An iterative algorithm for electrical impedance imaging using neural networks

The accuracy of iterative and non-iterative strain-gage balance load calculation methods was compared using data from the calibration of a force balance. Two iterative and one non-iterative method were investigated. In addition, transformations were applied to balance loads in order to process the calibration data in both direct read and force balance format. NASA's regression model optimization tool BALFIT was used to generate optimized regression models of the calibration data for each of the three load calculation methods. This approach made sure that the selected regression models met strict statistical quality requirements. The comparison of the standard deviation of the load residuals showed that the first iterative method may be applied to data in both the direct read and force balance format. The second iterative method, on the other hand, implicitly assumes that the primary gage sensitivities of all balance gages exist. Therefore, the second iterative method only works if the given balance data is processed in force balance format. The calibration data set was also processed using the non-iterative method. Standard deviations of the load residuals for the three load calculation methods were compared. Overall, the standard deviations show very good agreement. The load prediction accuracies of the three methods appear to be compatible as long as regression models used to analyze the calibration data meet strict statistical quality requirements. Recent improvements of the regression model optimization tool BALFIT are also discussed in the paper.

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

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.

Traditional lung electrical impedance imaging devices require the placement of measurement electrodes around the chest, which is inconvenient for clinical application in patients in a supine position. To address this issue, this study proposes a virtual electrode-based level set combined with GREIT algorithm (VESL-GREIT) for pulmonary impedance tomography. It utilizes measurement data from electrodes placed on only one side of the chest and generates full electrode measurement data using a deep learning network. The improved GREIT algorithm is then employed for image reconstruction. Acting as a “gray box,” VESL-GREIT not only takes advantage of prior information about the lungs but also enhances the robustness and generalizability of the algorithm. Simulation experiments demonstrate that with a measurement signal-to-noise ratio (SNR) above 20 dB, the imaging relative error (RE) and structure correlation coefficient (CC) change slowly, indicating high robustness of the algorithm. Physical model experiments show that compared to traditional dynamic electrical impedance imaging algorithms, the proposed algorithm achieves an RE of 0.167 and a structure CC of 0.886, enabling more accurate characterization of lung ventilation status. This research contributes to promoting the clinical application and dissemination of lung electrical impedance imaging monitoring methods and provides a new approach for open dynamic electrical impedance imaging.

Electrical impedance tomography (EIT) is a non-invasive imaging method that allows for the acquisition of resistivity distribution information within an object without the use of radiation. EIT is widely used in various fields, such as medical imaging, industrial imaging, geological exploration, etc. Presently, most electrical impedance imaging methods are restricted to uniform domains, such as pixelated pictures. These algorithms rely on model learning-based image reconstruction techniques, which often necessitate interpolation and embedding if the fundamental imaging model is solved on a non-uniform grid. EIT technology still confronts several obstacles today, such as insufficient prior information, severe pathological conditions, numerous imaging artifacts, etc. In this paper, we propose a new electrical impedance tomography algorithm based on the graph convolutional neural network model. Our algorithm transforms the finite-element model (FEM) grid data from the ill-posed problem of EIT into a network graph within the graph convolutional neural network model. Subsequently, the parameters in the non-linear inverse problem of the EIT process are updated by using the improved Levenberg—Marquardt (ILM) method. This method generates an image that reflects the electrical impedance. The experimental results demonstrate the robust generalizability of our proposed algorithm, showcasing its effectiveness across different domain shapes, grids, and non-distributed data.

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.

