OOEIT: An open source object oriented electrical impedance tomography software package

This paper presents a pure element-free Galerkin method (EFGM) forward model for image reconstruction in 2D and 3D electrical impedance tomography (EIT) using an adaptive current injection method. In EIT systems with the adapting current injection method, both static and dynamic images can be reconstructed; however, determination of electrode contact impedances in the complete electrode model is difficult and the Gap model is used. In this paper, in the EIT forward problem a weak form functional based on the Gap model and a pure EFGM approach are developed, and in the EIT inverse problem, Jacobian matrix is computed by the EFGM, and a fast integration technique is introduced to calculate the entries of the Jacobian matrix within an adequate computation time. The influence of increasing the density of nodes at and near the electrodes with steep electric potential gradients on the accuracy of FEM and EFGM forward solutions is investigated, and the performance of the image reconstruction algorithm with the proposed fast integration technique is examined. The numerical results reveal that the proposed EFGM forward model with the fast integration technique has an efficient performance both in terms of mean relative imaging errors and computational time.

A real-time EIT imaging system based on the split augmented Lagrangian shrinkage algorithm

Image reconstruction in electrical impedance tomography using the sensitivity theorem is generally based on the assumption that the initial conductivity distribution of the body being imaged is uniform. The technique of image reconstruction using this method is described and reconstructed images are presented. Improvements in image quality and accuracy are demonstrated when accurate a priori `anatomical' information, in the form of a model of the distribution of conductivity within the region to be imaged, is used to construct the sensitivity matrix. In practice correct a priori information is not available, for example the conductivity values within the various anatomical regions will not be known. An iterative algorithm is presented which allows the conductivity parameters of the a priori model to be determined during reconstruction. Keywords: electrical impedance tomography, image reconstruction, a priori information

This paper introduces a new method of simultaneous acquisition of electrocardiogram (ECG) signals and electrical impedance tomography (EIT) measurements on active electrodes, i.e. electrodes that are applying current, in an EIT system. Howland noise reduction followed by an integrate-and-dump filter and adaptive filtering are used to extract the ECG signal without loss of amplitude. Results of simultaneous ECG signals and EIT reconstructions on human subjects are shown. It is also demonstrated that the recovered ECG waveforms from 32 electrodes can be used to reconstruct the heart current source vectors during a cardiac cycle, yielding an approximate solution to the inverse problem of electrocardiography. Acquiring ECG signals from all electrodes attached to the body simultaneously with EIT data acquisition eliminates the need for a separate system to collect ECG signals. Time-aligned ECG signals are helpful in interpreting pulsatile perfusion images by providing exact timing of each image in relation to the cardiac cycle. Additionally, collecting ECG signals from many or all of the EIT electrodes enables reconstruction of the heart's moving dipole moment at the same time as the EIT images.

Image reconstruction in electrical impedance tomography (EIT) deals with an ill-posed and non-linear inverse problem. It has the objective of minimizing the difference between simulated (virtual) object data and electric voltage measurements performed on a non-simulated (real) object. In this paper, a new approach to the simulated annealing method applied to the reconstruction of EIT images is described. The main advantage of this approach is that all conductivity parameters are updated simultaneously. Most methods that employ simulated annealing to the problem of EIT usually evaluate each conductivity parameter individually resulting in high computational cost. The algorithm was tested both with computationally generated data and with measurements performed on a physical tank. In both cases, the method was able to make data inversion, determining the position, the dimensions and the conductivity of materials in an opaque object plane.

Impact of the Radial Basis Function Spread Factor onto Image Reconstruction in Electrical Impedance Tomography

