A new numerical approach to inverse transport equation with error analysis
This work provides a theoretically justified alternative to existing PDE-constrained optimization methods for inverse transport problems in medical imaging and other fields, addressing the lack of existence/uniqueness guarantees and error quantification.
The paper presents a new algorithm for the inverse radiative transfer problem that decomposes measurements into components to separately recover absorption and scattering coefficients, proving well-posedness and improved error quantification. The method is shown to have harder-to-control errors in the diffusive limit.
The inverse radiative transfer problem finds broad applications in medical imaging, atmospheric science, astronomy, and many other areas. This problem intends to recover the optical properties, denoted as absorption and scattering coefficient of the media, through the source-measurement pairs. A typical computational approach is to form the inverse problem as a PDE-constraint optimization, with the minimizer being the to-be-recovered coefficients. The method is tested to be efficient in practice, but lacks analytical justification: there is no guarantee of the existence or uniqueness of the minimizer, and the error is hard to quantify. In this paper, we provide a different algorithm by levering the ideas from singular decomposition analysis. Our approach is to decompose the measurements into three components, two out of which encode the information of the two coefficients respectively. We then split the optimization problem into two subproblems and use those two components to recover the absorption and scattering coefficients separately. In this regard, we prove the well-posedness of the new optimization, and the error could be quantified with better precision. In the end, we incorporate the diffusive scaling and show that the error is harder to control in the diffusive limit.