Bayesian Approximation Error Research Articles

Approximate marginalization of absorption and scattering in fluorescence diffuse optical tomography

In fluorescence diffuse optical tomography (fDOT), the reconstruction of the fluorophoreconcentration inside the target body is usually carried out using a normalized Born approximation model wherethe measured fluorescent emission data is scaled by measured excitation data. One of the benefits of the model is that it can tolerate inaccuracy in the absorption and scattering distributions that are used in the construction of the forward model to some extent. In this paper, we employ the recently proposed Bayesian approximation error approach to fDOT for compensating for themodeling errors caused by the inaccurately known optical properties of the target in combination with the normalized Born approximation model. The approach is evaluated using a simulated test case withdifferent amount of error in the optical properties. The results show that the Bayesian approximation error approach improvesthe tolerance of fDOT imaging against modeling errors caused by inaccurately known absorption and scattering of the target.

Contrast enhancement in EIT imaging of the brain

We consider electrical impedance tomography (EIT) imaging of the brain. The brain is surrounded by the poorly conducting skull which has low conductivity compared to the brain. The skull layer causes a partial shielding effect which leads to weak sensitivity for the imaging of the brain tissue. In this paper we propose an approach based on the Bayesian approximation error approach, to enhance the contrast in brain imaging. With this approach, both the (uninteresting) geometry and the conductivity of the skull are embedded in the approximation error statistics, which leads to a computationally efficient algorithm that is able to detect features such as internal haemorrhage with significantly increased sensitivity and specificity. We evaluate the approach with simulations and phantom data.

Estimating pipeline location using ground-penetrating radar data in the presence of model uncertainties

We study the inverse problem of estimating the pipeline location from ground-penetrating radar data in the context of Bayesian inversion. Maxwellʼs equations are used to model the electromagnetic wave propagation, and are solved using a high-order discontinuous Galerkin method. The uncertainties related to the wave propagation in inhomogeneous background are taken into account by the Bayesian approximation error (BAE) approach. The inverse problem is solved using the full waveform data. Numerical simulations suggest that by using the BAE the model uncertainties can be taken satisfactorily into account, while at the same time making a significant reduction in the computational burden. Furthermore, the estimates for the location of the pipeline are feasible in the sense that the posterior model supports the actual location.

Statistical inversion approach to estimating water content in an aquifer from seismic data

This study focuses on developing computational tools to estimate water content in an aquifer from seismic measurements. The poroelastic signature from an aquifer is simulated and methods that use this signature to estimate the water table level and aquifer thickness are investigated. In this work, the spectral-element method is used to solve the forward model that characterizes the propagation of seismic waves. The inverse problem is formulated in the Bayesian framework, so that all uncertainties are explicitly modelled as probability distributions, and the solution is given as summary statistics over the posterior distribution of parameters relative to data. For the inverse problem, we use the Bayesian approximation error method which reduces the overall computational demand. In this study, results in the two-dimensional case with simulated data are presented.

Bayesian approximation error approach in full-wave ultrasound tomography

In ultrasound tomography, the spatial distribution of the speed of sound in a region of interest is reconstructed based on transient measurements made around the object. The computation of the forward problem (the full-wave solution) within the object is a computationally intensive task and can often be prohibitive for practical purposes. The purpose of this paper is to investigate the feasibility of using approximate forward solvers and the partial recovery from the related errors by employing the Bayesian approximation error approach. In addition to discretization error, we also investigate whether the approach can be used to reduce the reconstruction errors that are due to the adoption of approximate absorbing boundary models. We carry out two numerical studies in which the objective is to reduce the computational times to around 3% of the time that would be required by a numerically accurate forward solver. The results show that the Bayesian approximation error approach improves the reconstructions.

Compensation of modeling errors due to unknown domain boundary in diffuse optical tomography.

Diffuse optical tomography is a highly unstable problem with respect to modeling and measurement errors. During clinical measurements, the body shape is not always known, and an approximate model domain has to be employed. The use of an incorrect model domain can, however, lead to significant artifacts in the reconstructed images. Recently, the Bayesian approximation error theory has been proposed to handle model-based errors. In this work, the feasibility of the Bayesian approximation error approach to compensate for modeling errors due to unknown body shape is investigated. The approach is tested with simulations. The results show that the Bayesian approximation error method can be used to reduce artifacts in reconstructed images due to unknown domain shape.

