The inverse radiative transfer problem finds broad applications in medicalimaging, atmospheric science, astronomy, and many other areas. This problemintends to recover the optical properties, denoted as absorption and scatteringcoefficient of the media, through the source-measurement pairs. A typicalcomputational approach is to form the inverse problem as a PDE-constraintoptimization, with the minimizer being the to-be-recovered coefficients. Themethod is tested to be efficient in practice, but lacks analyticaljustification: there is no guarantee of the existence or uniqueness of theminimizer, and the error is hard to quantify. In this paper, we provide adifferent algorithm by levering the ideas from singular decomposition analysis.Our approach is to decompose the measurements into three components, two out ofwhich encode the information of the two coefficients respectively. We thensplit the optimization problem into two subproblems and use those twocomponents to recover the absorption and scattering coefficients separately. Inthis regard, we prove the well-posedness of the new optimization, and the errorcould be quantified with better precision. In the end, we incorporate thediffusive scaling and show that the error is harder to control in the diffusivelimit.
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