Sensitivity matrix analysis of the back-projection algorithm in Electrical Impedance Tomography

In this study, a reconstruction algorithm for a 16-electrode interleaved-drive electrical impedance tomography (EIT) system is developed, based on inversion of an analytically calculated sensitivity matrix. The sensitivity matrix is calculated using Geselowitz's lead-sensitivity theorem. Eight interleaved electrodes out of 16 (equally spaced) electrodes are designated as current injection electrodes and the remaining eight electrodes are designated as measurement electrodes. The sensitivity matrix is singular, therefore singular value decomposition (SVD) of the sensitivity matrix, followed by pseudoinversion - with and without truncation - is used to reconstruct images. The algorithm is a single-pass algorithm. Data from a saline filled tank and in vivo data during respiration and the cardiac cycle, acquired by using a Sheffield multifrequency system, are used to reconstruct images. The effect of different truncation levels on the reconstructed images is investigated.

Sensitivity-matrix-independent electrical impedance tomography: Damage detection for composite materials

Image reconstruction in electrical impedance tomography (EIT) is a non-linear inverse problem. The linear model is always used in most of the reconstruction algorithm in dynamic EIT, which causes large errors of image reconstruction. In this paper, we proposed a new image reconstruction method in dynamic EIT. In the method the artificial neural network based on the error back propagation algorithm (BP ANN) is used to express the non-linear relation between the impedance change position inside the measured object and the voltage change value measured on the surface of the object. Thus, the location ofthe impedance change can be decided by the measured voltage change on the surface, and then the impedance change image will be reconstructed with linear approximated method. The reconstructed error will be decreased largely, because the impedance change position can be detected precisely by our proposed method. The experimental results indicate that the precision of the reconstructed image with our method is greatly higher than that with the back projection 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 emerging new imaging modality not yet routinely used in a clinical setting. The generation of EIT scans is based on the non-invasive measurement of electrical properties of tissues which does not require the use of radiation. Thoracic EIT examinations have been shown to track changes in regional lung volumes at scan rates of up to 40-50 scans/s. Experimental and clinical studies indicate the potential of EIT in monitoring regional lung ventilation and aeration at the bedside. Mechanically ventilated patients with acute lung injury and acute respiratory distress syndrome (ARDS) might benefit from possible continuous monitoring of their regional lung function by EIT which could be used to optimise the ventilator settings and minimise the ventilator-induced lung injury. This review explains the measuring principle of EIT and the analysis of regional EIT signals which allow the assessment of regional lung ventilation, aeration and potentially also lung perfusion. Protective lung ventilation using low tidal volumes and adequate positive end-expiratory pressures, recruitment manoeuvres, surfactant administration, prone posture, restricted fluid administration, assisted ventilation allowing spontaneous breathing, partial liquid ventilation, administration of pulmonary vasoactive, anti-inflammatory and antioxidant agents are used or considered in the management of ARDS patients. EIT is capable of monitoring the pulmonary effects of many of these therapeutic procedures. This is documented by examples from experimental and clinical EIT studies and by references to relevant EIT literature. The perspectives and limitations of EIT monitoring are also addressed.

Electrical impedance tomography (EIT) is a technique to reconstruct the conductivity distribution of an inhomogeneous medium. Sensitivity matrix of EIT is calculated with a selected reference conductivity distribution, which will change with the media distribution. Landweber iterative algorithm based on updated sensitivity matrix was presented in this paper. Reconstructed image based on conventional Landweber iteration was selected as the initial image for sensitivity matrix update, and the reconstructed images after sensitivity matrix update using different initial images were compared. The effect on the quality of reconstructed images for different times of sensitivity matrix update was also analyzed. Simulation and static test results showed that reconstructed images with higher quality can be obtained.

