Solving the electrical impedance tomography inverse problem for logarithmic conductivity

Electrical Impedance Tomography (EIT) is a noninvasive imaging method that reconstructs internal conductivity maps by applying low-frequency electrical currents through surface electrodes and measuring the resulting voltages. While EIT's inverse problem is inherently ill-posed, requiring regularization techniques to stabilize solutions, recent advances in neural networks offer promising alternatives for improving reconstruction accuracy. This study investigates the application of Deep Implicit Layers to EIT reconstruction, combining fixed-point iteration and Newton's method to solve implicit constraints. The forward problem is solved using the finite element method (FEM), and numerical phantoms are employed for training. To ensure a more realistic representation of measurement conditions, contact conductivity was modeled as a random variable sampled from a Gaussian distribution, improving the model's robustness and ability to handle uncertainties. The implicit neural network effectively distinguishes closely positioned objects; however, contrast limitations persist at the boundaries of the phantoms. The results highlight the potential of implicit models in EIT and suggest future improvements through advanced optimization techniques and expansion to 3D thorax reconstruction.

The conductivity profiles arising in medical applications of electrical impedance tomography (EIT) are often of `blocky' structure, i.e. they are relatively constant inside an organ and are rapidly varying at its boundary. Standard regularization methods for the inverse problem tend to blur these sharply defined edges. A penalty method is proposed which is especially designed for a better reconstruction of discontinuous conductivity profiles.

Electrical Impedance Tomography (EIT) is a non-invasive, radiation-free imaging method that has witnessed an intensive rise of interest in the biomedical field. To reconstruct an image a boundary voltages are measured using a set of electrodes, then based on these values the inner conductivity distribution is reconstructed. Nevertheless, EIT image reconstruction suffers from low spatial resolution as it represents a highly ill-conditioned, and nonlinear inverse problem. Besides, most of the existing population-based methods, used to solve this problem, get easily trapped in the local optima, which leads to an inaccurate image. This work proposes an optimized swarm intelligence algorithm based on Particle Swarm optimization (PSO) for EIT image reconstruction, as it considers the EIT inverse problem as an optimization problem and makes use of the whole search space to find the global optima. Two variations of PSO with a non-blind search have been developed and tested, namely an Advanced Particle Swarm optimization (APSO) and a Refined Advanced Particle Swarm optimization (RAPSO), to improve the performance of the PSO algorithm for EIT image reconstruction. The implementation of these algorithms is carried out on a circular mesh. After several tests, RAPSO proved a better outcome, with 0.024 fitness in 200 iterations and a population size equal to 100.

A geometric inverse problem identification procedure for detection of cavities

The inverse problem of Electrical Impedance Tomography (EIT), especially for open EIT which involves less measurement, is a non-linear ill-posed problem. In this paper, a novel method based on Particle Swarm Optimization (PSO) is proposed to solve the open EIT inverse problem. This method combines a modified Newton-Raphson algorithm, a conductivity-based clustering algorithm, with an adaptive PSO algorithm to enhance optimal search capability and improve the quality of the reconstructed image. The results of numerical simulations show that the proposed method has a faster convergence to optimal solution and higher spatial resolution on a reconstructed image than a Newton-Raphson type algorithm.

Imaging is a rapidly growing area in applied sciences. It has an interdisciplinary character and a wide range of applications such as medicine, nondestructive evaluation, microscopy and astronomy, as well as many industrial processes. The increasing demand on imaging is due to the change of role of vision. Today vision is not through eyes only but complemented, for instance, by ultrasound, x-ray computerized tomography (CT), electrical impedance tomography (EIT), to name but a few. Moreover, traditional imaging systems such as microscopes and telescopes are now equipped with detection instruments (CCD cameras) and the resulting digital images are currently processed and enhanced. Finally, the relevance of imaging for industry is best documented by a recent feature by Robert West (West R 2003 In industry, seeing is believing Physics World June 2003). Today the scope of imaging has broadened and plays a central role in many different areas ranging, for instance, from remote sensing to seismology.In most cases the new imaging techniques are based on indirect measurements of physical parameters; therefore they quite naturally lead to the demand of solving (linear or nonlinear) inverse problems. This indicates the central role that inverse problems have in imaging science.This special section highlights several topics of recent advances in imaging. The first five papers concern problems originating from medical imaging which can have important applications in other domains. The paper by Ji et al covers a new and promising diagnostic tool in medicine: the identification of abnormal tissues by elastic shear wave properties. The two subsequent papers by Louis and by Defrise et al concern 3D cone beam tomography which is the most recent and advanced technique in x-ray CT. Both the case of circular and helical scanning are considered. The paper by Natterer et al is also about tomography but is intended to exploit the mathematical analogies between x-ray CT and synthetic aperture radar, achieving a unified approach to the important problem of estimating resolution in these two completely different imaging techniques. Electrical impedance tomography is an imaging technique originally proposed for medical applications which can be usefully applied also to problems of nondestructive evaluation. Recent progress in the mathematical treatment of this problem is presented in the paper by Hanke and Brühl.The next three papers are about scattering problems, a fundamental topic in imaging techniques based for instance on ultrasound and microwave sounding. The papers by Kress and by Colton et al are concerned with inverse obstacle problems presenting two different concepts: Kress gives a survey of Newton methods while Colton et al discuss linear sampling methods. Borcea et al cover the problem of detecting and imaging small or extended objects embedded in inhomogeneous media.Finally Strong and Chan discuss the application of the total variation regularization method to denoising problems and present new results which enlighten the edge-preserving and scale-dependent properties of this method.

