A geometric inverse problem identification procedure for detection of cavities

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.

In this paper, we present theoretical developments and experimental results for the problem of estimating the conductivity map inside a volume using electrical impedance tomography (EIT) when the boundary locations of any internal inhomogeneities are known. We describe boundary element method (BEM) implementations of advanced electrode models for the forward problem of EIT. We then use them in the inverse problem with known internal boundaries and derive the associated Jacobians. We report on the results of two EIT phantom studies, one using a homogeneous cubical tank, and one using a cylindrical tank with agar conductivity inhomogeneities. We test both the accuracy of our BEM forward model, including the electrode models, as well as our inverse solution, against the measured data. Results show good agreement between measured values and both forward-computed tank voltages and inverse-computed conductivities; for instance, in a phantom experiment, we reconstructed the conductivities of three agar objects inside a cylindrical tank with an error less than 2% of their true value.

An inverse geometry problem in identifying irregular boundary configurations

The time-difference image reconstruction problem of electrical impedance tomography (EIT) refers to reconstructing the conductivity change in a human body part between two time points using the boundary impedance measurements. Conventionally, the problem can be formulated as a linear inverse problem. However, due to the physical property of the forward process, the inverse problem is seriously ill-posed. As a result, traditional regularized least-squares (LSs)-based methods usually produce low-resolution images that are difficult to interpret. This work proposes a framework that uses a deep generative model (DGM) to constrain the unknown conductivity. Specifically, this framework allows the inclusion of a constraint that describes a mathematical relationship between the generative model and the unknown conductivity. The resultant constrained minimization problem is solved using an extended alternating direction method of multipliers (ADMM). The effectiveness of the framework is demonstrated by the example of 3-D time-difference chest EIT imaging. Numerical experiment shows a significant improvement of the image quality compared with total variation (TV)-regularized LSs method (peak signal-to-noise ratio (PSNR) is improved by 4.3% for 10% noise and 4.6% for 30% noise; structural similarity (SSIM) is improved by 4.8% for 10% noise and 6.0% for 30% noise). Human experiments show improved correlation between the reconstructed images and images from reference techniques.

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.

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.

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.

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.

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.

In this paper, we use the theory of symmetric Dirichlet forms to derive Feynman–Kac formulae for the forward problem of electrical impedance tomography with possibly anisotropic, merely measurable conductivities corresponding to different electrode models on bounded Lipschitz domains. Subsequently, we employ these Feynman–Kac formulae to rigorously justify stochastic homogenization in the case of a stochastic boundary value problem arising from an inverse anomaly detection problem. Motivated by this theoretical result, we prove an estimate for the speed of convergence of the projected mean-square displacement of the underlying process which may serve as the theoretical foundation for the development of new scalable stochastic numerical homogenization schemes.

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.

The absolute image reconstruction problem of electrical impedance tomography (EIT) is ill-posed. Traditional methods usually solve a nonlinear least squares problem with some kind of regularization. These methods suffer from low accuracy, poor anti-noise performance, and long computation time. Besides, the integration of a priori information is not very flexible. This work tries to solve EIT inverse problem using a machine learning algorithm for the application of thorax imaging. We developed the supervised descent learning EIT (SDL-EIT) inversion algorithm based on the idea of supervised descent method (SDM). The algorithm approximates the mapping from measured data to the conductivity image by a series of descent directions learned from training samples. We designed a training data set in which the thorax contour, and some general structure of lungs, and heart are embedded. The algorithm is implemented in both two-, and three-dimensional cases, and is evaluated using synthetic, and measured thoracic data. Results, and conclusion: For synthetic data, SDL-EIT shows better accuracy, and anti-noise performance compared with traditional Gauss-Newton inversion (GNI) method. For measured data, the result of SDL-EIT is reasonable compared with computed tomography (CT) scan image. Using SDL-EIT, prior information can be easily integrated through the specifically designed training data set, and the image reconstruction process can be accelerated. The algorithm is effective in inverting measured thoracic data. It is a potential algorithm for human thorax imaging.

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.

We prove a new global stability estimate for the Gel’fand–Calderón inverse problem on a two-dimensional bounded domain. Specifically, the inverse boundary value problem for the equation ${- }\Delta \psi + v\hspace{0.167em} \psi = 0$ on $D$ is analysed, where $v$ is a smooth real-valued potential of conductivity type defined on a bounded planar domain $D$. The main feature of this estimate is that it shows that the smoother a potential is, the more stable its reconstruction is. Furthermore, the stability is proven to depend exponentially on the smoothness, in a sense to be made precise. The same techniques yield a similar estimate for the Calderón problem for electrical impedance tomography.