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.     

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.     

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.     

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.     

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.     

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.     

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     

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.     

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.     

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.     

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。     

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 parallel implementation of a modified Richardson-Lucy algorithm for image de-blurring

Passive imaging in the millimetre wave region of the spectrum is attractive compared with imaging in the visible and infrared because of its better penetration through cloud and rain. However, the diffraction limited spatial resolution is far worse and computer restoration of millimetre wave images is necessary. The Richardson-Lucy algorithm is frequently used for image restoration because compared with linear algorithms it has reduced sensitivity to error in a priori knowledge, and fewer restoration artefacts. However, it requires many iterations to converge and is very slow compared with linear methods. This paper investigates modifications of the Richardson-Lucy algorithm which speed up the execution of the algorthm. Software techniques investigated include incorporating a speedup parameter, using dual point spread functions and implementation in the Fourier domain. A parallel implementation of the algorithm has also been investigated using a tree topology network of between 1 and 10 transputers. Finally an application of the transputer system to the de-blurring of a millimetre-wave image obtained from a mechanical scanning system is described in the paper.     

A universal algorithm for parallel CRC computation and its implementation

Derived from a proposed universal mathematical expression, this paper investigates a novel algorithm for parallel Cyclic Redundancy Check （CRC） computation, which is an iterative algorithm to update the check-bit sequence step by step and suits to various argument selections of CRC computation. The algorithm proposed is quite suitable for hardware implementation. The simulation implementation and performance analysis suggest that it could efficiently speed up the computation compared with the conventional ones. The algorithm is implemented in hardware at as high as 21Gbps, and its usefulness in high-speed CRC computations is implied, such as Asynchronous Transfer Mode （ATM） networks and 10G Ethernet.     

Iterative reconstruction methods using regularization and optimal current patterns in electrical impedance tomography

An iterative reconstruction method which minimizes the effects of ill-conditioning is discussed. Based on the modified Newton-Raphson algorithm, a regularization method which integrates prior information into the image reconstruction was developed. This improves the conditioning of the information matrix in the modified Newton-Raphson algorithm. Optimal current patterns were used to obtain voltages with maximal signal-to-noise ratio (SNR). A complete finite element model (FEM) was used for both the internal and the boundary electric fields. Reconstructed images from phantom data show that the use of regularization optimal current patterns, and a complete FEM model improves image accuracy. The authors also investigated factors affecting the image quality of the iterative algorithm such as the initial guess, image iteration, and optimal current updating.     

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.     

Stationary positron emission tomography and its image reconstruction

Feasibility of stationary positron emission tomography (PET) using discrete detectors has been investigated by simulation studies. To enable full utilization of detector resolution, a "bank array" of detectors is proposed and an EM algorithm is adopted for image reconstruction. The bank array consists of an odd number of detector banks arranged on a circular ring with a gap equal to one half the detector width. The EM algorithm [11] is used with some modifications for reducing the quantity of computation, improving the convergence speed, and suppressing statistical noise, so as to meet the present purpose. Simulation studies involving several phantoms show that the stationary PET with the new detector array provides image quality which is good enough for clinical applications. For fast dynamic studies with low spatial resolution, the convolution-backprojection method is efficient, but for high-resolution static imaging, resolution enhancement by an iterative method is required. Problems arising in the corrections for attenuation of photons and detector sensitivity, etc., are also discussed. A totally stationary PET avoids the mechanical problems associated with accurate movement of heavy assemblies and is particularly advantageous in gated cardiac imaging or in fast dynamic studies. Elimination of a scan along the detector plane allows a quick scan in the axial direction to achieve three-dimensional imaging with a small number of detector rings.     

一种具视差鲁棒性的图像拼接算法

传统的图像拼接算法对视差较为敏感,要求不同帧的拍照位置必须相同。为了解决具视差的图像拼接问题,在对基于平滑变化的仿射拼接算法的分析研究上提出改进算法。通过设计合理规则筛选有效的特征点,再采用非局部平均滤波求解任意点仿射变换参数,最后设计特定的插值方法实现图像融合,从而提高传统拼接算法在时间和拼接效果上的不足。实验结果证明改进算法对具视差变化的图像拼接有较高鲁棒性,可高速实现较好质量的图像拼接。