Approximate marginalization of unknown scattering in quantitative photoacoustic tomography
Aki Pulkkinen Ville Kolehmainen Jari P. Kaipio Benjamin T. Cox Simon R. Arridge Tanja Tarvainen
Inverse Problems & Imaging 2014, 8(3): 811-829 doi: 10.3934/ipi.2014.8.811
Quantitative photoacoustic tomography is a hybrid imaging method, combining near-infrared optical and ultrasonic imaging. One of the interests of the method is the reconstruction of the optical absorption coefficient within the target. The measurement depends also on the uninteresting but often unknown optical scattering coefficient. In this work, we apply the approximation error method for handling uncertainty related to the unknown scattering and reconstruct the absorption only. This way the number of unknown parameters can be reduced in the inverse problem in comparison to the case of estimating all the unknown parameters. The approximation error approach is evaluated with data simulated using the diffusion approximation and Monte Carlo method. Estimates are inspected in four two-dimensional cases with biologically relevant parameter values. Estimates obtained with the approximation error approach are compared to estimates where both the absorption and scattering coefficient are reconstructed, as well to estimates where the absorption is reconstructed, but the scattering is assumed (incorrect) fixed value. The approximation error approach is found to give better estimates for absorption in comparison to estimates with the conventional measurement error model using fixed scattering. When the true scattering contains stronger variations, improvement of the approximation error method over fixed scattering assumption is more significant.
keywords: quantitative photoacoustic tomography uncertainty quantification numerical methods. approximation error parameter estimation Inverse problems
Approximate marginalization of absorption and scattering in fluorescence diffuse optical tomography
Meghdoot Mozumder Tanja Tarvainen Simon Arridge Jari P. Kaipio Cosimo D'Andrea Ville Kolehmainen
Inverse Problems & Imaging 2016, 10(1): 227-246 doi: 10.3934/ipi.2016.10.227
In fluorescence diffuse optical tomography (fDOT), the reconstruction of the fluorophore concentration inside the target body is usually carried out using a normalized Born approximation model where the 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 the modeling 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 with different amount of error in the optical properties. The results show that the Bayesian approximation error approach improves the tolerance of fDOT imaging against modeling errors caused by inaccurately known absorption and scattering of the target.
keywords: Bayesian methods inverse problems tomography fluorescence diffuse optical tomography. Image reconstruction techniques
Recovering boundary shape and conductivity in electrical impedance tomography
Ville Kolehmainen Matti Lassas Petri Ola Samuli Siltanen
Inverse Problems & Imaging 2013, 7(1): 217-242 doi: 10.3934/ipi.2013.7.217
Electrical impedance tomography (EIT) aims to reconstruct the electric conductivity inside a physical body from current-to-voltage measurements at the boundary of the body. In practical EIT one often lacks exact knowledge of the domain boundary, and inaccurate modeling of the boundary causes artifacts in the reconstructions. A novel method is presented for recovering the boundary shape and an isotropic conductivity from EIT data. The first step is to determine the minimally anisotropic conductivity in a model domain reproducing the measured EIT data. Second, a Beltrami equation is solved, providing shape-deforming reconstruction. The algorithm is applied to simulated noisy data from a realistic electrode model, demonstrating that approximate recovery of the boundary shape and conductivity is feasible.
keywords: conformal deformation. electrical impedance tomography minimally anisotropic conductivity quasiconformal maps Inverse conductivity problem
Estimation of conductivity changes in a region of interest with electrical impedance tomography
Dong liu Ville Kolehmainen Samuli Siltanen Anne-maria Laukkanen Aku Seppänen
Inverse Problems & Imaging 2015, 9(1): 211-229 doi: 10.3934/ipi.2015.9.211
This paper proposes a novel approach to reconstruct changes in a target conductivity from electrical impedance tomography measurements. As in the conventional difference imaging, the reconstruction of the conductivity change is based on electrical potential measurements from the exterior boundary of the target before and after the change. In this paper, however, images of the conductivity before and after the change are reconstructed simultaneously based on the two data sets. The key feature of the approach is that the conductivity after the change is parameterized as a linear combination of the initial state and the change. This allows for modeling independently the spatial characteristics of the background conductivity and the change of the conductivity - by separate regularization functionals. The approach also allows in a straightforward way the restriction of the conductivity change to a localized region of interest inside the domain. While conventional difference imaging reconstruction is based on a global linearization of the observation model, the proposed approach amounts to solving a non-linear inverse problem. The feasibility of the proposed reconstruction method is tested experimentally and with a simulation which demonstrates a potential new medical application of electrical impedance tomography: imaging of vocal folds in voice loading studies.
keywords: absolute imaging imaging of vocal folds. region of interest electrical impedance tomography Inverse problem
Incorporating structural prior information and sparsity into EIT using parallel level sets
Ville Kolehmainen Matthias J. Ehrhardt Simon R. Arridge
Inverse Problems & Imaging 2019, 13(2): 285-307 doi: 10.3934/ipi.2019015

EIT is a non-linear ill-posed inverse problem which requires sophisticated regularisation techniques to achieve good results. In this paper we consider the use of structural information in the form of edge directions coming from an auxiliary image of the same object being reconstructed. In order to allow for cases where the auxiliary image does not provide complete information we consider in addition a sparsity regularization for the edges appearing in the EIT image. The combination of these approaches is conveniently described through the parallel level sets approach. We present an overview of previous methods for structural regularisation and then provide a variational setting for our approach and explain the numerical implementation. We present results on simulations and experimental data for different cases with accurate and inaccurate prior information. The results demonstrate that the structural prior information improves the reconstruction accuracy, even in cases when there is reasonable uncertainty in the prior about the location of the edges or only partial edge information is available.

keywords: Electrical impedance tomography structural prior computational inverse problem regularization finite element method

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