Electrical Impedance Tomography Inverse Problem Research Articles

On the influence of spread constant in radial basis networks for electrical impedance tomography

Electrical impedance tomography (EIT) is a non-invasive imaging technique. The main task of this work is to solve a non-linear inverse problem, for which several techniques have been suggested, but none of which gives a very high degree of accuracy. This paper introduces a novel approach, based on radial basis function (RBF) artificial neural networks (ANNs), to solve this problem, and uses several ANNs to obtain the best solution to the EIT inverse problem. ANNs have the potential to directly estimate the solution of the inverse problem with a high degree of accuracy. While different radial basis neural networks do not always perform well on different problems, they usually give good results on some specific problems. This paper evidences a strong correlation between the area of the target and the spread constant of the RBF network that gives the best reconstruction. A solution to automatically estimate the size of the target and pick the best neural network directly from voltage measurements is presented, making the reconstruction process automatic. By automatically selecting the best ANN for each specific set of voltage measurements, the proposed solution gives a more accurate reconstruction of both small and large targets.

EIT Image Reconstruction Using Iterative TV Regularized PD-IPM Algorithm

Biological tissues have electrical conductivity and permittivity properties which depend on tissue composition, structure and health status. Bioelectrical properties can be utilized for non invasive diagnosis. Electrical Impedance Tomography (EIT) is a method for reconstructing the image of distribution of electrical conductivity and permittivity inside a volume from measurements made at the surface of the volume. EIT image reconstruction is an ill-posed problem that requires a priori information called regularization. The Total Variation (TV) regularization is often used in solving EIT inverse problem. In this paper, simulation has been carried out in noise free and noisy cases and TV regularized iterative Primal Dual Interior Point Method (PD-IPM) has been used to reconstruct the difference conductivity image.

인체 흉부 영상 복원을 위한 행렬 적응 조정 방법의 적용

전기 임피던스 단층촬영법(EIT)에서 역문제는 매우 높은 비정치성이므로 이것을 완화시키기 위해서 사전정보가 사용되고 EIT 역문제를 푸는 과정에서 만족스러운 복원성능을 갖기 위해 조정 기법은 적용된다. 반복적 Gauss-Newton 방법은 정확성과 빠른 수렴속도로 인해서 일반적으로 역문제를 푸는데 사용되지만 항상 좋은 성능을 내는 것은 아니며 조정 인자 선택에 따라 성능이 좌지우지된다. 비록 L-곡선과 같이 조정 인자를 결정하는데 이용할 수 있는 여러 가지 방법들이 존재하지만 이러한 방법들이 모든 경우에 적용할 수 있는 것은 아니다. 게다가 조정 인자는 스칼라이고 반복 연산동안 변하지 않는다. 그러므로 이 논문에서는 복원 성능을 향상시키기 위해서 조정 인자를 결정해주는 새로운 방법을 사용하였다. 각각의 반복 연산과정에서 도전율의 norm을 구하고 이것을 대각 행렬형태인 조정 인자를 구하는데 사용한다. 제안한 방법을 인체 흉부 영상 복원에 적용하였고, 기존의 방법들과 복원 성능을 비교하였다. 모의실험 결과, 기존의 방법들과 비교해서 개선된 성능을 확인할 수 있었다. Inverse problem in electrical impedance tomography (EIT) is highly ill-posed therefore prior information is used to mitigate the ill-posedness. Regularization methods are often adopted in solving EIT inverse problem to have satisfactory reconstruction performance. In solving the EIT inverse problem, iterative Gauss-Newton method is generally used due to its accuracy and fast convergence. However, its performance is still suboptimal and mainly depends on the selection of regularization parameter. Although, there are few methods available to determine the regularization parameter such as L-curve method they are sometimes not applicable for all cases. Moreover, regularization parameter is a scalar and it is fixed during iteration process. Therefore, in this paper, a novel method is used to determine the regularization parameter to improve reconstruction performance. Conductivity norm is calculated at each iteration step and it used to obtain the regularization parameter which is a diagonal matrix in this case. The proposed method is applied to human thorax imaging and the reconstruction performance is compared with traditional methods. From numerical results, improved performance of proposed method is seen as compared to conventional methods.

