TY - JOUR

T1 - Bayesian Instability of Optical Imaging

T2 - Ill Conditioning of Inverse Linear and Nonlinear Radiative Transfer Equation in the Fluid Regime

AU - Li, Qin

AU - Newton, Kit

AU - Wang, Li

N1 - Publisher Copyright:
© 2022 by the authors. Licensee MDPI, Basel, Switzerland.

PY - 2022/2

Y1 - 2022/2

N2 - For the inverse problem in physical models, one measures the solution and infers the model parameters using information from the collected data. Oftentimes, these data are inadequate and render the inverse problem ill-posed. We study the ill-posedness in the context of optical imaging, which is a medical imaging technique that uses light to probe (bio-)tissue structure. Depending on the intensity of the light, the forward problem can be described by different types of equations. High-energy light scatters very little, and one uses the radiative transfer equation (RTE) as the model; low-energy light scatters frequently, so the diffusion equation (DE) suffices to be a good approximation. A multiscale approximation links the hyperbolic-type RTE with the parabolic-type DE. The inverse problems for the two equations have a multiscale passage as well, so one expects that as the energy of the photons diminishes, the inverse problem changes from well-to ill-posed. We study this stability deterioration using the Bayesian inference. In particular, we use the Kullback– Leibler divergence between the prior distribution and the posterior distribution based on the RTE to prove that the information gain from the measurement vanishes as the energy of the photons decreases, so that the inverse problem is ill-posed in the diffusive regime. In the linearized setting, we also show that the mean square error of the posterior distribution increases as we approach the diffusive regime.

AB - For the inverse problem in physical models, one measures the solution and infers the model parameters using information from the collected data. Oftentimes, these data are inadequate and render the inverse problem ill-posed. We study the ill-posedness in the context of optical imaging, which is a medical imaging technique that uses light to probe (bio-)tissue structure. Depending on the intensity of the light, the forward problem can be described by different types of equations. High-energy light scatters very little, and one uses the radiative transfer equation (RTE) as the model; low-energy light scatters frequently, so the diffusion equation (DE) suffices to be a good approximation. A multiscale approximation links the hyperbolic-type RTE with the parabolic-type DE. The inverse problems for the two equations have a multiscale passage as well, so one expects that as the energy of the photons diminishes, the inverse problem changes from well-to ill-posed. We study this stability deterioration using the Bayesian inference. In particular, we use the Kullback– Leibler divergence between the prior distribution and the posterior distribution based on the RTE to prove that the information gain from the measurement vanishes as the energy of the photons decreases, so that the inverse problem is ill-posed in the diffusive regime. In the linearized setting, we also show that the mean square error of the posterior distribution increases as we approach the diffusive regime.

KW - Asymptotic analysis

KW - Bayesian approach

KW - Inverse problems

KW - Multiscale modeling

KW - Stability deterioration

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U2 - 10.3390/computation10020015

DO - 10.3390/computation10020015

M3 - Article

AN - SCOPUS:85123675642

SN - 2079-3197

VL - 10

JO - Computation

JF - Computation

IS - 2

M1 - 15

ER -