Magnetic induction tomography (MIT) of biological tissue is used for the reconstruction of the complex conductivity distribution κ inside the object under investigation. It is based on the perturbation of an alternating magnetic field caused by the object and can be used in all applications of electrical impedance tomography (EIT) such as functional lung monitoring and assessment of tissue fluids. In contrast to EIT, MIT does not require electrodes and magnetic fields can also penetrate non-conducting barriers such as the skull. As in EIT, the reconstruction of absolute conductivity values is very difficult because of the method's sensitivity to numerical errors and noise. To overcome this problem, image reconstruction in EIT is often done differentially. Analogously, this concept has been adopted for MIT. Two different methods for differential imaging are applicable. The first one is state-differential, for example when the conductivity change between inspiration and expiration in the lung regions is being detected. The second one is frequency-differential, which is of high interest in motionless organs like the brain, where a state-differential method cannot be applied. An equation for frequency-differential MIT was derived taking into consideration the frequency dependence of the sensitivity matrix. This formula is valid if we can assume that only small conductivity changes occur. In this way, the non-linear inverse problem of MIT can be approximated by a linear one (depending only on the frequency), similar to in EIT. Keeping this limitation in mind, the conductivity changes between one or more reference frequencies and several measurement frequencies were reconstructed, yielding normalized conductivity spectra. Due to the differential character of the method, these spectra do not provide absolute conductivities but preserve the shape of the spectrum. The validity of the method was tested with artificial data generated with a spherical perturbation within a conducting cylinder as well as for real measurement data. The measurement data were obtained from a potato immersed in saline. The resulting spectra were compared with reference measurements and the preservation of the shape of the spectra was analyzed.

Image reconstruction in electrical impedance tomography is, generally, an ill-posed nonlinear inverse problem. Regularization methods are widely used to ensure a stable solution. In this paper we present a case study, which uses a novel electrical impedance tomography method for reconstruction of layered biological tissues with piece-wise continuous plane-stratified profiles. The algorithm implements the recently proposed reconstruction scheme for piecewise constant conductivity profiles, based on an improved Prony method in conjunction with Bessel transform. It is shown that the proposed algorithm is capable of successfully reconstructing piece-wise continuous conductivity profiles with moderate (WKB) slop. This reconstruction procedure, which calculate both the locations and the conductivities, repetitively provides inhomogeneous depth discretization, i.e., the depths grid is not equispaced. Incorporation of this specific inhomogeneous grid, the main result of our investigation, in the widely used MLS reconstruction procedure results in a stable reconstruction, whereas, utilization of equispaced depth grid leads to inaccurate reconstruction. The effectiveness and the significance of inhomogeneous discretization in EIT reconstruction procedure is further demonstrated and verified via numerical simulations.

This paper describes a new approach to reconstruction of the conductivity field in electrical impedance tomography. Our goal is to improve the tradeoff between the quality of the images and the numerical complexity of the reconstruction method. In order to reduce the computational load, we adopt a linearized approximation to the forward problem that describes the relationship between the unknown conductivity and the measurements. In this framework, we focus on finding a proper way to cope with the ill-posed nature of the problem, mainly caused by strong attenuation phenomena; this is done by devising regularization techniques well suited to this particular problem. First, we propose a solution which is based on Tikhonov regularization of the problem. Second, we introduce an original regularized reconstruction method in which the regularization matrix is determined by space-uniformization of the variance of the reconstructed condictivities. Both methods are nonsupervised, i.e., all tuning parameters are automatically determined from the measured data. Tests performed on simulated and real data indicate that Tikhonov regularization provides results similar to those obtained with iterative methods, but with a much smaller amount of computations. Regularization using a variance uniformization constraint yields further improvements, particularly in the central region of the unknown object where attenuation is most severe. We anticipate that the variance uniformization approach could be adapted to iterative methods that preserve the nonlinearity of the forward problem. More generally, it appears as a useful tool for solving other severely ill-posed reconstruction problems such as eddy current tomography.

Resistivity reconstruction of practical phantoms is studied with Projection Error Propagation-based Regularization (PEPR) method to improve the image quality in Electrical Impedance Tomography (EIT). PEPR method calculates the regularization parameter as a function of the projection error produced due to the mismatch between experimental measurements and the calculated data. The regularization parameter in the reconstruction algorithm automatically adjusts its magnitude depending on the noise level present in measured data as well as the ill-posedness of the Hessian matrix. Resistivity images are reconstructed from the practical phantom data using the Electrical Impedance Diffuse Optical Reconstruction Software (EIDORS). The resistivity images reconstructed with PEPR method are compared with the Single step Tikhonov Regularization (STR) and Modified Levenberg Regularization (LMR) techniques. The results show that, the PEPR technique reduces the reconstruction errors in each iteration and improves the reconstructed images with better contrast to noise ratio (CNR), percentage of contrast recovery (PCR), coefficient of contrast (COC) and diametric resistivity profile (DRP).KeywordsElectrical Impedance Tomography (EIT)Projection Error Propagation-based Regularization (PEPR)Practical PhantomsBoundary potentialsEIDORSResistivity reconstructionSTRLMR