The objective of this research is to improve upon existing reconstruction methods for static impedance imaging in electrical impedance tomography (EIT), both in speed and quality. A multigrid (MG) method is implemented to speed up the solution of the forward problem in EIT. Compared with direct solution methods and other iterative algorithms, it is more efficient, especially for large system models. On the other hand, due to the nonlinearity of the underlying problem, all existing methods suffer from local optima. In order to find global or at least near optima, simulated annealing (SA) is used to solve the inverse problem of EIT. Some strategies are also proposed, including the insertion of the forward problem in the annealing process, and the discretization of the configuration space. Simulation experiments show the advantages and feasibility of the proposed method.

This thesis is a reconciliation between multiple bodies of work with a shared objective that is ``to identify the active support" under different environment settings. The applications of focus within this thesis will be placed on electrical impedance tomography (EIT) and direction-of-arrival (DOA) estimation. The term ``active support" or only ``support" is commonly used in compressed sensing and sparse signal recovery communities which is essentially referring to ``the most contributing parameters" in a system of linear equations. Although with a common objective shared among the works that have been done within this thesis, this thesis can be divided into two major parts based on applications’ environment. The first part is the support identification under a process expressible by Fredholm integral equation of the first kind. For this problem, a research topic that is known as ``Electrical impedance tomography (EIT)," is chosen as a representative application for a collection of problems expressible by this integral equation. The physical laws (Physics) of EIT can be explained by Maxwell's equations under quasi-static environment. Under such condition, the energy dissipated from the source is spread throughout the region (domain) in a non-linear fashion. Together with the absent of phase or time delay information, the image reconstruction task for EIT usually faces the issue of low signal-to-noise-ratio (SNR) and support ambiguity. Multiple real-life applications typically fall under this category of problems such as medical and geophysical subsurface imaging, deconvolution, and image deblurring. The EIT is an imaging technique that estimates a distribution of electrical admittivity $\gamma$ of a ``subject under test (SUT)" sometimes called as ``medium" based on an inference from data captured from the medium's boundary. Although the name would suggests the value of `impedance' to be measured, in most practical scenario its reciprocal which is `admittance' (admittivity value) is commonly calculated instead. The admittivity value $\gamma$ encapsulates the information of medium's conductivity ($\sigma$) and permittivity ($\epsilon$) with the following expression $\gamma = \sigma + i\omega\epsilon$, where $\omega=2\pi f$ is the angular frequency (rad/s) and $f$ is frequency in Hz. The ``tomographic reconstruction" or ``the process of estimating the medium's internal admittivity distribution" can be accomplished by solving a mathematical problem that is commonly known as an ``inverse problem." Due to a quasi-static nature of this imaging modality, the image reconstruction of the EIT is known to be one of the most severely ill-posed inverse problems. The image reconstruction problem of EIT will be analyzed through two different viewpoints. Firstly, this problem will be analyzed through the lens of a discrete ill-posed inverse problem from the applied mathematics community viewpoint. Secondly, this problem will be re-analyzed through the lens of array signal processing/beamforming framework from the signal processing community viewpoint. While exploring the EIT image reconstruction techniques, it was observed that although the EIT system can be implemented economically, the implementation process is not completely straightforward. Therefore, a framework of a simple and reliable EIT prototyping system that could provide means to collect different data format (e.g. single measurement vector (SMV) and multiple measurement vectors (MMV)) were implemented in this thesis. This robust, economical, and straightforward implementation framework was developed with an aim that it could be beneficial for those who are interested in image reconstruction and inverse algorithms without the access of a full functioning commercialized EIT system. The second part is the support identification for DOA estimation under Gaussian and $\alpha-$stable distributed noises. Several improvements of a robust beamforming technique called MUSIC-like algorithm had been further investigated which helps improve the algorithm's performance. The main focus of this part is the performance improvement of the algorithm that can be used for support identification task under different noise scenarios. Three different approaches for estimating a relaxation parameter, an essential parameter used in the MUSIC-like algorithm, are presented. By experimenting with different approaches, an adaptive framework for relaxation parameter was observed to have the potential for further improvement. The performance of the MUSIC-like algorithm was then further explored through the directional adaptive framework under both Gaussian and $\alpha-$stable distributed noise. It was also compared with the conventional MUSIC algorithm coupled with a pre-conditioned covariance matrix.

