scholarly journals Diffusion Equation-Assisted Markov Chain Monte Carlo Methods for the Inverse Radiative Transfer Equation

Entropy ◽  
2019 ◽  
Vol 21 (3) ◽  
pp. 291 ◽  
Author(s):  
Qin Li ◽  
Kit Newton
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Radiative Transfer ◽  
Diffusion Equation ◽  
Posterior Distribution ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Optical Tomography ◽  
Scattering Coefficient

Optical tomography is the process of reconstructing the optical properties of biological tissue using measurements of incoming and outgoing light intensity at the tissue boundary. Mathematically, light propagation is modeled by the radiative transfer equation (RTE), and optical tomography amounts to reconstructing the scattering coefficient in the RTE using the boundary measurements. In the strong scattering regime, the RTE is asymptotically equivalent to the diffusion equation (DE), and the inverse problem becomes reconstructing the diffusion coefficient using Dirichlet and Neumann data on the boundary. We study this problem in the Bayesian framework, meaning that we examine the posterior distribution of the scattering coefficient after the measurements have been taken. However, sampling from this distribution is computationally expensive, since to evaluate each Markov Chain Monte Carlo (MCMC) sample, one needs to run the RTE solvers multiple times. We therefore propose the DE-assisted two-level MCMC technique, in which bad samples are filtered out using DE solvers that are significantly cheaper than RTE solvers. This allows us to make sampling from the RTE posterior distribution computationally feasible.

Markov Chain Monte Carlo

2015 ◽  
Author(s):  
N. Thompson Hobbs ◽  
Mevin B. Hooten
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Bayesian Analysis ◽  
Posterior Distribution ◽  
Mcmc Algorithm ◽  
Bayesian Analyses ◽  
Seminal Paper ◽  
Formal Treatment ◽  
High Level

This chapter explains how to implement Bayesian analyses using the Markov chain Monte Carlo (MCMC) algorithm, a set of methods for Bayesian analysis made popular by the seminal paper of Gelfand and Smith (1990). It begins with an explanation of MCMC with a heuristic, high-level treatment of the algorithm, describing its operation in simple terms with a minimum of formalism. In this first part, the chapter explains the algorithm so that all readers can gain an intuitive understanding of how to find the posterior distribution by sampling from it. Next, the chapter offers a somewhat more formal treatment of how MCMC is implemented mathematically. Finally, this chapter discusses implementation of Bayesian models via two routes—by using software and by writing one's own algorithm.

scholarly journals Image Reconstruction for Diffuse Optical Tomography Based on Radiative Transfer Equation

2015 ◽  
Vol 2015 ◽  
pp. 1-23 ◽  
Author(s):  
Bo Bi ◽  
Bo Han ◽  
Weimin Han ◽  
Jinping Tang ◽  
Li Li
Keyword(s):  
Image Reconstruction ◽  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Optical Tomography ◽  
Measurement Data ◽  
Diffuse Optical Tomography ◽  
Forward Model ◽  
Sparsity Regularization ◽  
Reconstruction Methods

Diffuse optical tomography is a novel molecular imaging technology for small animal studies. Most known reconstruction methods use the diffusion equation (DA) as forward model, although the validation of DA breaks down in certain situations. In this work, we use the radiative transfer equation as forward model which provides an accurate description of the light propagation within biological media and investigate the potential of sparsity constraints in solving the diffuse optical tomography inverse problem. The feasibility of the sparsity reconstruction approach is evaluated by boundary angular-averaged measurement data and internal angular-averaged measurement data. Simulation results demonstrate that in most of the test cases the reconstructions with sparsity regularization are both qualitatively and quantitatively more reliable than those with standardL2regularization. Results also show the competitive performance of the split Bregman algorithm for the DOT image reconstruction with sparsity regularization compared with other existingL1algorithms.

scholarly journals Mixed Total Variation and L1 Regularization Method for Optical Tomography Based on Radiative Transfer Equation

2017 ◽  
Vol 2017 ◽  
pp. 1-15 ◽  
Author(s):  
Jinping Tang ◽  
Bo Han ◽  
Weimin Han ◽  
Bo Bi ◽  
Li Li
Keyword(s):  
Radiative Transfer ◽  
Total Variation ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Imaging Modality ◽  
Optical Tomography ◽  
Identification Problem ◽  
Split Bregman ◽  
Computation Efficiency ◽  
Reconstruction Methods

Optical tomography is an emerging and important molecular imaging modality. The aim of optical tomography is to reconstruct optical properties of human tissues. In this paper, we focus on reconstructing the absorption coefficient based on the radiative transfer equation (RTE). It is an ill-posed parameter identification problem. Regularization methods have been broadly applied to reconstruct the optical coefficients, such as the total variation (TV) regularization and the L1 regularization. In order to better reconstruct the piecewise constant and sparse coefficient distributions, TV and L1 norms are combined as the regularization. The forward problem is discretized with the discontinuous Galerkin method on the spatial space and the finite element method on the angular space. The minimization problem is solved by a Jacobian-based Levenberg-Marquardt type method which is equipped with a split Bregman algorithms for the L1 regularization. We use the adjoint method to compute the Jacobian matrix which dramatically improves the computation efficiency. By comparing with the other imaging reconstruction methods based on TV and L1 regularizations, the simulation results show the validity and efficiency of the proposed method.