State estimation and inverse problems in electrical impedance tomography: observability, convergence and regularization

Solving electrical impedance tomography (EIT) inverse problems in real-time is a challenging task due to their dimension, the nonlinearities involved and the fact that they are ill-posed. Thus, efficient algorithms are required to address the application of tomographic technologies in process industry. In practical applications the EIT inverse problem is often linearized for fast and robust reconstruction. The aim of this paper is to analyse the solution of linearized EIT inverse problem from the perspective of a state estimation problem, providing links between regularization, observability and convergence of the algorithms. In addition, also a new way to define the fictitious outputs is proposed, leading to observers with fewer parameters than with the approach widely used in literature. Simulation of EIT examples illustrate the main ideas and algorithmic improvements of the proposed approaches.

Randomize-Then-Optimize for Sampling and Uncertainty Quantification in Electrical Impedance Tomography

In a typical inverse problem, a spatially distributed parameter in a physical model is estimated from measurements of model output. Since measurements are stochastic in nature, so is any parameter estimate. Moreover, in the Bayesian setting, the choice of regularization corresponds to the definition of the prior probability density function, which in turn is an uncertainty model for the unknown parameters. For both of these reasons, significant uncertainties exist in the solution of an inverse problem. Thus to fully understand the solution, quantifying these uncertainties is important. When the physical model is linear and the error model and prior are Gaussian, the posterior density function is Gaussian with a known mean and covariance matrix. However, the electrical impedance tomography inverse problem is nonlinear, and hence no closed form expression exists for the posterior density. The typical approach for such problems is to sample from the posterior and then use the samples to compute statistics (such as the mean and variance) of the unknown parameters. Sampling methods for electrical impedance tomography have been studied by various authors in the inverse problems community. However, up to this point the focus has been on the development of increasingly sophisticated implementations of the Gibbs sampler, whose samples are known to converge very slowly to the correct density for large-scale problems. In this paper, we implement a recently developed sampling method called randomize-then-optimize (RTO), which provides nearly independent samples for each application of an appropriate numerical optimization algorithm. The sample density for RTO is not the posterior density, but RTO can be used as a very effective proposal within a Metropolis--Hastings algorithm to obtain samples from the posterior. Here our focus is on implementing the method on synthetic examples from electrical impedance tomography, and we show that it is both computationally efficient and provides good results. We also compare RTO performance with the Metropolis adjusted Langevin algorithm and find RTO to be much more efficient.

