Abstract. Diffuse optical tomography (DOT) is an imaging modality in which images of the optical properties of a biological tissue, specifically diffusion [Formula: see text] and absorption [Formula: see text], are estimated based on measurements of near-infrared light on the surface of the body. In practical applications, one often lacks exact knowledge of the measurement domain boundary. This poses a significant challenge, as inaccuracies in the boundary shape of the computational domain may result in substantial artifacts in the reconstructed images. In this study, the following two results are achieved in a two-dimensional setting: (i) when the measurement domain [Formula: see text] is known, it is shown that knowledge of the Robin-to-Neumann map for two modulation frequencies uniquely determines [Formula: see text] and [Formula: see text], both assumed to be isotropic. (ii) When the measurement domain [Formula: see text] is not exactly known, a method is proposed for simultaneous reconstruction of [Formula: see text] and [Formula: see text] as well as the boundary [Formula: see text]. For this, DOT measurements are needed for two modulation frequencies. This approach yields reconstructed coefficients that are a conformal deformation of the true coefficients in the exact domain. The new method is demonstrated using simulated noisy data.

Abstract. This work introduces a method for preprocessing measurements of electrical impedance tomography to considerably reduce the effect uncertainties in the electrode contacts have on the reconstruction quality, without a need to explicitly estimate the contacts. The idea is to compute the Jacobian matrix of the forward map with respect to the contact strengths and project the electrode measurements and the forward map onto the orthogonal complement of the range of this Jacobian. Using the smoothened complete electrode model as the forward model, it is demonstrated that inverting the resulting projected equation with respect to only the internal conductivity of the examined body results in good quality reconstructions both when resorting to a single step linearization with a smoothness prior and when combining lagged diffusivity iteration with total variation regularization. The quality of the reconstructions is further improved if the range of the employed projection is also orthogonal to that of the Jacobian with respect to the electrode positions. These results hold even if the projections are formed at internal and contact conductivities that significantly differ from the true ones; it is numerically demonstrated that the orthogonal complement of the range of the contact Jacobian is almost independent of the conductivity parameters at which it is evaluated. In particular, our observations introduce a numerical technique for inferring whether a change in the electrode measurements is caused by a change in the internal conductivity or alterations in the electrode contacts, which has potential applications, e.g., in bedside monitoring of stroke patients. The ideas are tested both on simulated data and on real-world water tank measurements with adjustable contact resistances.

The objective of electrical impedance tomography (EIT) is to reconstruct the internal conductivity of a physical body based on current and voltage measurements at the boundary of the body. In many medical applications the exact shape of the domain boundary and contact impedances are not available. This is problematic as even small errors in the boundary shape of the computation domain or in the contact impedance values can produce large artifacts in the reconstructed images, which results in a loss of relevant information. A method is proposed that simultaneously reconstructs the conductivity, the contact impedances, and the boundary shape from EIT data. The approach consists of three steps: first, the unknown contact impedances and an anisotropic conductivity reproducing the measured EIT data in a model domain are computed. Second, using isothermal coordinates, a deformation is constructed that makes the conductivity isotropic. The final step minimizes the error of true and reconstructed known geometric properties (like the electrode lengths) using conformal deformations. The feasibility of the method is illustrated with experimental EIT data, with robust and accurate reconstructions of both conductivity and boundary shape.

A computational method is introduced for choosing the regularization parameter for total variation (TV) regularization. A partial understanding of the properties of the method is provided by rigorously proving that the TV norms of the reconstructions converge with any choice of regularization parameter. The computational approach is based on computing reconstructions at a few different resolutions and various values of regularization parameter. The chosen parameter is the smallest one resulting in approximately discretization-invariant TV norms of the reconstructions. The method is tested with simulated and experimental X-ray tomography data and compared to the S-curve method. The results are comparable to those of the S-curve method. However, the S-curve method needs quantitative a priori information about the expected sparsity (TV norm) of the unknown, while the proposed method does not require such input parameters.

The combination of positron emission tomography (PET) and magnetic resonance imaging (MRI) offers unique possibilities. In this paper we aim to exploit the high spatial resolution of MRI to enhance the reconstruction of simultaneously acquired PET data. We propose a new prior to incorporate structural side information into a maximum a posteriori reconstruction. The new prior combines the strengths of previously proposed priors for the same problem: it is very efficient in guiding the reconstruction at edges available from the side information and it reduces locally to edge-preserving total variation in the degenerate case when no structural information is available. In addition, this prior is segmentation-free, convex and no a priori assumptions are made on the correlation of edge directions of the PET and MRI images. We present results for a simulated brain phantom and for real data acquired by the Siemens Biograph mMR for a hardware phantom and a clinical scan. The results from simulations show that the new prior has a better trade-off between enhancing common anatomical boundaries and preserving unique features than several other priors. Moreover, it has a better mean absolute bias-to-mean standard deviation trade-off and yields reconstructions with superior relative$\ell ^{2}$-error and structural similarity index. These findings are underpinned by the real data results from a hardware phantom and a clinical patient confirming that the new prior is capable of promoting well-defined anatomical boundaries.