Propagation of Ultra-Short-Pulse Radiation in Participating Media: A Laguerre-Galerkin Solution

2007 ◽  
Author(s):  
Tuba Okutucu ◽  
Yaman Yener
Keyword(s):  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Transfer Problem ◽  
Short Pulse ◽  
Optical Tomography ◽  
Time Derivative ◽  
Product Analysis ◽  
Participating Media ◽  
Radiative Transfer Problem

Transient analysis of the radiative transfer problem in participating media has become essential due to the recent applications involving extremely small time scales. In classical radiation problems, the time derivative term in the radiative transfer equation has a negligible order of magnitude compared to the others. Lasers of pico- to femtosecond pulse durations are now being used to investigate the properties of scattering and absorbing media in such applications as, optical tomography, combustion product analysis, and remote sensing. For such applications, the time derivative in the radiative transfer equation can no longer be neglected. Numerous approaches such as, integral formulation, direct numerical approach, discrete ordinates method, Monte Carlo simulations, and Galerkin technique have been introduced for the solution of transient radiative transfer problems in participating media. In the present work, Laguerre-Galerkin solutions for both rectangular and Gaussian incident pulse profiles are presented.

Efficient History Matching Using the Markov-Chain Monte Carlo Method by Means of the Transformed Adaptive Stochastic Collocation Method

SPE Journal ◽  
2019 ◽  
Vol 24 (04) ◽  
pp. 1468-1489 ◽  
Author(s):  
Qinzhuo Liao ◽  
Lingzao Zeng ◽  
Haibin Chang ◽  
Dongxiao Zhang
Keyword(s):  
Monte Carlo ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Collocation Method ◽  
Posterior Distribution ◽  
History Matching ◽  
Stochastic Collocation ◽  
Mcmc Method ◽  
Stochastic Collocation Method ◽  
Surrogate System

Summary Bayesian inference provides a convenient framework for history matching and prediction. In this framework, prior knowledge, system nonlinearity, and measurement errors can be directly incorporated into the posterior distribution of the parameters. The Markov-chain Monte Carlo (MCMC) method is a powerful tool to generate samples from the posterior distribution. However, the MCMC method usually requires a large number of forward simulations. Hence, it can be a computationally intensive task, particularly when dealing with large-scale flow and transport models. To address this issue, we construct a surrogate system for the model outputs in the form of polynomials using the stochastic collocation method (SCM). In addition, we use interpolation with the nested sparse grids and adaptively take into account the different importance of parameters for high-dimensional problems. Furthermore, we introduce an additional transform process to improve the accuracy of the surrogate model in case of strong nonlinearities, such as a discontinuous or unsmooth relation between the input parameters and the output responses. Once the surrogate system is built, we can evaluate the likelihood with little computational cost. Numerical results demonstrate that the proposed method can efficiently estimate the posterior statistics of input parameters and provide accurate results for history matching and prediction of the observed data with a moderate number of parameters.

scholarly journals A Hybrid Inversion Scheme Combining Markov Chain Monte Carlo and Iterative Methods for Determining Optical Properties of Random Media

2019 ◽  
Vol 9 (17) ◽  
pp. 3500
Author(s):  
Yu Jiang ◽  
Yoko Hoshi ◽  
Manabu Machida ◽  
Gen Nakamura
Keyword(s):  
Monte Carlo ◽  
Optical Properties ◽  
Markov Chain ◽  
Markov Chain Monte Carlo ◽  
Hybrid Method ◽  
Random Media ◽  
Optical Tomography ◽  
Diffuse Optical Tomography ◽  
Marquardt Algorithm ◽  
Levenberg Marquardt

Near-infrared spectroscopy (NIRS) including diffuse optical tomography is an imaging modality which makes use of diffuse light propagation in random media. When optical properties of a random medium are investigated from boundary measurements of reflected or transmitted light, iterative inversion schemes such as the Levenberg–Marquardt algorithm are known to fail when initial guesses are not close enough to the true value of the coefficient to be reconstructed. In this paper, we investigate how this weakness of iterative schemes is overcome using Markov chain Monte Carlo. Using time-resolved measurements performed against a polyurethane-based phantom, we present a case that the Levenberg–Marquardt algorithm fails to work but the proposed hybrid method works well. Then, with a toy model of diffuse optical tomography we illustrate that the Levenberg–Marquardt method fails when it is trapped by a local minimum but the hybrid method can escape from local minima by using the Metropolis–Hastings Markov chain Monte Carlo algorithm until it reaches the valley of the global minimum. The proposed hybrid scheme can be applied to different inverse problems in NIRS which are solved iteratively. We find that for both numerical and phantom experiments, optical properties such as the absorption and reduced scattering coefficients can be retrieved without being trapped by a local minimum when Monte Carlo simulation is run only about 100 steps before switching to an iterative method. The hybrid method is compared with simulated annealing. Although the Metropolis–Hastings MCMC arrives at the steady state at about 10,000 Monte Carlo steps, in the hybrid method the Monte Carlo simulation can be stopped way before the burn-in time.

Utilizing the Radiative Transfer Equation in Optical Tomography

PIERS Online ◽  
2008 ◽  
Vol 4 (6) ◽  
pp. 655-660 ◽  
Author(s):  
Tanja Tarvainen ◽  
Marko Vauhkonen ◽  
Ville Kolehmainen ◽  
Jari P. Kaipio ◽  
Simon R. Arridge
Keyword(s):  
Radiative Transfer ◽  
Transfer Equation ◽  
Radiative Transfer Equation ◽  
Optical Tomography
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