전기 임피던스 단층촬영법에서 잔류오차 기반의 반복적 조정기법을 이용한 영상 복원

전기 임피던스 단층촬영법을 이용한 정적 영상 복원에서 대표적으로 사용되고 있는 복원 알고리즘은 modified Newton-Raphson(mNR) 알고리즘으로 수렴 속도 및 추정 정확도 측면에서 비교적 다른 알고리즘들에 비해 좋은 성능을 나타낸다. mNR 알고리즘에서는 측정 전압과 계산 전압과의 차이, 즉 잔류오차를 최소화하도록 목적함수를 설정하고 이를 반복 연산하여 내부의 저항률 분포를 추정한다. 이때 EIT 역문제의 비정치성을 완화시키기 위해 조정방법을 사용하며 조정인자에 따라 서로 다른 영상 복원 성능을 나타낸다. 기존 기법에서는 반복 연산마다 일정한 상수 값의 조정인자를 사용하기 때문에 대상 물체의 내부 상태가 변하거나 측정 잡음 등이 있는 경우 때때로 조정인자에 따라 영상 복원이 수렴되지 않는다. 따라서 본 논문에서는 영상 복원 수렴 및 성능을 개선하기 위하여 잔류오차에 기반하여 반복 연산마다 자동적으로 조정인자를 수정하는 기법을 제안하였다. 시뮬레이션과 실험을 수행하여 제안된 기법의 영상 복원성능을 평가한 결과 비교적 양호한 성능을 나타내었다. In electrical impedance tomography (EIT), modified Newton Raphson (mNR) method is widely used inverse algorithm for static image reconstruction due to its convergence speed and estimation accuracy. The unknown conductivity distribution is estimated iteratively by minimizing a cost functional such that the residual error namely the difference in measured and calculated voltages is reduced. Although, mNR method has good estimation performance, EIT inverse problem still suffers from ill-conditioned and ill-posedness nature. To mitigate the ill-posedness, generally, regularization methods are adopted. The inverse solution is highly dependent on the choice of regularization parameter. In most cases, the regularization parameter has a constant value and is chosen based on experience or trail and error approach. In situations, when the internal distribution changes or with high measurement noise, the solution does not get converged with the use of constant regularization parameter. Therefore, in this paper, in order to improve the image reconstruction performance, we propose a new scheme to determine the regularization parameter. The regularization parameter is computed based on residual error and updated every iteration. The proposed scheme is tested with numerical simulations and laboratory phantom experiments. The results show an improved reconstruction performance when using the proposed regularization scheme as compared to constant regularization scheme.

Choice of reconstructed tissue properties affects interpretation of lung EIT images

Electrical impedance tomography (EIT) estimates an image of change in electrical properties within a body from stimulations and measurements at surface electrodes. There is significant interest in EIT as a tool to monitor and guide ventilation therapy in mechanically ventilated patients. In lung EIT, the EIT inverse problem is commonly linearized and only changes in electrical properties are reconstructed. Early algorithms reconstructed changes in resistivity, while most recent work using the finite element method reconstructs conductivity. Recently, we demonstrated that EIT images of ventilation can be misleading if the electrical contrasts within the thorax are not taken into account during the image reconstruction process. In this paper, we explore the effect of the choice of the reconstructed electrical properties (resistivity or conductivity) on the resulting EIT images. We show in simulation and experimental data that EIT images reconstructed with the same algorithm but with different parametrizations lead to large and clinically significant differences in the resulting images, which persist even after attempts to eliminate the impact of the parameter choice by recovering volume changes from the EIT images. Since there is no consensus among the most popular reconstruction algorithms and devices regarding the parametrization, this finding has implications for potential clinical use of EIT. We propose a program of research to develop reconstruction techniques that account for both the relationship between air volume and electrical properties of the lung and artefacts introduced by the linearization.

Boundary element method and simulated annealing algorithm applied to electrical impedance tomography image reconstruction

Physics has played a fundamental role in medicine sciences, specially in imaging diagnostic. Currently, image reconstruction techniques are already taught in Physics courses and there is a growing interest in new potential applications. The aim of this paper is to introduce to students the electrical impedance tomography, a promising technique in medical imaging. We consider a numerical example which consists in finding the position and size of a non-conductive region inside a conductive wire. We review the electrical impedance tomography inverse problem modeled by the minimization of an error functional. To solve the boundary value problem that arises in the direct problem, we use the boundary element method. The simulated annealing algorithm is chosen as the optimization method. Numerical tests show the technique is accurate to retrieve the non-conductive inclusion.

