A Deficiency Problem of the Least Squares Finite Element Method for Solving Radiative Transfer in Strongly Inhomogeneous Media

The accuracy and stability of the least squares finite element method (LSFEM) and the Galerkin finite element method (GFEM) for solving radiative transfer in homogeneous and inhomogeneous media are studied theoretically via a frequency domain technique. The theoretical result confirms the traditional understanding of the superior stability of the LSFEM as compared to the GFEM. However, it is demonstrated numerically and proved theoretically that the LSFEM will suffer a deficiency problem for solving radiative transfer in media with strong inhomogeneity. This deficiency problem of the LSFEM will cause a severe accuracy degradation, which compromises too much of the performance of the LSFEM and makes it not a good choice to solve radiative transfer in strongly inhomogeneous media. It is also theoretically proved that the LSFEM is equivalent to a second order form of radiative transfer equation discretized by the central difference scheme

Incorporating reflection boundary conditions in the Neumann series radiative transport equation: Application to photon propagation and reconstruction in diffuse optical imaging

We propose a formalism to incorporate boundary conditions in a Neumann-series-based radiative transport equation. The formalism accurately models the reflection of photons at the tissue-external medium interface using Fresnel’s equations. The formalism was used to develop a gradient descent-based image reconstruction technique. The proposed methods were implemented for 3D diffuse optical imaging. In computational studies, it was observed that the average root-mean-square error (RMSE) for the output images and the estimated absorption coefficients reduced by 38% and 84%, respectively, when the reflection boundary conditions were incorporated. These results demonstrate the importance of incorporating boundary conditions that model the reflection of photons at the tissue-external medium interface

Diffusive optical tomography in the Bayesian framework

Many naturally-occuring 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 infering 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 Kullback-Leibler divergence

Gradient-based quantitative image reconstruction in ultrasound-modulated optical tomography: first harmonic measurement type in a linearised diffusion formulation

Ultrasound-modulated optical tomography is an emerging biomedical imaging modality which uses the spatially localised acoustically-driven modulation of coherent light as a probe of the structure and optical properties of biological tissues. In this work we begin by providing an overview of forward modelling methods, before deriving a linearised diffusion-style model which calculates the first-harmonic modulated flux measured on the boundary of a given domain. We derive and examine the correlation measurement density functions of the model which describe the sensitivity of the modality to perturbations in the optical parameters of interest. Finally, we employ said functions in the development of an adjoint-assisted gradient based image reconstruction method, which ameliorates the computational burden and memory requirements of a traditional Newton-based optimisation approach. We validate our work by performing reconstructions of optical absorption and scattering in two- and three-dimensions using simulated measurements with 1% proportional Gaussian noise, and demonstrate the successful recovery of the parameters to within +/-5% of their true values when the resolution of the ultrasound raster probing the domain is sufficient to delineate perturbing inclusions.Comment: 12 pages, 6 figure