Electrical impedance tomography (EIT) is an emerging technique for medical imaging. According to reconstructed images, the human disease can be identified. Note that the reconstruction of EIT is an ill-posed inverse problem. To solve this problem, various sensitivity-based methods have been proposed. Nevertheless, a sensitivity matrix is affected by a number of factors. Therefore, classification reflected by images is sometimes not very reliable. In this work, a novel method based on a residual neural network is proposed for stroke classification. It is realized by directly processing measured difference data from EIT. After training of the residual neural network, the proposed method is quantitatively evaluated when noise is exerted, shape deviation occurs, and conductivity varies in three different layers. Also, impacts of contact impedance and small-sized inclusion are studied. Furthermore, phantom experiments are conducted. Compared with fully connected neural networks and shallow convolution neural networks, the results demonstrate that the proposed method has a better performance in classification for various cases. This article offers an effective alternative for stroke classification in EIT.

In the aqueous-based two-phase flow, if the void fraction of dispersed phase exceeds 0.25, conventional electrical impedance tomography (EIT) produces a considerable error due to the linear approximation of the sensitivity back-projection (SBP) method, which limits the EIT’s wider application in the process industry. In this paper, an EIT sensing system which is able to handle full void fraction range in two-phase flow is reported. This EIT system employs a voltage source, conducts true mutual impedance measurement and reconstructs an online image with the modified sensitivity back-projection (MSBP) algorithm. The capability of the Maxwell relationship to convey full void fraction is investigated. The limitation of the linear sensitivity back-projection method is analysed. The MSBP algorithm is used to derive relative conductivity change in the evaluation. A series of static and dynamic experiments demonstrating the mean void fraction obtained using this EIT system has a good agreement with reference void fractions over the range from 0 to 1. The combination of the new EIT system and MSBP algorithm would significantly extend the applications of EIT in industrial process measurement.

Non-linear image reconstruction methods are desirable for applications in electrical impedance tomography (EIT) such as brain or breast imaging where the assumptions of linearity are violated. We present a novel non-linear Newton-Krylov method for solving large-scale EIT inverse problems, which has the potential advantages of improved robustness and computational efficiency over previous methods. This combines Krylov-subspace efficiency in the production of an implicit Hessian inverse together with the Newton-type search direction effectiveness. The computational cost was assessed by comparing the objective function value and image error norm with respect to run-time, iteration count and memory consumption with six other non-linear methods, including Damped Newton-Gauss, Levenberg-Marquardt, Variable Metric and non-linear Conjugated Gradients, using realistic layered head models with meshes of 4, 12 and 31K elements. For the small-scale model, Newton-type methods slightly outperformed the Krylov-Newton approach, while the other large-scale methods performed poorly. For the larger two models, the Newton-Krylov approach converged much more rapidly than the Krylov-subspace and quasi-Newton methods; Newton-type methods failed to converge in the time available. This approach opens a new frontier for non-linear EIT image reconstruction, as it allows production of accurate solutions of large-scale realistic models using modest computational resources.

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

Electrical impedance tomography (EIT) is a promising agile imaging modality that allows estimation of the electrical conductivity distribution at the interior of an object from boundary measurements, which is attractive in a broad spectrum of biomedical, geophysical, and industrial continuous monitoring applications. As the inverse problem of EIT image reconstruction generally suffers from severe ill-posedness, various regularization techniques have been adopted to impose moderate constraints so as to obtain desired, usually sparse solutions. Extensive investigation showed that an enhanced recover performance can be achieved by exploiting a priori knowledge on the sparse structures. For regularization incorporating such knowledge, Bayesian approach is considered a natural mechanism. In this paper, we introduce the concept of sparse Bayesian learning to EIT imaging, which yields improved accuracy and robustness in the image reconstruction performance by exploiting signal structures in terms of their block sparsity and intra-block correlation. The effectiveness of the proposed algorithm is confirmed by the phantom test results.

