Marko Vauhkonen

Tikhonov regularization and prior information in electrical impedance tomography

The solution of impedance distribution in electrical impedance tomography is a nonlinear inverse problem that requires the use of a regularization method. The generalized Tikhonov regularization methods have been popular in the solution of many inverse problems. The regularization matrices that are usually used with the Tikhonov method are more or less ad hoc and the implicit prior assumptions are, thus, in many cases inappropriate. In this paper, the authors propose an approach to the construction of the regularization matrix that conforms to the prior assumptions on the impedance distribution. The approach is based on the construction of an approximating subspace for the expected impedance distributions. It is shown by simulations that the reconstructions obtained with the proposed method are better than with two other schemes of the same type when the prior is compatible with the true object. On the other hand, when the prior is incompatible with the true object, the method will still give reasonable estimates.

Statistical inversion and Monte Carlo sampling methods in electrical impedance tomography

This paper discusses the electrical impedance tomography (EIT) problem: electric currents are injected into a body with unknown electromagnetic properties through a set of contact electrodes. The corresponding voltages that are needed to maintain these currents are measured. The objective is to estimate the unknown resistivity, or more generally the impedivity distribution of the body based on this information. The most commonly used method to tackle this problem in practice is to use gradient-based local linearizations. We give a proof for the differentiability of the electrode boundary data with respect to the resistivity distribution and the contact impedances. Due to the ill-posedness of the problem, regularization has to be employed. In this paper, we consider the EIT problem in the framework of Bayesian statistics, where the inverse problem is recast into a form of statistical inference. The problem is to estimate the posterior distribution of the unknown parameters conditioned on measurement data. From the posterior density, various estimates for the resistivity distribution can be calculated as well as a posteriori uncertainties. The search of the maximum a posteriori estimate is typically an optimization problem, while the conditional expectation is computed by integrating the variable with respect to the posterior probability distribution. In practice, especially when the dimension of the parameter space is large, this integration must be done by Monte Carlo methods such as the Markov chain Monte Carlo (MCMC) integration. These methods can also be used for calculation of a posteriori uncertainties for the estimators. In this paper, we concentrate on MCMC integration methods. In particular, we demonstrate by numerical examples the statistical approach when the prior densities are non-differentiable, such as the prior penalizing the total variation or the L1 norm of the resistivity.

Three-dimensional electrical impedance tomography based on the complete electrode model

In electrical impedance tomography an approximation for the internal resistivity distribution is computed based on the knowledge of the injected currents and measured voltages on the surface of the body. It is often assumed that the injected currents are confined to the two-dimensional (2-D) electrode plane and the reconstruction is based on 2-D assumptions. However, the currents spread out in three dimensions and, therefore, off-plane structures have significant effect on the reconstructed images. In this paper we propose a finite element-based method for the reconstruction of three-dimensional resistivity distributions. The proposed method is based on the so-called complete electrode model that takes into account the presence of the electrodes and the contact impedances. Both the forward and the inverse problems are discussed and results from static and dynamic (difference) reconstructions with real measurement data are given. It is shown that in phantom experiments with accurate finite element computations it is possible to obtain static images that are comparable with difference images that are reconstructed from the same object with the empty (saline filled) tank as a reference.

Inverse problems with structural prior information

In this paper we propose a method for the regularization of inverse problems whose solutions are known to exhibit anisotropic characteristics. The method is based on the generalized Tikhonov regularization and on the spatial prior information on the underlying solution. We allow the prior information to be only of approximate nature. In the proposed method, the prior information is incorporated into the regularization operator with the aid of a properly constructed matrix-valued field. Although the approach is deterministic it also has a clear statistical interpretation that will be discussed from the Bayesian viewpoint. The method is applied to two examples, the first is the inversion of a Fredholm integral equation of the first kind and the second is a case study of electrical impedance tomography (EIT).

A MATLAB package for the EIDORS project to reconstruct two-dimensional EIT images.

The EIDORS (electrical impedance and diffuse optical reconstruction software) project aims to produce a software system for reconstructing images from electrical or diffuse optical data. MATLAB is a software that is used in the EIDORS project for rapid prototyping, graphical user interface construction and image display. We have written a MATLAB package (http://venda.uku.fi/ vauhkon/) which can be used for two-dimensional mesh generation, solving the forward problem and reconstructing and displaying the reconstructed images (resistivity or admittivity). In this paper we briefly describe the mathematical theory on which the codes are based on and also give some examples of the capabilities of the package.

