A direct reconstruction algorithm for electrical impedance tomography

A direct (noniterative) reconstruction algorithm for electrical impedance tomography in the two-dimensional (2-D), cross-sectional geometry is reviewed. New results of a reconstruction of a numerically simulated phantom chest are presented. The algorithm is based on the mathematical uniqueness proof by A. I. Nachman [1996] for the 2-D inverse conductivity problem. In this geometry, several of the clinical applications include monitoring heart and lung function, diagnosis of pulmonary embolus, diagnosis of pulmonary edema, monitoring for internal bleeding, and the early detection of breast cancer.     

An image reconstruction algorithm for three-dimensional electrical impedance tomography

Electrical impedance tomography (EIT) has been studied by many authors and in most of this work it has been considered to be a two-dimensional problem. Many groups are now turning their attention to the full three-dimensional case in which the computational demands become much greater. It is interesting to look for ways to reduce this demand and in this paper we describe an implementation of an algorithm that is able to achieve this by precomputing many of the quantities needed in the image reconstruction. The algorithm is based on a method called NOSER introduced some years ago by Cheney et al. [3]. In this paper we have significantly extended the method by introducing a more realistic electrode model into the analysis. We have given explicit formulae for the quantities involved so that the reader can reproduce our results.     

A reconstruction algorithm for breast cancer imaging with electrical impedance tomography in mammography geometry

The conductivity and permittivity of breast tumors are known to differ significantly from those of normal breast tissues, and electrical impedance tomography (EIT) is being studied as a modality for breast cancer imaging to exploit these differences. At present, X-ray mammography is the primary standard imaging modality used for breast cancer screening in clinical practice, so it is desirable to study EIT in the geometry of mammography. This paper presents a forward model of a simplified mammography geometry and a reconstruction algorithm for breast tumor imaging using EIT techniques. The mammography geometry is modeled as a rectangular box with electrode arrays on the top and bottom planes. A forward model for the electrical impedance imaging problem is derived for a homogeneous conductivity distribution and is validated by experiment using a phantom tank. A reconstruction algorithm for breast tumor imaging based on a linearization approach and the proposed forward model is presented. It is found that the proposed reconstruction algorithm performs well in the phantom experiment, and that the locations of a 5-mm-cube metal target and a 6-mm-cube agar target could be recovered at a target depth of 15 mm using a 32 electrode system.     

A robust image reconstruction algorithm and its parallel implementation in electrical impedance tomography

An efficient and robust image reconstruction algorithm for static impedance imaging using Hachtel's augmented matrix method was developed. This improved Newton-Raphson method produced more accurate images by reducing the undesirable effects of the ill-conditioned Hessian matrix. It is demonstrated that the electrical impedance tomography (EIT) system could produce two-dimensional static images from a physical phantom with 7% spatial resolution at the center and 5% at the periphery. Static EIT image reconstruction requires a large amount of computation. In order to overcome the limitations on reducing the computation time by algorithmic approaches, the improved Newton-Raphson algorithm was implemented on a parallel computer system. It is shown that the parallel computation could reduce the computation time from hours to minutes.     

A neural network image reconstruction technique for electrical impedance tomography

Reconstruction of images in electrical impedance tomography requires the solution of a nonlinear inverse problem on noisy data. This problem is typically ill-conditioned and requires either simplifying assumptions or regularization based on a priori knowledge. The authors present a reconstruction algorithm using neural network techniques which calculates a linear approximation of the inverse problem directly from finite element simulations of the forward problem. This inverse is adapted to the geometry of the medium and the signal-to-noise ratio (SNR) used during network training. Results show good conductivity reconstruction where measurement SNR is similar to the training conditions. The advantages of this method are its conceptual simplicity and ease of implementation, and the ability to control the compromise between the noise performance and resolution of the image reconstruction.     

A reconstruction algorithm of electrical impedance tomography with optimal configuration of the driven electrodes

A reconstruction algorithm for electrical impedance tomography (EIT) is presented. The least-squares (LS) method is applied and a formulation similar to that of the perturbation method is found. The main difference from perturbation lies with the sensitivity matrix, which here is replaced by the Jacobian matrix, defined in terms of the partial derivatives of every sensing electrode pair voltage difference with respect to every element's conductivity. The mutual position between the active electrodes is chosen to give optimum sensitivity. The results shown that the algorithm presented here has a better convergence and needs fewer iterations than the perturbation method.     

Three-dimensional electrical impedance tomography based on thecomplete 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.     