Stochastic modeling error reduction using Bayesian approach coupled with an adaptive Kriging-based model

Purpose– Magnetic material properties of an electromagnetic device (EMD) can be recovered by solving a coupled experimental numerical inverse problem. In order to ensure the highest possible accuracy of the inverse problem solution, all physics of the EMD need to be perfectly modeled using a complex numerical model. However, these fine models demand a high computational time. Alternatively, less accurate coarse models can be used with a demerit of the high expected recovery errors. The purpose of this paper is to present an efficient methodology to reduce the effect of stochastic modeling errors in the inverse problem solution.Design/methodology/approach– The recovery error in the electromagnetic inverse problem solution is reduced using the Bayesian approximation error approach coupled with an adaptive Kriging-based model. The accuracy of the forward model is assessed and adapted a priori using the cross-validation technique.Findings– The adaptive Kriging-based model seems to be an efficient technique for modeling EMDs used in inverse problems. Moreover, using the proposed methodology, the recovery error in the electromagnetic inverse problem solution is largely reduced in a relatively small computational time and memory storage.Originality/value– The proposed methodology is capable of not only improving the accuracy of the inverse problem solution, but also reducing the computational time as well as the memory storage. Furthermore, to the best of the authors knowledge, it is the first time to combine the adaptive Kriging-based model with the Bayesian approximation error approach for the stochastic modeling error reduction.

Estimation of fixed charge density and diffusivity profiles in cartilage using contrast enhanced computer tomography

SUMMARYContrast enhanced computer tomography (CT) imaging of articular cartilage has been proposed for diagnostics of cartilage degeneration, that is, osteoarthritis. Previous studies also indicate that acute cartilage damage can be detected by measuring diffusion of contrast agent into cartilage using CT. However, currently, there is no reliable method to measure spatial diffusion rates within cartilage tissue, and only average bulk values have been reported. In this paper, we develop a method to determine depthwise diffusivity of contrast agents in cartilage tissue using contrast enhanced CT. The triphasic mechano‐electrochemical theory of cartilage is modified to include diffusion of contrast agents. By applying statistical inversion theory and Bayesian approximation error approach, the method allows us to estimate a fixed charge density distribution in the cartilage tissue, an important determinant for mechanical competence of articular cartilage. The method is tested by using a one‐dimensional simulation study. Preliminary tests with experimental data on diffusion of anionic iodine contrast agent in bovine articular cartilage indicate that the method can provide realistic estimates for depth dependent fixed charge density. Thereby, the present study can improve our understanding on the feasibility of contrast enhanced CT for cartilage diagnostics. Copyright © 2014 John Wiley & Sons, Ltd.

Estimation of aquifer dimensions from passive seismic signals with approximate wave propagation models

Recently, it has been proposed that spontaneous seismic activity could be used in the estimation of hydrological parameters of aquifers such as permeability and storage. Approximate wave propagation models such as ray tracing, which are commonly used in hydrological parameter estimation with active sources and backscattering geometry, are not feasible with passive seismological imaging. With respect to full wave propagation models, the most accurate known model for aquifers is the poroelastic model while bedrock is usually modelled as an elastic medium. Using a poroelastic model in the forward model can be a computationally impractical choice. In this paper, we carry out a feasibility study in which we attempt to estimate the aquifer depth and water table using a highly approximate elastic model also for the aquifer. We adopt the Bayesian approximation error approach in which a statistical model is constructed for the errors that are induced by using model approximations such as sparse meshing and simplified physical models. We consider the problem in a simple two-dimensional geometry and show that straightforward adoption of approximate models leads to inconsistent parameter estimates, that is, the true parameters have essentially vanishing posterior density. On the other hand, using the Bayesian approximation error approach, the parameter estimates are consistent.

Bayesian Image Reconstruction in Quantitative Photoacoustic Tomography

Quantitative photoacoustic tomography is an emerging imaging technique aimed at estimating chromophore concentrations inside tissues from photoacoustic images, which are formed by combining optical information and ultrasonic propagation. This is a hybrid imaging problem in which the solution of one inverse problem acts as the data for another ill-posed inverse problem. In the optical reconstruction of quantitative photoacoustic tomography, the data is obtained as a solution of an acoustic inverse initial value problem. Thus, both the data and the noise are affected by the method applied to solve the acoustic inverse problem. In this paper, the noise of optical data is modelled as Gaussian distributed with mean and covariance approximated by solving several acoustic inverse initial value problems using acoustic noise samples as data. Furthermore, Bayesian approximation error modelling is applied to compensate for the modelling errors in the optical data caused by the acoustic solver. The results show that modelling of the noise statistics and the approximation errors can improve the optical reconstructions.

Approximation error method can reduce artifacts due to scalp blood flow in optical brain activation imaging

Diffuse optical tomography can image the hemodynamic response to an activation in the human brain by measuring changes in optical absorption of near-infrared light. Since optodes placed on the scalp are used, the measurements are very sensitive to changes in optical attenuation in the scalp, making optical brain activation imaging susceptible to artifacts due to effects of systemic circulation and local circulation of the scalp. We propose to use the Bayesian approximation error approach to reduce these artifacts. The feasibility of the approach is evaluated using simulated brain activations. When a localized cortical activation occurs simultaneously with changes in the scalp blood flow, these changes can mask the cortical activity causing spurious artifacts. We show that the proposed approach is able to recover from these artifacts even when the nominal tissue properties are not well known.

