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Electrical impedance tomography (EIT) is a technique for determining the electrical conductivity and permittivity distribution inside a medium from measurements made on its surface. The impedance distribution reconstruction in EIT is a nonlinear inverse problem that requires the use of a regularization method. The generalized Tikhonov regularization methods are often used in solving inverse problems. However, for EIT image reconstruction, the generalized Tikhonov regularization methods may lose the boundary information due to its smoothing operation. In this paper, we propose an iterative Lavrentiev regularization and L-curve-based algorithm to reconstruct EIT images. The regularization parameter should be carefully chosen, but it is often heuristically selected in the conventional regularization-based reconstruction algorithms. So, an L-curve-based optimization algorithm is used for selecting the Lavrentiev regularization parameter. Numerical analysis and simulation results are performed to illustrate EIT image reconstruction. It is shown that choosing the appropriate regularization parameter plays an important role in reconstructing EIT images.

Electrical impedance tomography (EIT) is an imaging technique which determines the electrical conductivity and permittivity distribution within a medium using electrical measurement from a series of electrodes on its surface [1–3]. Electrodes are brought into contact with the surface of the object being imaged. A set of voltage (or current) are applied and the corresponding currents (or voltage) are measured [

However, the process of property estimation in EIT is a highly nonlinear, ill-conditioned, and ill-posed problem. The sensitivity matrix, which relates interior admittivity perturbations to perturbations in the boundary data, is heavily ill-conditioned with respect to inversion. So, it requires special treatment in the form of regularization or a truncation of a singular value expansion [

Some papers on image reconstruction algorithms have been published [10–14], but little work on EIT image reconstruction is published. The first proposed EIT reconstruction algorithm is the equipotential back-projection [

In this paper, we propose an iterative Lavrentiev regularization-based algorithm to reconstruct EIT images using knowledge of the noise variance of the measurements and the covariance of the conductivity distribution. As the regularization parameter should be carefully chosen, an L-curve-based optimization algorithm is applied. Numerical analysis and simulation results are provided. The remaining sections are organized as follows. Section 2 forms the problem of EIT image reconstruction and outlines the motivations of this paper. Section 3 details the iterative Lavrentiev regularization-based EIT reconstruction algorithm, followed by the simulation examples in Section 4. Finally, Section 5 concludes the whole paper.

Taking EIT imaging in medical applications as an example [18,19], different tissues of the body are shown to have different electrical characteristics. Most tissues can be considered isotropic with the exception of muscles and brain tissue which are anisotropic. It is usually assumed to be homogenous and isotropic, where the constitute parameters such as the conductivity and permittivity are independent of position and direction. Therefore, the underlying relationships that govern the interaction between EIT electricity and magnetism are the Maxwell’s equations. The medium () is modeled as a closed and bounded subset of three-dimensional space with smooth boundary () and uniform conductivity (). The electric field () enclosed in is expressed in terms of the scalar potential

where

is the vector differential operator, and is the potential in the medium. The current () is given by the multiplication of the conductivity. The electric field can then be computed as

As there are no interior current sources, EIT simulation in the medium can be described in terms of a scalar voltage potential satisfying Kirchoff’s voltage law

The boundary current density () can then be represented by

According to this relationship, the problem of determining the potential inside the medium from boundary measurements can then be carried out [

To apply regularization-based image reconstruction methods, EIT problems can be formulated as a system of linear equations [

where is the nodal potential distribution, is the potential values on the boundary electrodes, , and are the local matrices. This equation can be represented by

where is the error in the data with and. One standard approach used to solve linear estimation problems is the least square estimation, but this estimate is unsatisfactory because the calculated independent conductivity region is too small.

The inverse of the A cannot be directly computed because the singular values will grow without bound. This will amplify the noise components in the solution associated with the numerical null space of. That is to say, small measurement perturbations may produce large variations in such that the 3-norm residual error is unbounded. Therefore, the matrix is poor conditioned because EIT makes current injection and measurement on the medium surface including higher current densities near the surface where conductivity contrasts will result in more signal than for contrasts in the center. This problem can be resolved by regularization [

Correspondingly, Equation (7) can be expressed by

As instability may arises due to division by small singular values, to overcome this problem we apply an iterative Lavrentiev-based regularization algorithm.

Lavrentiev regularization replaces Equation (7) by [

where is the regularization parameter. Correspondingly, the iterative Lavrentiev regularization is

This equation can be changed into

Suppose the solution of the Equation (7) is

where is the normalized function with. Note that refers to the smoothness source conditions. Considering the function

when, it arrives its maximum

Hence, we have

where is a constant parameter for the specific. Therefore, this algorithm is converged with a convergence rate of

In fact, the Lavrentiev regularization just uses the filter function

with a regularization parameter of, as shown in

L-curve is a parametric plot of the squared norm of the regularized solution against the squared norm of the regularized residual for a range of values of regularization parameter. The L-curve criterion for regularization parameter selection is to pick the parameter value corresponding to the “corner” of this curve. Let denote the regularized solution and let denote the regularized residual. Define

We can then select the value of that maximizes the curvature function

where and are represented, respectively, by

The, , and denote the first and second derivatives of, with respect to. As shown in

In many industrial process and biomedical EIT applications, it is often the case that one knows in advance what materials (tissues) are included within the measurement domain. As mixtures of materials are known to have intermediate conductivities, this effectively restricts the admissible solutions, i.e., the pixels of the reconstructed images, to lie within a set of known values. In this sense, the bound-constrained (with bounds on the values of the admittivity distribution) is to locate the detected inhomogeneities.

As an example, one can allow that the admittivity distribution to be reconstructed is mainly homogenous with an unknown number of shaped inclusions. In EIT applications, the current with a frequency of 10-100kHz is widely used. These patterns are similar to those appearing in

image reconstruction. For the reconstructions, simulated measurements contaminated with 2% Gaussian noise have been assumed. Regularization parameter is often used in conventional regularizationbased reconstruction methods, e.g., [

EIT is a technique used to create images of the electrical properties in the interior of a medium from measurements on its boundary, which is particularly important for medical and industrial applications [26,27]. Usually a set of voltage or current measurements is acquired from the boundaries of a conductive volume. In this paper, we presented an iterative Lavrentiev regularization and Lcurve-based algorithm to reconstruct EIT images using knowledge of the noise variance of the measurements and the covariance of the conductivity distribution. The regularization parameter should be carefully selected, but it is often heuristically selected in conventional regularization-based reconstruction algorithms. So, an L-curvebased optimization algorithm is applied to choose the Lavrentiev regularization parameter. The method is validated with numerical analysis and simulation results. Further research efforts are planned to focus on experimental investigations [28,29] and other computational electromagnetic-based algorithms [