A Kalman filter approach to track fast impedance changes in electrical impedance tomography

In electrical impedance tomography (EIT), an estimate for the cross-sectional impedance distribution is obtained from the body by using current and voltage measurements made from the boundary. All well-known reconstruction algorithms use a full set of independent current patterns for each reconstruction. In some applications, the impedance changes may be so fast that information on the time evolution of the impedance distribution is either lost or severely blurred. Here, the authors propose an algorithm for EIT reconstruction that is able to track fast changes in the impedance distribution. The method is based on the formulation of EIT as a state-estimation problem and the recursive estimation of the state with the aid of the Kalman filter. The performance of the proposed method is evaluated with a simulation of human thorax in a situation in which the impedances of the ventricles change rapidly. The authors show that with optimal current patterns and proper parameterization, the proposed approach yields significant enhancement of the temporal resolution over the conventional reconstruction strategy.

Approximation errors and model reduction with an application in optical diffusion tomography

Model reduction is often required in several applications, typically due to limited available time, computer memory or other restrictions. In problems that are related to partial differential equations, this often means that we are bound to use sparse meshes in the model for the forward problem. Conversely, if we are given more and more accurate measurements, we have to employ increasingly accurate forward problem solvers in order to exploit the information in the measurements. Optical diffusion tomography (ODT) is an example in which the typical required accuracy for the forward problem solver leads to computational times that may be unacceptable both in biomedical and industrial end applications. In this paper we review the approximation error theory and investigate the interplay between the mesh density and measurement accuracy in the case of optical diffusion tomography. We show that if the approximation errors are estimated and employed, it is possible to use mesh densities that would be unacceptable with a conventional measurement model.

Recovery of region boundaries of piecewise constant coefficients of an elliptic PDE from boundary data

In this study we consider the recovery of smooth region boundaries of piecewise constant coefficients of an elliptic PDE, - a+b = f, from data on the exterior boundary . The assumption made is that the values of the coefficients (a,b) are known a priori but the information about the geometry of the smooth region boundaries where a and b are discontinous is missing. For the full characterization of (a,b) it is then sufficient to find the region boundaries separating different values of the coefficients. This leads to a nonlinear ill-posed inverse problem. In this study we propose a numerical algorithm that is based on the finite-element method and subdivision of the discretization elements. We formulate the forward problem as a mapping from a set of coefficients representing boundary shapes to data on , and derive the Jacobian of this forward mapping. Then an iterative algorithm which seeks a boundary configuration minimizing the residual norm between measured and predicted data is implemented. The method is illustrated first for a general elliptic PDE and then applied to optical tomography where the goal is to find the diffusion and absorption coefficients of the object by transilluminating the object with visible or near-infrared light. Numerical test results for this specific application are given with synthetic data.

Simultaneous reconstruction of electrode contact impedances and internal electrical properties: I. Theory

In electrical impedance tomography (EIT) currents are applied through the electrodes attached on the surface of the object and the resulting voltages are measured using the same or additional electrodes. The internal admittivity distribution is estimated based on the current and voltage data. When the voltages are measured on the current carrying electrodes the contact impedance that exists in the electrode–surface interface causes a voltage drop. In some cases this effect of the electrodes is known. However, this is not always the case and the contact impedance has to be taken into account in the image reconstruction. In this paper we propose an approach for estimating the contact impedance of the electrodes simultaneously with the estimation of the admittivity of the object. The complete electrode model (CEM) is used in the estimation procedure. We compare the proposed approach to a simple method which is based on the well known definition of the sample resistivity. The proposed approach is tested with real measurements by estimating the admittivity of isotonic saline solution in a cylindrical test cell and with simulations in a three-dimensional cylindrical domain. The CEM-based approach is shown to produce results that are similar to the results obtained with the simple approach in the test cell case. The advantage of the CEM-based approach over the simple approach is that the complete electrode model does not have any geometrical constraints, which makes it possible to utilize it in EIT studies. The results show that the CEM-based approach works well and can be used in practical contact impedance estimation with real measurements. This will be further studied in part II of this paper.

Assessment of errors in static electrical impedance tomography with adjacent and trigonometric current patterns

In electrical impedance tomography (EIT), difference imaging is often preferred over static imaging. This is because of the many unknowns in the forward modelling which make it difficult to obtain reliable absolute resistivity estimates. However, static imaging and absolute resistivity values are needed in some potential applications of EIT. In this paper we demonstrate by simulation the effects of different error components that are included in the reconstruction of static EIT images. All simulations are carried out in two dimensions with the so-called complete electrode model. Errors that are considered are the modelling error in the boundary shape of an object, errors in the electrode sizes and localizations and errors in the contact impedances under the electrodes. Results using both adjacent and trigonometric current patterns are given.