Reconstruction of three-dimensional data for electrical impedance tomography

A two-dimensional reconstruction algorithm based on a modified version of the method of sensitivity regions is used to reconstruct data obtained from a three-dimensional finite element model. By using data obtained from off-drive-plane measurements an improved image of changes in resistivity on the drive plane is obtained.<>     

An experimental study in electrical impedance tomography usingbackprojection reconstruction

This paper reports on experiments designed to evaluate the performance of the equipotentials backprojection method under conditions modeling those of proposed applications of electrical impedance tomography. Small spherical targets were placed inside a saline-filled tank with dimensions similar to a human torso. Data were acquired with a computer-based instrument that applies current to pairs of electrodes located on two horizontal planes and records potential differences between electrodes of a third plane. The relative contrast produced by nonconducting spheres in a uniform saline background was measured on the reconstructed images and used to determine system sensitivity to target volume and to the radial and vertical positions of single spheres. Results show that for radial positions within a critical radius sensitivity is always maximum when the spheres center is on the recording plane and decreases gradually when the target is moved outside this plane. Localization of simple targets in 3-D, with data acquired from multiple recording planes, appears feasible. The results provide guidelines for the interpretation of images with complex 3-D conductivity distributions.     

MPSO-MNR算法及噪声对该算法在EIT成像中的影响

针对现有电阻抗成像算法的局限,在此将修正的粒子群算法与牛顿拉夫孙结合算法,形成MPSO-MNR算法。对二维圆形求解区域,采用有限元剖分,在三角电流驱动模式下,应用提出的MPSO-MNR算法进行电阻抗重构,并研究了噪声对重构结果的影响。数值仿真结果表明：MPSO-MNR算法能够准确重构解域内电阻抗分布;噪声影响成像的质量,随着噪声的增加（信噪比的减少）,重构目标的边界、背景变得逐渐模糊。MPSO-MNR算法避免要求迭代初值接近真值,并具有较快收敛的特点,在一定的噪声范围内,可用于电阻抗重构。     

Regularized reconstruction in electrical impedance tomography usinga variance uniformization constraint

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 conductivities. 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     

Ramp-preserving denoising for conductivity image reconstruction in magnetic resonance electrical impedance tomography

In magnetic resonance electrical impedance tomography, among several conductivity image reconstruction algorithms, the harmonic B(z) algorithm has been successfully applied to B(z) data from phantoms and animals. The algorithm is, however, sensitive to measurement noise in B(z) data. Especially, in in vivo animal and human experiments where injection current amplitudes are limited within a few milliampere at most, measured B(z) data tend to have a low SNR. In addition, magnetic resonance (MR) signal void in outer layers of bones and gas-filled organs, for example, produces salt-pepper noise in the MR phase and, consequently, B(z) images. The B(z) images typically present areas of sloped transitions, which can be assimilated to ramps. Conductivity contrasts change ramp slopes in B(z) images and it is critical to preserve positions of those ramps to correctly recover edges in conductivity images. In this paper, we propose a ramp-preserving denoising method utilizing a structure tensor. Using an eigenvalue analysis, we identified local regions of salt-pepper noise. Outside the identified local regions, we applied an anisotropic smoothing to reduce noise while preserving their ramp structures. Inside the local regions of salt-pepper noise, we used an isotropic smoothing. After validating the proposed denoising method through numerical simulations, we applied it to in vivo animal imaging experiments. Both numerical simulation and experimental results show significant improvements in the quality of reconstructed conductivity images.     

Image reconstruction using interval simulated annealing in electrical impedance tomography

Electrical impedance tomography (EIT) is an imaging technique that attempts to reconstruct the impedance distribution inside an object from the impedance between electrodes placed on the object surface. The EIT reconstruction problem can be approached as a nonlinear nonconvex optimization problem in which one tries to maximize the matching between a simulated impedance problem and the observed data. This nonlinear optimization problem is often ill-posed, and not very suited to methods that evaluate derivatives of the objective function. It may be approached by simulated annealing (SA), but at a large computational cost due to the expensive evaluation process of the objective function, which involves a full simulation of the impedance problem at each iteration. A variation of SA is proposed in which the objective function is evaluated only partially, while ensuring boundaries on the behavior of the modified algorithm.     

Walsh function current patterns and data synthesis for electrical impedance tomography

A data collection method which uses Walsh functions as injection current patterns is presented. This method can satisfy two conditions: the optimality of current patterns in every iteration and the single-time data measurement condition. The use of Walsh functions simplifies the design of current sources since only two levels of current (+1 and -1) are required, whereas sinusoidal injection requires a digital-to-analog converter to produce many different values of currents. Compared to diagonal or neighboring type of pulses as injection current patterns, Walsh injection current patterns provide more information about the interior of the subject since Walsh function simulate low and high spatial frequency patterns. Therefore, Walsh function injection uses the simplicity of pulse type injection and yields the better distinguishability or SNR of sinusoidal injection.     

Impact of model shape mismatch on reconstruction quality in electrical impedance tomography

Electrical impedance tomography (EIT) is a low-cost, noninvasive and radiation free medical imaging modality for monitoring ventilation distribution in the lung. Although such information could be invaluable in preventing ventilator-induced lung injury in mechanically ventilated patients, clinical application of EIT is hindered by difficulties in interpreting the resulting images. One source of this difficulty is the frequent use of simple shapes which do not correspond to the anatomy to reconstruct EIT images. The mismatch between the true body shape and the one used for reconstruction is known to introduce errors, which to date have not been properly characterized. In the present study we, therefore, seek to 1) characterize and quantify the errors resulting from a reconstruction shape mismatch for a number of popular EIT reconstruction algorithms and 2) develop recommendations on the tolerated amount of mismatch for each algorithm. Using real and simulated data, we analyze the performance of four EIT reconstruction algorithms under different degrees of shape mismatch. Results suggest that while slight shape mismatch is well tolerated by all algorithms, using a circular shape severely degrades their performance.     

