P. J. Vauhkonen

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.

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.

Fluid dynamical models and state estimation in process tomography: Effect due to inaccuracies in flow fields

In this paper we consider the reconstruction of rapidly varying objects in process tomography. The evolution of the physical parameters is approximated with stochastic convection diffusion and fluid dynamics models. The actual time-varying reconstruction is carried out as a state estimation problem. As the boundary observations we use the voltage data of electrical impedance tomography. We have previously shown that state estimation works well in process tomography in the cases in which the fluid dynamics of the system are modeled correctly. In the real case, however, the velocity field cannot usually be determined accurately. This may be caused, for example, by complex nature of the flow, the turbulence, discretization, etc. In adopting the first proposed approach, it is essential to know how much the inaccuracies in the fluid dynamical model affect the state estimates in process tomography. In this paper we consider the tolerance of the approach with respect to these inaccuracies. We show that the estimation scheme is relatively tolerant to modeling errors in the flow field. Thus relatively reliable estimates can be obtained, for example, in a case in which a laminar flow model is used in turbulent flow conditions. However, the degradation that is due to incorrect flow fields is not insignificant and it is also conjectured that it could be possible that an extension of the proposed method could be used to estimate some flow field parameters.

Dynamic electrical impedance tomography - phantom studies

In electrical impedance tomography (EIT) 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. In some applications the resistivity changes may be so fast that the information on the time evolution of the resistivity distribution is either lost or severely blurred. In this paper we study with phantom experiments the capabilities of the earlier proposed approaches, Kalman filter and Kalman smoother. We show that in “two-dimensional” phantom these approaches are capable of reconstructing a sequence of absolute images of a moving object. By an absolute image we mean that no additional reference voltage measurement is needed for the image reconstruction. An image is obtained after each current injection. Also, when compared to traditional reconstructions, the dynamic approaches reveal much more information about the dynamic behaviour of the object.

Errors due to the truncation of the computational domain in static three-dimensional electrical impedance tomography.

In electrical impedance tomography (EIT), 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. The currents spread out in three dimensions and therefore off-plane structures have a significant effect on the reconstructed images. A question arises: how far from the current carrying electrodes should the discretized model of the object be extended? If the model is truncated too near the electrodes, errors are produced in the reconstructed images. On the other hand if the model is extended very far from the electrodes the computational time may become too long in practice. In this paper the model truncation problem is studied with the extended finite element method. Forward solutions obtained using so-called infinite elements, long finite elements and separable long finite elements are compared to the correct solution. The effects of the truncation of the computational domain on the reconstructed images are also discussed and results from the three-dimensional (3D) sensitivity analysis are given. We show that if the finite element method with ordinary elements is used in static 3D EIT, the dimension of the problem can become fairly large if the errors associated with the domain truncation are to be avoided.

Sensitivity matrix and reconstruction algorithm for EIT assuming axial uniformity

In electrical impedance tomography (EIT) two-dimensional models continue to be applied despite their known inability to provide correct reconstruction. In this paper, a reconstruction algorithm that assumes a translationally invariant conductivity distribution is described. A more precise forward solver is obtained by taking off-slice currents into consideration. An appropriate sensitivity matrix is derived. Numerical evidence for the improvement in precision compared to two-dimensional reconstruction is given.

Three-dimensional electrical impedance tomography using complete electrode model

In Electrical Impedance Tomography (EIT) 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 2D electrode plane and the reconstruction is based on the 2D 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 3D resistivity distributions. Both the forward and the inverse problems are discussed and the results from reconstructions with simulated an real measurement data are presented. 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. This makes it possible to apply the proposed method also for static EIT with complicated geometries.