A new numerical approach to inverse transport equation with error analysis

TL;DR

This paper introduces a novel numerical method for inverse radiative transfer problems that decomposes measurements to separately recover optical properties, providing analytical guarantees and improved error quantification.

Contribution

The paper proposes a new algorithm based on singular decomposition analysis that guarantees well-posedness and allows for precise error quantification in inverse transport equations.

Findings

The new method guarantees the existence and uniqueness of the solution.

Error quantification is improved compared to traditional approaches.

Error control becomes more challenging in the diffusive limit.

Abstract

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.