Electrical impedance tomography (EIT) is an imaging technique that reconstructs the conductivity distribution in the interior of an object using electrical measurements from the electrodes that are attached around the boundary. The Complete Electrode Model (CEM) accurately incorporates the electrode size, shape, and effective contact impedance into the forward problem for EIT. In this work, the effect of the conductivity distribution and the electrode contact impedance on the solution of the forward problem is addressed. In particular, the sensitivity of the electric potential with respect to a small-amplitude perturbation in the conductivity, and with respect to some defective electrodes is studied. The Gâteaux derivative is introduced as a tool for the sensitivity analysis and the Gâteaux differentiability of the electric potential with respect to the conductivity and to the contact impedance of the electrodes is proved. The derivative is then expressed as the unique solution to a variational problem and the discretization is performed with Finite Elements of type P1. Numerical simulations for different 2D and 3D configurations are presented. This study illustrates the impact of the presence of perturbations in the parameters of CEM on EIT measurements. Finally, the 2D inverse conductivity problem for EIT is numerically solved for some configurations and the results confirm the conclusions of the numerical sensitivity analysis.

It is important to get reliable measurement data in Electrical impedance tomography (EIT) for getting satisfactory reconstruction result. Because EIT inverse problem is non-linear and ill-posed, incorrect data such as measured data through fault electrode can effect the estimation of resistivity distribution. So, EIT measurement system with methods for checking reliability has been developed. However, most existing methods require extra cost to evaluate reliability of EIT system. This paper presents modified Gauss-Newton (GN) method based on random sample consensus (RANSAC) algorithm for finding fault electrodes and getting good reconstructed image with faulty data for two-phase flow application. Also, suitable residual equation is proposed for determining the threshold to use RANSAC algorithm. Numerical simulations are performed to validate the performance of the proposed method. The results show that the proposed method has a good reconstruction performance compared to the conventional GN method.

This paper presents a pure element-free Galerkin method (EFGM) forward model for image reconstruction in 2D and 3D electrical impedance tomography (EIT) using an adaptive current injection method. In EIT systems with the adapting current injection method, both static and dynamic images can be reconstructed; however, determination of electrode contact impedances in the complete electrode model is difficult and the Gap model is used. In this paper, in the EIT forward problem a weak form functional based on the Gap model and a pure EFGM approach are developed, and in the EIT inverse problem, Jacobian matrix is computed by the EFGM, and a fast integration technique is introduced to calculate the entries of the Jacobian matrix within an adequate computation time. The influence of increasing the density of nodes at and near the electrodes with steep electric potential gradients on the accuracy of FEM and EFGM forward solutions is investigated, and the performance of the image reconstruction algorithm with the proposed fast integration technique is examined. The numerical results reveal that the proposed EFGM forward model with the fast integration technique has an efficient performance both in terms of mean relative imaging errors and computational time.

In this paper, we develop a non‐iterative way to solve biharmonic boundary value problems by using a radial basis meshless method. This is an original application of meshless method to solving inverse problems without any iteration, since traditional numerical methods for inverse boundary value problems mainly are iterative and hence very time‐consuming. Numerical examples are presented for inverse biharmonic boundary value problems and corresponding direct problems, since solving direct problems is a preliminary step for inverse problems. All our examples of direct and inverse problems are solved within seconds in CPU time on a standard PC, which makes our proposed technique a great potential candidate for wide‐spread applications to other inverse problems. Copyright © 2004 John Wiley & Sons, Ltd.

Chapter eight - Electrical impedance tomography image reconstruction based on autoencoders and extreme learning machines

Electrical impedance tomography is an emerging imaging modality based on reconstruction of the resistivity distribution of the interior of the body, using non-invasive surface electrical measurements. Image reconstruction is a nonlinear, ill-posed, inverse problem. A new approach to the reconstruction problem is proposed, using multi-layer feed forward neural networks. The technique was tested using numerical simulations and actual measurements from a physical model. When compared with iterative techniques, superior performance was observed vis-a-vis image quality, reconstruction time and robustness to noise. This method can be easily adapted for different forward models of resistivity distribution used in existing reconstruction algorithms.

: With the funding provided by AFOSR, the PI was able to take his sabbatical leave for the period of 1 Sep 91 - 31 Aug 92 at Rutgers University. At Rutgers, the PI collaborated with Prof. Michael Vogelius on several problems. The research resulted in several reports, some of which have already been accepted for publication. The main research proposed was to investigate various aspects of the problem of electrical impedance tomography. In electrical impedance tomography, the problem is to image the interior of an object from static electrical measurements conducted on the boundary. The techniques of electrical impedance tomography may be applied to nondestructive evaluation of structures. The grant was directly responsible for two major accomplishments - (1) Inversion from real data and (2) Sensitivity of electrical impedance tomography.