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Diffusive optical tomography in the Bayesian framework

Newton, Kit and Li, Qin and Stuart, Andrew (2020) Diffusive optical tomography in the Bayesian framework. Multiscale Modeling and Simulation, 18 (2). pp. 589-611. ISSN 1540-3459. doi:10.1137/19M1247346.

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Many naturally occurring models in the sciences are well approximated by simplified models using multiscale techniques. In such settings it is natural to ask about the relationship between inverse problems defined by the original problem and by the multiscale approximation. We develop an approach to this problem and exemplify it in the context of optical tomographic imaging. Optical tomographic imaging is a technique for inferring the properties of biological tissue via measurements of the incoming and outgoing light intensity; it may be used as a medical imaging methodology. Mathematically, light propagation is modeled by the radiative transfer equation (RTE), and optical tomography amounts to reconstructing the scattering and the absorption coefficients in the RTE from boundary measurements. We study this problem in the Bayesian framework, focussing on the strong scattering regime. In this regime the forward RTE is close to the diffusion equation (DE). We study the RTE in the asymptotic regime where the forward problem approaches the DE and prove convergence of the inverse RTE to the inverse DE in both nonlinear and linear settings. Convergence is proved by studying the distance between the two posterior distributions using the Hellinger metric and using the Kullback--Leibler divergence.

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Additional Information:© 2020 Society for Industrial and Applied Mathematics. Received by the editors February 28, 2019; accepted for publication January 30, 2020; published electronically April 22, 2020. The work of the third author was supported by AFOSR grant FA9550-17-1-0185, and the work of the first and second authors was supported by NSF DMS 1619778 and 1750488 and NSF TRIPODS 1740707.
Funding AgencyGrant Number
Air Force Office of Scientific Research (AFOSR)FA9550-17-1-0185
Subject Keywords:inverse problems, radiative transfer equation, Bayesian
Issue or Number:2
Classification Code:AMS subject classifications: 62F15, 35R30
Record Number:CaltechAUTHORS:20190722-155900728
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:97333
Deposited By: Tony Diaz
Deposited On:22 Jul 2019 23:29
Last Modified:16 Nov 2021 17:30

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