Reducing computational costs in large scale 3D EIT by using a sparse Jacobian matrix with block-wise CGLS reconstruction

Electrical impedance tomography (EIT) is a fast and cost-effective technique to provide a tomographic conductivity image of a subject from boundary current–voltage data. This paper proposes a time and memory efficient method for solving a large scale 3D EIT inverse problem using a parallel conjugate gradient (CG) algorithm. The 3D EIT system with a large number of measurement data can produce a large size of Jacobian matrix; this could cause difficulties in computer storage and the inversion process. One of challenges in 3D EIT is to decrease the reconstruction time and memory usage, at the same time retaining the image quality. Firstly, a sparse matrix reduction technique is proposed using thresholding to set very small values of the Jacobian matrix to zero. By adjusting the Jacobian matrix into a sparse format, the element with zeros would be eliminated, which results in a saving of memory requirement. Secondly, a block-wise CG method for parallel reconstruction has been developed. The proposed method has been tested using simulated data as well as experimental test samples. Sparse Jacobian with a block-wise CG enables the large scale EIT problem to be solved efficiently. Image quality measures are presented to quantify the effect of sparse matrix reduction in reconstruction results.

Reconstruction of electrical impedance tomography (EIT) images based on the expectation maximum (EM) method

Electrical impedance tomography (EIT) calculates the internal conductivity distribution within a body using electrical contact measurements. The image reconstruction for EIT is an inverse problem, which is both non-linear and ill-posed. The traditional regularization method cannot avoid introducing negative values in the solution. The negativity of the solution produces artifacts in reconstructed images in presence of noise. A statistical method, namely, the expectation maximization (EM) method, is used to solve the inverse problem for EIT in this paper. The mathematical model of EIT is transformed to the non-negatively constrained likelihood minimization problem. The solution is obtained by the gradient projection-reduced Newton (GPRN) iteration method. This paper also discusses the strategies of choosing parameters. Simulation and experimental results indicate that the reconstructed images with higher quality can be obtained by the EM method, compared with the traditional Tikhonov and conjugate gradient (CG) methods, even with non-negative processing.

Left and right preconditioning for electrical impedance tomography with structural information

A common problem in computational inverse problems is to find an efficient way of solving linear or nonlinear least-squares problems. For large-scale problems, iterative solvers are the method of choice for solving the associated linear systems, and for nonlinear problems, an additional effective local linearization method is required. In this paper, we discuss an efficient preconditioning scheme for Krylov subspace methods, based on the Bayesian analysis of the inverse problem. The model problem to which we apply this methodology is electrical impedance tomography (EIT) augmented with prior information coming from a complementary modality, such as x-ray imaging. The particular geometry considered here models the x-ray-guided EIT for breast imaging. The interest in applying EIT concurrently with x-ray breast imaging arises from the experimental observation that the impedivity spectra of certain types of malignant and benign tissues differ significantly from each other, thus offering a possibility of diagnosis without more invasive tissue sampling. After setting up the EIT inverse problem within a Bayesian framework, we present an inner and outer iteration scheme for computing a maximum a posteriori estimate. The prior covariance provides a right preconditioner and the modeling error covariance provides a left preconditioner for the iterative method used to solve the linear least-squares problem at each outer iteration of the optimization problem. Moreover, the stopping criterion for the inner iterations is coupled with the progress of the solution of the outer iteration. Besides the preconditioning scheme, the computational efficiency relies on a very efficient method to compute the Jacobian, obtained by carefully organizing the forward computation. Computed examples illustrate the robustness and computational efficiency of the proposed algorithm.

Hp-Finite element discretisation of the electrical impedance tomography problem

This paper is concerned with the accurate recovery of the permittivity and conductivity from the solution of the electrical impedance tomography (EIT) inverse problem, given the knowledge of either the Neumann to Dirichlet or the Dirichlet to Neumann map. The linearised EIT inverse problem for the Fourier regime is discretised by hp-finite elements, which allows the accurate calculation of the sensitivity matrix entries. The variational algorithm of Knowles (1998) [31] is extended to allow the recovery of both material parameters by solution of the EIT problem in the Laplace regime. The associated forward problem is again discretised by hp-finite elements. A series of numerical results are included to demonstrate the effectiveness of the proposed techniques.