Experimental evaluation of two iterative reconstruction methods for induced current electrical impedance tomography

The Newton-Raphson (N-R) with two different regularization methods: the Levenberg-Marquardt (N-R-LM) and the Hachtel's Augmented Matrix (N-R-HAM), were used to reconstruct images of conductivity changes in a cylindrical medium by Induced Current Electrical Impedance Tomography (ic-EIT). Experimental data were obtained from an 8-cm high, 19.2-cm diameter tank with 16 electrodes on the boundary surface and surrounded by eight 50-cm diameter coils. The coils were angularly displaced by 45 degrees and offset 12.4 cm from the center of the tank. They were driven by a 150-mA (peak) 20-kHz sine wave. Potential differences between adjacent electrodes were measured with phase-sensitive demodulators. The scalar potential field in the electrode plane of the conducting medium, resulting from eddy currents generated by each coil, was computed by the Finite Element Method. Image reconstruction by the N-R-HAM method was found to provide higher resolution and better noise immunity than the N-R-LM method. Two 2.2-cm diameter nonconducting rods located 3.9 cm from the center of the tank, 180 degrees from each other, were clearly resolved. Spatial resolution is estimated at 15% of the tank diameter and is comparable to the resolution obtained by conventional EIT using the Sheffield protocol. Higher resolution could be achieved with more coils and/or electrodes. A 16-coil system should present no construction problems. However, voltages induced by stray magnetic flux through the electrode leads and measurement circuits are significant and may limit the ability of ic-EIT to perform static imaging of conductivity distributions.     

Finite element implementation of Maxwell's equations for image reconstruction in electrical impedance tomography

Traditionally, image reconstruction in electrical impedance tomography (EIT) has been based on Laplace's equation. However, at high frequencies the coupling between electric and magnetic fields requires solution of the full Maxwell equations. In this paper, a formulation is presented in terms of the Maxwell equations expressed in scalar and vector potentials. The approach leads to boundary conditions that naturally align with the quantities measured by EIT instrumentation. A two-dimensional implementation for image reconstruction from EIT data is realized. The effect of frequency on the field distribution is illustrated using the high-frequency model and is compared with Laplace solutions. Numerical simulations and experimental results are also presented to illustrate image reconstruction over a range of frequencies using the new implementation. The results show that scalar/vector potential reconstruction produces images which are essentially indistinguishable from a Laplace algorithm for frequencies below 1 MHz but superior at frequencies reaching 10 MHz.     

Propagation of measurement noise through backprojection reconstruction in electrical impedance tomography

A framework to analyze the propagation of measurement noise through backprojection reconstruction algorithms in electrical impedance tomography (EIT) is presented. Two measurement noise sources were considered: noise in the current drivers and in the voltage detectors. The influence of the acquisition system architecture (serial/semi-parallel) is also discussed. Three variants of backprojection reconstruction are studied: basic (unweighted), weighted and exponential backprojection. The results of error propagation theory have been compared with those obtained from simulated and experimental data. This comparison shows that the approach provides a good estimate of the reconstruction error variance. It is argued that the reconstruction error in EIT images obtained via backprojection can be approximately modeled as a spatially nonstationary Gaussian distribution. This methodology allows us to develop a spatial characterization of the reconstruction error in EIT images.     

Weighted least-squares criteria for electrical impedance tomography

Methods are developed for the design of electrical impedance tomographic reconstruction algorithms with specified properties. Assuming a starting model with constant conductivity or some other specified background distribution, an algorithm with the following properties is found. (1) The optimum constant for the starting model is determined automatically. (2) The weighted least-squares error between the predicted and measured power dissipation data is as small as possible. (3) The variance of the reconstructed conductivity from the starting model is minimized. (4) Potential distributions with the largest volume integral of gradient squared have the least influence on the reconstructed conductivity, and therefore distributions most likely to be corrupted by contact impedance effects are deemphasized. (5) Cells that dissipate the most power during the current injection tests tend to deviate least from the background value. For a starting model with nonconstant conductivity, the reconstruction algorithm has analogous properties.     

A simplified first-order reconstruction algorithm for microwave tomography

A first-order reconstruction algorithm for microwave tomographic imaging has been developed considering the wave nature of the probing signal. The algorithm has been simplified by neglecting mutual interaction terms. The traditional ray theory which is successfully applied in X-ray and ultrasound tomography, is totally inadequate to explain different electromagnetic phenomena, for example, scattering, diffraction and boundary reflections. A beam concept has been introduced in developing the algorithm. The results of applying the algorithm on a small biological model containing bone, muscle and muscle-like materials have been extremely encouraging indicating the possibility of development of a microwave imaging scheme. The image, however, at present is not as perfect as it should have been from clinical point of view, most probably due to the simplifying assumptions made in developing the algorithm.