We read with interest the recent article by Zarantonello et al.1 on the physiologic effects of prone positioning on ventilation and perfusion distribution, oxygenation, and lung mechanics in COVID-19 acute respiratory distress syndrome patients. We would like to first congratulate the authors for the nicely conducted study that once again proved the clinical applicability of electrical impedance tomography for the assessment of ventilation–perfusion matching at the bedside. We acknowledge the clinical significance of the findings, but we wish to point out that several parts of the electrical impedance tomography methodology were incorrectly described in the article.In the Methods section, the authors described how the lung areas that were (1) perfused but not ventilated, (2) ventilated but not perfused, and (3) both ventilated and perfused were “divided by the sum of ventilated and perfused pixels” to calculate the corresponding relative fractions of the lung area in percentages, i.e., (1) “% shunt area,” (2) “% dead space area,” and (3) “% matched area,” respectively. However, the denominator must be the number of pixels that were ventilated or perfused, i.e., using the logical operator “or,” otherwise the calculated relative areas would be incorrect. The authors also determined “the percentage of total ventilated lung area that was located in the dorsal half of the thorax,” which they inaccurately termed as “dorsal ventilation.” According to the consensus electrical impedance tomography terminology and definitions,2 regional ventilation is determined as the sum of all pixel values of tidal impedance variation not as the pixel count, which defines just the ventilated area.The authors further introduced a new electrical impedance tomography parameter, the “homogeneity index,” which they “calculated as 1 – global inhomogeneity index,” based on the assumption of the global inhomogeneity index ranging from 1 to 0. However, the global inhomogeneity index values are commonly higher than 1 in patients with pronounced ventilation inhomogeneity, as shown already in some of the examined mechanically ventilated subjects in the original description of the global inhomogeneity index3 or recently in patients with exacerbated chronic lung disease.4 An example patient case is presented in figure 1, in which a global inhomogeneity index value of 2.59 was noted in the presence of pendelluft, which is a frequent cause of out-of-phase impedance variation both in spontaneously breathing and ventilated patients.5 The “homogeneity index” proposed by the authors would be negative in this case and assume the confusing value of –1.59. We therefore strongly advise against using the “homogeneity index” as a surrogate for the well established global inhomogeneity index.The authors pointed out that electrical impedance tomography perfusion assessment with saline bolus injection had been validated only in two animal studies. In fact, this technique of contrast-enhanced electrical impedance tomography has been studied in several other animal studies as well.6–9 Last, we agree with the authors that electrical impedance tomography images do not cover the whole lungs, but as summarized in the consensus paper on chest electrical impedance tomography,2 an electrical impedance tomography image does not originate from “only the area of the lung surrounded by the belt.”Electrical impedance tomography receives increasing attention due to its noninvasive bedside monitoring features. Correct description and implementation of the electrical impedance tomography technique and analysis procedures are crucial for further development and clinical use.Dr. Zhao receives a consulting fee from Dräger Medical (Lübeck, Germany). Dr. Frerichs reports funding from the European Union’s Framework program for Research and Innovation Horizon 2020 (WELMO, grant No. 825572).

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.

A review of electrical impedance tomography in lung applications: Theory and algorithms for absolute images

Detection of Heterogeneous Phase Generated Under the Accidental Condition by Employing Electrical Tomography Technique

PurposeIn the context of electrical impedance tomography (EIT), this paper aims to evaluate limitations of estimating conductivity or resistivity, as well as the improvements achieved with the use of an alternate description of the solution space, the logarithmic conductivity.Design/methodology/approachA quantitative analysis is performed, solving the inverse EIT problem by using the Gauss–Newton and non-linear conjugate gradient methods for a numerical phantom of 15 elements. A property of symmetry is studied for the direct EIT problem for a phantom of 385,601 elements.FindingsSolving the inverse EIT problem in logarithmic conductivity is more robust to the initial guess, as solutions are kept within physical bounds (conductivity positiveness). Also, convergence is faster and less dependent on the final values of the estimates.Research limitations/implicationsLogarithmic conductivity provides an advantageous description of the solution space for the EIT inverse problem. Similar estimation problems might be subject to analogous conclusions.Originality/valueThis study provides a novel analysis, quantitatively comparing the effect of different variables to solve the inverse EIT problem.

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.

An inverse problem is solved to obtain the particle concentration profile in suspension under pressuredriven flow with electrical impedance tomography (EIT). The finite element method (FEM) is employed in the forward problem and the regularized Newton-Raphson iterative method is used in the inverse problem. Different FEM meshes are used in the forward and the inverse problem not to commit inverse crime. To avoid post-calibration of measurement data, the complete electrode model is introduced. For the evaluation of the robustness of the reconstruction algorithm, several testing cases with measurement error are considered. The proposed algorithm can be used to reconstruct the particle concentration in suspension.

A real-time EIT imaging system based on the split augmented Lagrangian shrinkage algorithm

Microdisk modulator-assisted optical nonlinear activation functions for photonic neural networks