Image reconstruction in electrical impedance tomography (EIT) is a non-linear inverse problem. The linear model is always used in most of the reconstruction algorithm in dynamic EIT, which causes large errors of image reconstruction. In this paper, we proposed a new image reconstruction method in dynamic EIT. In the method the artificial neural network based on the error back propagation algorithm (BP ANN) is used to express the non-linear relation between the impedance change position inside the measured object and the voltage change value measured on the surface of the object. Thus, the location ofthe impedance change can be decided by the measured voltage change on the surface, and then the impedance change image will be reconstructed with linear approximated method. The reconstructed error will be decreased largely, because the impedance change position can be detected precisely by our proposed method. The experimental results indicate that the precision of the reconstructed image with our method is greatly higher than that with the back projection method.

Realistic forward and inverse model mesh generation for rapid three-dimensional thoracic electrical impedance imaging

This research has been aimed to carry out a study of peculiarities that arise in a numerical simulation of the electrical impedance tomography (EIT) problem. Static EIT image reconstruction is sensitive to a measurement noise and approximation error. A special consideration has been given to reducing of the approximation error, which originates from numerical implementation drawbacks. This paper presents in detail two numerical approaches for solving EIT forward problem. The finite volume method (FVM) on unstructured triangular mesh is introduced. In order to compare this approach, the finite element (FEM) based forward solver was implemented, which has gained the most popularity among researchers. The calculated potential distribution with the assumed initial conductivity distribution has been compared to the analytical solution of a test Neumann boundary problem and to the results of problem simulation by means of ANSYS FLUENT commercial software. Two approaches to linearized EIT image reconstruction are discussed. Reconstruction of the conductivity distribution is an ill-posed problem, typically requiring a large amount of computation and resolved by minimization techniques. The objective function to be minimized is constructed of measured voltage and calculated boundary voltage on the electrodes. A classical modified Newton type iterative method and the stochastic differential evolution method are employed. A software package has been developed for the problem under investigation. Numerical tests were conducted on simulated data. The obtained results could be helpful to researches tackling the hardware and software issues for medical applications of EIT.

Electrical impedance tomography (EIT) is an imaging modality that can estimate a visualization of the conductivity distribution inside the human body. However, the spatial resolution of EIT is limited because measurements are sensitive to noise. We investigate a technique to incorporate a priori information into the EIT reconstructions of the D-Bar algorithm. Our paper aims to help engineers understand the behavior of the D-Bar algorithm and its implementation. The a priori information is provided by a radar setup and a one-dimensional reconstruction of the radar data. The EIT reconstruction is carried out with a D-Bar algorithm. An intermediate step in the D-Bar algorithm is the scattering transform. The a priori information is added in this exact step to increase the spatial resolution of the reconstruction. As the D-Bar algorithm is widely used in the mathematical community and thus far has limited usage in the engineering domain, we also aim to explain the implementation of the algorithm and give an intuitive understanding where possible. Different parameters of the reconstruction algorithm are analyzed systematically with the help of the GREIT figures of merit. Even a limited one-dimensional a priori information can increase the reconstruction quality considerably. Artifacts from noisy EIT measurements are reduced. However, the selection of the amount of a priori information and the estimation of its value can worsen the reconstruction results again.

Studying of elemental resistivity profile of reconstructed images in Electrical Impedance Tomography (EIT) is essential to assess its image quality, reconstruction process and the systems performance. Visual assessment of the impedance images must not be accepted as the ultimate and sufficient judgment criteria for reconstruction efficiency of the tomograph. To identify the best image quality in EIT reconstruction, resistivity images are reconstructed from the simulated data using Electrical Impedance Diffuse Optical Reconstruction Software (EIDORS) and their elemental resistivity profiles are analyzed with image analyzing parameters. Results show that the image analyzing parameters are essential to assess the reconstructed images more technically, qualitatively and quantitatively.KeywordsElectrical Impedance Tomography (EIT)EIDORSImage AnalysisImage Analyzing ParametersImpedance ImagesPhantomResistivity Images