Electrical impedance tomography is a highly unstable problem with respect to measurement and modeling errors. This instability is especially severe when absolute imaging is considered. With clinical measurements, accurate knowledge about the body shape is usually not available, and therefore an approximate model domain has to be used in the computational model. It has earlier been shown that large reconstruction artefacts result if the geometry of the model domain is incorrect. In this paper, we adapt the so-called approximation error approach to compensate for the modeling errors caused by inaccurately known body shape. This approach has previously been shown to be applicable to a variety of modeling errors, such as coarse discretization in the numerical approximation of the forward model and domain truncation. We evaluate the approach with a simulated example of thorax imaging, and also with experimental data from a laboratory setting, with absolute imaging considered in both cases. We show that the related modeling errors can be efficiently compensated for by the approximation error approach. We also show that recovery from simultaneous discretization related errors is feasible, allowing the use of computationally efficient reduced order models.

In electrical impedance tomography (EIT) electric currents are injected into a body with unknown electromagnetic properties through a set of contact electrodes at the boundary of the body. The resulting voltages are measured on the same electrodes and the objective is to reconstruct the unknown conductivity function inside the body based on these data. All the traditional approaches to the reconstruction problem assume that the boundary of the body and the electrode-skin contact impedances are known a priori. However, in clinical experiments one usually lacks the exact knowledge of the boundary and contact impedances, and therefore, approximate model domain and contact impedances have to be used in the image reconstruction. However, it has been noticed that even small errors in the shape of the computation domain or contact impedances can cause large systematic artefacts in the reconstructed images, leading to loss of diagnostically relevant information. In a recent paper (Kolehmainen , 2006), we showed how in the 2-D case the errors induced by the inaccurately known boundary can be eliminated as part of the image reconstruction and introduced a novel method for finding a deformed image of the original isotropic conductivity using the theory of TeichmUller mappings. In this paper, the theory and reconstruction method are extended to include the estimation of unknown contact impedances. The method is implemented numerically and tested with experimental EIT data. The results show that the systematic errors caused by inaccurately known boundary and contact impedances can efficiently be eliminated by the reconstruction method.

In conventional tomography, the interior of an object is reconstructed from tomographic projections such as X-ray or transmission electron microscope images. All the current reconstruction methods assume that projection geometry of the imaging device is either known or solved in advance by using e.g., fiducial or nonfiducial feature points in the images. In this paper, we propose a novel approach where the imaging geometry is solved simultaneously with the volume reconstruction problem while no correspondence information is needed. Our approach is a direct application of Bayesian inversion theory and produces the maximum likelihood or maximum a posteriori estimates for the motion parameters under the selected noise and prior distributions. In this paper, the method is implemented for a two-dimensional model problem with one-dimensional affine projection data. The performance of the method is tested with simulated and measured X-ray projection data.

Diagnostic and operational tasks based on dental radiology often require three-dimensional (3-D) information that is not available in a single X-ray projection image. Comprehensive 3-D information about tissues can be obtained by computerized tomography (CT) imaging. However, in dental imaging a conventional CT scan may not be available or practical because of high radiation dose, low-resolution or the cost of the CT scanner equipment. In this paper, we consider a novel type of 3-D imaging modality for dental radiology. We consider situations in which projection images of the teeth are taken from a few sparsely distributed projection directions using the dentist's regular (digital) X-ray equipment and the 3-D X-ray attenuation function is reconstructed. A complication in these experiments is that the reconstruction of the 3-D structure based on a few projection images becomes an ill-posed inverse problem. Bayesian inversion is a well suited framework for reconstruction from such incomplete data. In Bayesian inversion, the ill-posed reconstruction problem is formulated in a well-posed probabilistic form in which a priori information is used to compensate for the incomplete information of the projection data. In this paper we propose a Bayesian method for 3-D reconstruction in dental radiology. The method is partially based on Kolehmainen et al. 2003. The prior model for dental structures consist of a weighted l1 and total variation (TV)-prior together with the positivity prior. The inverse problem is stated as finding the maximum a posteriori (MAP) estimate. To make the 3-D reconstruction computationally feasible, a parallelized version of an optimization algorithm is implemented for a Beowulf cluster computer. The method is tested with projection data from dental specimens and patient data. Tomosynthetic reconstructions are given as reference for the proposed method.