A Bayesian approach for the stochastic modeling error reduction of magnetic material identification of an electromagnetic device

Magnetic material properties of an electromagnetic device can be recovered by solving an inverse problem where measurements are adequately interpreted by a mathematical forward model. The accuracy of these forward models dramatically affects the accuracy of the material properties recovered by the inverse problem. The more accurate the forward model is, the more accurate recovered data are. However, the more accurate ‘fine’ models demand a high computational time and memory storage. Alternatively, less accurate ‘coarse’ models can be used with a demerit of the high expected recovery errors. This paper uses the Bayesian approximation error approach for improving the inverse problem results when coarse models are utilized. The proposed approach adapts the objective function to be minimized with the a priori misfit between fine and coarse forward model responses. In this paper, two different electromagnetic devices, namely a switched reluctance motor and an EI core inductor, are used as case studies. The proposed methodology is validated on both purely numerical and real experimental results. The results show a significant reduction in the recovery error within an acceptable computational time.

An approximation error approach for compensating for modelling errors between the radiative transfer equation and the diffusion approximation in diffuse optical tomography

In this paper, we investigate the applicability of the Bayesian approximation error approach to compensate for the discrepancy of the diffusion approximation in diffuse optical tomography close to the light sources and in weakly scattering subdomains. While the approximation error approach has earlier been shown to be a feasible approach to compensating for discretization errors, uncertain boundary data and geometry, the ability of the approach to recover from using a qualitatively incorrect physical model has not been contested. In the case of weakly scattering subdomains and close to sources, the radiative transfer equation is commonly considered to be the most accurate model for light scattering in turbid media. In this paper, we construct the approximation error statistics based on predictions of the radiative transfer and diffusion models. In addition, we investigate the combined approximation errors due to using the diffusion approximation and using a very low-dimensional approximation in the forward problem. We show that recovery is feasible in the sense that with the approximation error model the reconstructions with a low-dimensional diffusion approximation are essentially as good as with using a very high-dimensional radiative transfer model.

Approximation errors and model reduction in three-dimensional diffuse optical tomography

Model reduction is often required in diffuse optical tomography (DOT), typically because of limited available computation time or computer memory. In practice, this means that one is bound to use coarse mesh and truncated computation domain in the model for the forward problem. We apply the (Bayesian) approximation error model for the compensation of modeling errors caused by domain truncation and a coarse computation mesh in DOT. The approach is tested with a three-dimensional example using experimental data. The results show that when the approximation error model is employed, it is possible to use mesh densities and computation domains that would be unacceptable with a conventional measurement error model.

The Bayesian approximation error approach for electrical impedance tomography—experimental results

Inverse problems can be characterized as problems that tolerate measurement and modelling errors poorly. While the measurement error issue has been widely considered as a solved problem, the modelling errors have remained largely untreated. The approximation and modelling errors can, however, be argued to dominate the measurement errors in most applications. There are several applications in which the temporal and memory requirements dictate that the computational complexity of the forward solver be radically reduced. For example, in process tomography the reconstructions have to be carried out typically in a few tens of milliseconds. Recently, a Bayesian approach for the treatment of approximation and modelling errors for inverse problems has been proposed. This approach has proven to work well in several classes of problems, but the approach has not been verified in any problem with real data. In this paper, we study two different types of modelling errors in the case of electrical impedance tomography: one related to model reduction and one concerning partially unknown geometry. We show that the approach is also feasible in practice and may facilitate the reduction of the computational complexity of the nonlinear EIT problem at least by an order of magnitude.

Effect of a thin layer on the measurement of the thermal diffusivity of a material by a flash method

The flash method has been the standard technique for the measurement of thermal diffusivity since it was first developed by Parker and co-workers. Several modifications of the original method have been proposed in the literature, to deal with different physical situations and materials. In this work, we extend the front-face flash method for the simultaneous estimation of the thermal diffusivity and thermal conductivity of aerodynamically levitated spherical solid metal samples at high temperatures, with laser excitation and contactless temperature measurements taken with a pyrometer. The mathematical model associated to the physical problem is formulated as an axisymmetric heat conduction problem, with heat losses by radiation and convection. Since the measurements of a pyrometer are in fact radiative fluxes, the transfer function of the pyrometer is integrated in the mathematical model. A sensitivity analysis is performed and reveals that the thermal diffusivity and the thermal conductivity of the sample can be simultaneously estimated, provided that the other model parameters are known. The solution of the inverse problem is obtained with algorithms within the Bayesian framework of statistics, by using a heat conduction reduced model with linear boundary conditions, for which an analytical solution is developed for fast computations. The Bayesian Approximation Error Model is used to compensate for errors between a complete heat conduction model with nonlinear boundary conditions and a reduced heat conduction model with linear boundary conditions. Moreover, errors related to the inaccuracy and the uncertainties of all model parameters are accounted for in the inverse analysis. Simulated transient flux measurements are assumed available from a multi-spectral pyrometer and are used for the inverse problem solution.