Electrical impedance tomography is a non-invasive imaging modality that uses the electrical conductivity distribution to reconstruct images based on potential measurements from the object's surface. The proposed study was to design and fabricate a low-cost and simple reconstruction method for 3D electrical impedance tomography imaging. In this study, we have been successfully developed 3 Dimensional Electrical Impedance Tomography (3D-EIT) system using 16 copper electrodes (Cu) to detect and reconstruct the presence of objects in the Phantom. 3D-EIT was developed using Phantom as a test object with PVC pipe material, with an inner diameter of 7.2 cm and a height of 5.4 cm. Electrodes were arranged in two rings, with each ring having eight electrodes arranged in a planar line. Furthermore, the Gauss-Newton algorithm and Laplace prior regularization were used to image reconstruction of objects inside the Phantom using voltage measurement produced from sequential pairs of neighboring electrodes. The voltage is obtained from the injection of a constant current of 1 mA at 20 kHz into the system's electrode pairs. The objects used in this research are PVC pipe, solid aluminum, PVC pipes filled with wax glue, and copper trusses. The results obtained show that the reconstruction results can provide information about the position, the electrical properties, and the shape of real objects. Finally, the system can detect the location, height, and electrical properties of objects for all variations of angle and height variations.

An image reconstruction method is proposed based on Lagrange-Newton method for electrical impedance tomography (EIT) and ultrasound tomography (UT) dual-modality imaging. Since the change in conductivity distribution is usually accompanied with the change in acoustic impedance distribution, the reconstruction targets of EIT and UT are unified to the conductivity difference using the same mesh model. Some background medium distribution information obtained from ultrasound transmission and reflection measurements can be used to construct a hard constraint about the conductivity difference distribution. Then, the EIT/UT dual-modality inverse problem is constructed by an equality constraint equation, and the Lagrange multiplier method combining Newton-Raphson iteration is used to solve the EIT/UT dual-modality inverse problem. The numerical and experimental results show that the proposed dual-modality image reconstruction method has a better performance than the single-modality EIT method and is more robust to the measurement noise.

Electrical impedance tomography (EIT) imaging modality has great potential on industrial applications with the advantages of being high temporal resolution. It is especially useful in cases, such as geophysical detection, landmine detection, and detections on non-transparent region, where the measurement data are only available from single surface, for data acquisition. Instead of the circular EIT model that uses the traditional circular electrode model, in this paper, planar array EIT is implemented, aiming to visualize a pipeline transporting a two-phase flow. The planar array can explore spatial information within its detectable region by producing 3D images, which have a higher spatial resolution in the axis direction than a traditional EIT with a dual-plane electrode sensor. However, in solving the inverse problem of a 3D subsurface EIT using a planar array, the images may be degraded, especially in cases where the location of the target is relatively deep. The total variation (TV) algorithm as block prior assumption-based regularization method has the potential to improve the image quality, and some works have shown that TV reconstructs sharper images, which provides an advantage when representing spatial information. In this paper, the performance of subsurface EIT using the TV algorithm for 3D visualization is presented based on simulations and experiments, and the results of quantitative measurement of depth are discussed.

Electrical brain imaging (EBI) refers to a set of techniques that exploit either the spontaneous electrical activity of the central nervous system, as in electroencephalographic (EEG) source reconstruction, or make use of external current injections, as in electrical impedance tomography (EIT) , to image the structure or function of the brain. When compared to other brain imaging methods used in research or in the clinical setting, such as computed tomography (CT), magnetic resonance imaging (MRI), functional MRI (fMRI), positron emission tomography (PET) and single photon emission computed tomography (SPECT), EIT and EEG source localization instrumentation offer the advantages of portability, low cost, high temporal resolution [ms] and quick setup. The downsides are a low spatial resolution [cm], high computational cost of the image reconstruction process and high sensitivity to imperfections of the electrical model of the head.
\nIn this work, a new special purpose reconstruction algorithm for EIT is presented and validated wth experimental measurements performed on a cylindrical phantom and on a simulated human head. The algorithm focuses on the quick detection of compact conductivity contrasts in imperfectly known in 3D domains. The performance of the proposed algorithm is then compared to the one of a benchmark reconstruction method in the EIT field, Tikhonov regularized reconstruction, with stroke detection and classification as a case study. Moreover, the possible application of EIT imaging to the detection of epileptic foci with intracranial deep electrodes (stereoelectroencephalography or SEEG) is explored.
\nFinally, EEG source reconstruction algorithms are implemented on a heterogeneous multi-CPU and multi-GPU computing system to significantly reduce the reconstruction time.