An analysis of electrical impedance tomography with applications to Tikhonov regularization

This paper analyzes the continuum model/complete electrode model in the electrical impedance tomography inverse problem of determining the conductivity parameter from boundary measurements. The continuity and differentiability of the forward operator with respect to the con- ductivity parameter in Lp-norms are proved. These analytical results are applied to several popular regularization formulations, which incorporate ap rioriinformation of smoothness/sparsity on the inho- mogeneity through Tikhonov regularization, for both linearized and nonlinear models. Some important properties, e.g., existence, stability, consistency and convergence rates, are established. This provides some theoretical justifications of their practical usage.

QUADRANT-SEARCHING: A NEW TECHNIQUE TO REDUCE THE SEARCH SPACE OF THE INVERSE PROBLEM OF EIT USING BEM TO THE DIRECT PROBLEM

The image reconstruction using the EIT (Electrical Impedance Tomography) technique is a nonlinear and ill-posed inverse problem which demands a powerful direct or iterative method. A typical approach for solving the problem is to minimize an error functional using an iterative method. In this case, an initial solution close enough to the global minimum is mandatory to ensure the convergence to the correct minimum in an appropriate time interval. The aim of this paper is to present a new, simple and low cost technique (quadrant-searching) to reduce the search space and consequently to obtain an initial solution of the inverse problem of EIT. This technique calculates the error functional for four different contrast distributions placing a large prospective inclusion in the four quadrants of the domain. Comparing the four values of the error functional it is possible to get conclusions about the internal electric contrast. For this purpose, initially we performed tests to assess the accuracy of the BEM (Boundary Element Method) when applied to the direct problem of the EIT and to verify the behavior of error functional surface in the search space. Finally, numerical tests have been performed to verify the new technique.

Maximum entropy regularization method for electrical impedance tomography combined with a normalized sensitivity map

Electrical impedance tomography (EIT) aims to estimate the electrical properties at the interior of an object from current–voltage measurements on its boundary. To overcome ill-posedness, regularization techniques such as Tikhonov regularization as well as some iterative methods were developed. In difference imaging between two different conductivity distributions, a conductivity change can be seen relatively non-negative to the medium with lower conductivity through some safeguard techniques. Therefore, the concept of maximum entropy from information theory and statistic mechanics can be used for this purpose. Furthermore, because the sensing field is “soft-field” and non-uniform, the same anomaly may produce different reconstruction signatures depending on its location within the image plane. Therefore, in this paper, maximum entropy based on general Tikhonov regularization, combined with normalized sensitivity map, is proposed to solve the inverse problem of EIT. Image reconstruction was carried out by maximum entropy regularization (MER) with a normalized sensitivity map and compared with the results from conjugate gradient method (CG), Tikhonov regularization, and CG with a normalized sensitivity map accordingly. Simulation and experiment results indicate that reconstructed images with higher quality can be obtained by MER with a normalized sensitivity map.

Exact Shape-Reconstruction by One-Step Linearization in Electrical Impedance Tomography

For electrical impedance tomography (EIT), the linearized reconstruction method using the Fréchet derivative of the Neumann-to-Dirichlet map with respect to the conductivity has been widely used in the last three decades. However, few rigorous mathematical results are known regarding the errors caused by the linear approximation. In this work we prove that linearizing the inverse problem of EIT does not lead to shape errors for piecewise-analytic conductivities. If a solution of the linearized equations exists, then it has the same outer support as the true conductivity change, no matter how large the latter is. Under an additional definiteness condition we also show how to approximately solve the linearized equation so that the outer support converges toward the right one. Our convergence result is global and also applies for approximations by noisy finite-dimensional data. Furthermore, we obtain bounds on how well the linear reconstructions and the true conductivity difference agree on the boundary of the outer support.

