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 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.     

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.     

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.     

Multiobjective neural network for image reconstruction

The authors propose a multiobjective neural network model and algorithm for image reconstruction from projections. This model combines the Hopfield model and multiobjective decision making approach. A weighted sum optimisation based neural network algorithm is developed. The dynamic process of the net is based on minimisation of a weighted sum energy function and Euler's iteration and this algorithm is applied to image reconstruction from computer-generated noisy projections and Siemens Somaton DR scanner data, respectively. Reconstructions based on this method are shown to be superior to those based on conventional iterative reconstruction algorithms such as MART (multiplicate algebraic reconstruction technique) and convolution from the point of view of accuracy of reconstruction. Computer simulation using the multiobjective method shows a significant improvement in image quality and convergence behaviour over conventional algorithms     

Increasing image resolution in electrical impedance tomography

An effective approach to increase the image resolution in static electrical impedance tomography is proposed, in which the image with local high resolution is reconstructed by fine meshing only the impedance abnormal element in the finite element model based on a genetic algorithm. Experimental results from a laboratory phantom are presented     

基于并行BP神经网络的近红外光断层图像重建方法基础研究

提出一种基于并行BP神经网络的近红外光断层成像(Near-infrared optical tomography,NIR OT)图像重建算法,利用BP神经网络来表征生物组织内部光学参数的空间分布和边界光强之间的非线性映射关系.该方法将一个复杂的模型分解成简单的模型分别建立并行的神经网络.利用Femlab软件完成基于有限元的稳态扩散方程的两个简单模型的正向问题求解,根据提出的平均优化散射系数和正向问题训练的大量数据集合,建立并训练并行神经网络,通过对两个网络结果的分析,实现快速获得更复杂模型的光学参数的重构.算法能够快速识别特异组织的位置和准确反映热疗过程中生物组织的优化散射系数的变化趋势.     

A reconstruction algorithm for electrical impedance tomography data collected on rectangular electrode arrays

A three-dimensional reconstruction algorithm in electrical impedance imaging is presented for determining the conductivity distribution beneath the surface of a medium, given surface voltage data measured on a rectangular array of electrodes. Such an electrode configuration may be desirable for using electrical impedence tomography to detect tumors in the human breast. The algorithm is based on linearizing the conductivity about a constant value. Here, we describe a simple implementation of the algorithm on a four-electrode--by-four-electrode array and the reconstructions obtained from numerical and experimental tank data. The results demonstrate significantly better spatial resolution in the plane of the electrodes than with respect to depth.     

An artificial neural network for SPECT image reconstruction

An artificial neural network has been developed to reconstruct quantitative single photon emission computed tomographic (SPECT) images. The network is trained with an ideal projection-image pair to learn a shift-invariant weighting (filter) for the projections. Once trained, the network produces weighted projections as a hidden layer when acquired projection data are presented to its input. This hidden layer is then backprojected to form an image as the network output. The learning algorithm adjusts the weighting coefficients using a backpropagation algorithm which minimizes the mean squared error between the ideal training image and the reconstructed training image. The response of the trained network to an impulse projection resembles the ramp filter typically used with backprojection, and reconstructed images are similar to filtered backprojection images.     

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.     

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.     

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     

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.     

Harmonic decomposition in PDE-based denoising technique for magnetic resonance electrical impedance tomography

Recent progress in magnetic resonance electrical impedance tomography (MREIT) research via simulation and biological tissue phantom studies have shown that conductivity images with higher spatial resolution and accuracy are achievable. In order to apply MREIT to human subjects, one of the important remaining problems to be solved is to reduce the amount of the injection current such that it meets the electrical safety regulations. However, by limiting the amount of the injection current according to the safety regulations, the measured MR data such as the z-component of magnetic flux density Bz in MREIT tend to have low SNR and get usually degraded in their accuracy due to the nonideal data acquisition system of an MR scanner. Furthermore, numerical differentiations of the measured Bz required by the conductivity image reconstruction algorithms tend to further deteriorate the quality and accuracy of the reconstructed conductivity images. In this paper, we propose a denoising technique that incorporates a harmonic decomposition. The harmonic decomposition is especially suitable for MREIT due to the physical characteristics of Bz. It effectively removes systematic and random noises, while preserving important key features in the MR measurements, so that improved conductivity images can be obtained. The simulation and experimental results demonstrate that the proposed denoising technique is effective for MREIT, producing significantly improved quality of conductivity images. The denoising technique will be a valuable tool in MREIT to reduce the amount of the injection current when it is combined with an improved MREIT pulse sequence.     

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.     

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.     

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.     

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.     

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.<>     

On optimal current patterns for electrical impedance tomography

We develop a statistical criterion for optimal patterns in planar circular electrical impedance tomography. These patterns minimize the total variance of the estimation for the resistance or conductance matrix. It is shown that trigonometric patterns (Isaacson, 1986), originally derived from the concept of distinguishability, are a special case of our optimal statistical patterns. New optimal random patterns are introduced. Recovering the electrical properties of the measured body is greatly simplified when optimal patterns are used. The Neumann-to-Dirichlet map and the optimal patterns are derived for a homogeneous medium with an arbitrary distribution of the electrodes on the periphery. As a special case, optimal patterns are developed for a practical EIT system with a finite number of electrodes. For a general nonhomogeneous medium, with no a priori restriction, the optimal patterns for the resistance and conductance matrix are the same. However, for a homogeneous medium, the best current pattern is the worst voltage pattern and vice versa. We study the effect of the number and the width of the electrodes on the estimate of resistivity and conductivity in a homogeneous medium. We confirm experimentally that the optimal patterns produce minimum conductivity variance in a homogeneous medium. Our statistical model is able to discriminate between a homogenous agar phantom and one with a 2 mm air hole with error probability (p-value) 1/1000.