dense media is usually modelled by the Boltzmann equation, in which the scattering and absorption properties of the tissue under investigation enter as coefficients.

For solving the inverse problem of determining the tissue properties from transmission measurements, the so-called diffusion approximation of the Boltzmann equation is most widely used. The inverse problem then reduces to a parameter estimation problem in an elliptic respectively parabolic pde. While on the one hand, the diffusion approximation facilitates the solution of the inverse problem, it also introduces a substantial modeling error on the other hand. As a result only qualitative reconstruction can be obtained.

In order to obtain quantitative results, we investigate better approximations of the Boltzmann equation by higher order moment methods and other Galerkin methods in the velocity space, and compare to results obtained by the diffusion approximation. Once we have established better approximations, we will use our methods with experimental data to ensure the quality of the reconstruction in practical relevant settings.