Inverse transport with angularly averaged measurements
The inverse problem in radiative transfer is considered. The measurement setup involves controlling incoming radiation at the boundary of a convex domain in Rn and measuring the outgoing radiation in an attempt to reconstruct (actually, prove uniqueness and stability theorems about the reconstruction of) the total cross section and scattering coefficients. Varying degrees of radiation production/measurement accuracy lead to different results. Assuming the index of refraction is constant, if the incoming radiation is directionally dependent, and the outgoing radiation is averaged over angle, then the total cross section is determined. With an additional smallness on the scattering kernel, these measurements also determine the scattering kernel (assuming we know a-priori its angular dependence). These results are generalized to the case of a spatially varying, though possibly anisotropic, index of refraction. Finally, we consider the case where the incoming radiation is isotropic, and the measurements are averaged over angle. In this setup, the contribution to the measurement is an integral of the scattering kernel against a product of harmonic functions, plus an additional term that is small when absorption and scattering are small. The linearized problem is severely ill-posed. We construct a regularized inverse that allows for reconstruction of the low frequency content of the scattering kernel, up to quadratic error, from the nonlinear map. An iterative scheme is used to improve this error so that it is small when the high frequency content of the scattering kernel is small.
- Mathematics