Erkki Somersalo

Existence and uniqueness for electrode models for electric current computed tomography

The following experiment is considered. To a body of given conductivity, a number of electrodes are attached, through which current is sent. On the same electrodes, the resulting voltages are measured. This experiment can be described by a number of mathematical models [K.-S. Cheng et al., IEEE Transactions on Biomedical Engineering, 36 (1989), pp. 918–924]. These models are discussed and their predictions compared with experiment. In particular, a model is exhibited that is capable of predicting the experimentally measured voltages to within 0.1 percent. For this model, existence and uniqueness of the associated electrical potential is proved.

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.

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).

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.

Layer stripping: a direct numerical method for impedance imaging

An impedance imaging problem is to find the electrical conductivity and permittivity distributions inside a body from measurements made on the boundary. The following experiment is considered: a set of electric currents are applied to the surface of the body and the resulting voltages are measured on that surface. The authors describe the performance of a direct numerical method for approximating the conductivity in the interior. The algorithm proceeds via two steps: first the conductivity is found near the bounding surface of the body from the data having the highest available spatial frequency; next the boundary data on an interior surface are synthesized using a nonlinear differential equation of Riccati type. The process is then repeated, and an estimate of the conductivity is found, layer by layer. They establish the theoretical basis for the algorithm and report on numerical tests.

Statistical inversion for medical x-ray tomography with few radiographs: I. General theory

In x-ray tomography, the structure of a three-dimensional body is reconstructed from a collection of projection images of the body. Medical CT imaging does this using an extensive set of projections from all around the body. However, in many practical imaging situations only a small number of truncated projections are available from a limited angle of view. Three-dimensional imaging using such data is complicated for two reasons: (i) typically, sparse projection data do not contain sufficient information to completely describe the 3D body, and (ii) traditional CT reconstruction algorithms, such as filtered backprojection, do not work well when applied to few irregularly spaced projections. Concerning (i), existing results about the information content of sparse projection data are reviewed and discussed. Concerning (ii), it is shown how Bayesian inversion methods can be used to incorporate a priori information into the reconstruction method, leading to improved image quality over traditional methods. Based on the discussion, a low-dose three-dimensional x-ray imaging modality is described.

On the existence and convergence of the solution of PML equations

In this article we study the mesh termination method in computational scattering theory known as the method of Perfectly Matched Layer (PML). This method is based on the idea of surrounding the scatterer and its immediate vicinity with a fictitious absorbing non-reflecting layer to damp the echoes coming from the mesh termination surface. The method can be formulated equivalently as a complex stretching of the exterior domain. The article is devoted to the existence and convergence questions of the solutions of the resulting equations. We show that with a special choice of the fictitious absorbing coefficient, the PML equations are solvable for all wave numbers, and as the PML layer is made thicker, the PML solution converge exponentially towards the actual scattering solution. The proofs are based on boundary integral methods and a new type of near-field version of the radiation condition, called here the double surface radiation condition.

Electrical impedance tomography with basis constraints

In this paper, we consider the impedance tomography problem of estimating the conductivity distribution within the body from static current/voltage measurements on the bodys surface. We present a new method of implementing prior information of the conductivities in the optimization algorithm. The method is based on the approximation of the prior covariance matrix by simulated samples of feasible conductivities. The reduction of the dimensionality of the optimization problem is performed by principal component analysis (PCA).

State estimation with fluid dynamical evolution models in process tomography - an application to impedance tomography

In this paper we consider the reconstruction of rapidly varying objects in process tomography. The evolution of the physical parameters can often be approximated with stochastic convection-diffusion and fluid dynamics models. We use the state estimation approach to obtain the tomographic reconstructions and show how these flow models can be exploited with the actual observation models that by themselves induce ill-posed problems. The state estimation problem can be stated in different ways based on the available temporal information. We concentrate on such cases in which continuous monitoring is essential but a small delay for the reconstructions is allowable. The state estimation problem is solved with the fixed-lag Kalman smoother algorithm. As the boundary observations we use the voltage data of electrical impedance tomography. We also give a numerical illustration of the approach in a case in which we track a bolus that moves rapidly through a pipeline.