A TRUST REGION SUBPROBLEM FOR 3D ELECTRICAL IMPEDANCE TOMOGRAPHY INVERSE PROBLEM USING EXPERIMENTAL DATA

Image reconstruction in electrical impedance tomography (EIT) is an ill-posed nonlinear inverse problem. Regularization methods are needed to solve this problem. The results of the ill- posed EIT problem strongly depends on noise level in measured data as well as regularization parameter. In this paper, we present trust region subproblem (TRS), with the use of L-curve maximum curvature criteria to flnd a regularization parameter. Currently Krylov subspace methods especially conjugate gradient least squares (CGLS) are used for large scale 3D problem. CGLS is an e-cient technique when the norm of measured noise is exactly known. This paper demonstrates that CGLS and TRS converge to the same point on the L-curve with the same noise level. TRS can be implemented e-ciently for large scale inverse EIT problem as CGLS with no need a priori knowledge of the noise level.

Linearized solution to electrical impedance tomography based on the Schur conjugate gradient method

Electrical impedance tomography (EIT) is a technique for reconstructing the conductivity distribution of an inhomogeneous medium, usually by injecting a current at the periphery of an object and measuring the resulting changes in voltage. The conjugate gradient (CG) method is one of the most popular methods applied for image reconstruction, although its convergence rate is low. In this paper, an advanced version of the CG method, i.e. the Schur conjugate gradient (Schur CG) method, is used to solve the inverse problem for EIT. The solution space is divided into two subspaces. The main part of the solution lies in the coarse subspace, which can be calculated directly and its corresponding correction term with a small norm can be solved in the Schur complement subspace. This paper discusses the strategies of choosing parameters. Simulation results using the Schur CG algorithm are presented and compared with the conventional CG algorithm. Experimental results obtained by the Schur CG algorithm are also presented, indicating that the Schur CG algorithm can reduce the computational time and improve the quality of image reconstruction with the selected parameters.

Electrical impedance tomography through constrained sequential linear programming: a topology optimization approach

Electrical impedance tomography (EIT) is an imaging method that estimates conductivity distribution inside a body. In EIT, images are obtained by applying a sequence of low intensity electrical currents through electrodes attached to the body. Although in EIT there are serious difficulties to obtain a high-quality conductivity image, for medical applications this technology is safer and cheaper than other tomography techniques. The EIT deals with an inverse problem in which given the measured voltages on electrodes and a finite element (FE) model, it estimates the conductivity distribution, which are parameters of the FE model. In this work, the topology optimization method is applied as a reconstruction algorithm to obtain absolute images in EIT. It is an optimization method that has been applied successfully to structural mechanical applications and consists of systematically finding a conductivity distribution (or material distribution) in the domain that minimizes the difference between measured voltages and voltages calculated by using a computational model. This algorithm combines the finite element method and sequential linear programming (SLP) to solve the inverse problem of EIT. The SLP allows us to easily apply some regularization schemes based on included constraints in the topology optimization problem. Constraints based on image tuning control and weighted distance interpolation (WDI) are proposed, while a material model is applied to ensure the relaxation of the optimization problem. A new formulation to analytically perform the sensitivity analysis is proposed, using Maxwell's reciprocity theorem. To illustrate, the implemented algorithm is applied to obtain conductivity image distributions of some 2D examples using numerical and experimental data.

A hybrid reconstruction algorithm for electrical impedance tomography

Electrical impedance tomography (EIT) is a technique for reconstructing the conductivity distribution inside an inhomogeneous distribution by injecting currents at the boundary of a subject and measuring the resulting changes in voltage. A hybrid method is proposed for solving the inverse problem for EIT, which combines the Krylov subspace and the Tikhonov regularization for double levels of regularization to the ill-posed problem. Numerical simulation results using the hybrid method are presented and compared to those from truncated singular value decomposition (TSVD) regularization and the Tikhonov regularization. Experimental results with the hybrid method are also presented, indicating that the hybrid method can reduce the computation time, and improve the resolution of reconstructed images with the regularization parameter automatically chosen by the L-curve method.