A novel deep neural network method for electrical impedance tomography

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.

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

The trust region (TR) method is an effective algorithm for image reconstruction of electrical impedance tomography (EIT). Aiming at the drawback of the traditional TR method, an improved method named the nonmonotonic and self-adaptive trust region (NSTR) method is proposed in this paper, in which three kinds of modified techniques are shown that help in improving the computational precision and convergence speed of the algorithm. The comparisons with image reconstruction are carried out between the NSTR, Levenberd–Marquardt (LM) and TR methods. Simulation experiment results indicate that for both the LM and TR methods it is difficult to accurately reconstruct concave and multiple boundaries. However the NSTR method cannot only realize accurate reconstruction, but also provide faster convergence speed. A noise simulation test is carried out and results show that the NSTR method also has strong stability in image reconstruction for EIT. This new method presents a feasible and effective way to research on image reconstruction for EIT.

Image reconstruction in electrical impedance tomography (EIT) deals with an ill-posed and non-linear inverse problem. It has the objective of minimizing the difference between simulated (virtual) object data and electric voltage measurements performed on a non-simulated (real) object. In this paper, a new approach to the simulated annealing method applied to the reconstruction of EIT images is described. The main advantage of this approach is that all conductivity parameters are updated simultaneously. Most methods that employ simulated annealing to the problem of EIT usually evaluate each conductivity parameter individually resulting in high computational cost. The algorithm was tested both with computationally generated data and with measurements performed on a physical tank. In both cases, the method was able to make data inversion, determining the position, the dimensions and the conductivity of materials in an opaque object plane.

Magnetic induction tomography (MIT) of biological tissue is used for the reconstruction of the complex conductivity distribution κ inside the object under investigation. It is based on the perturbation of an alternating magnetic field caused by the object and can be used in all applications of electrical impedance tomography (EIT) such as functional lung monitoring and assessment of tissue fluids. In contrast to EIT, MIT does not require electrodes and magnetic fields can also penetrate non-conducting barriers such as the skull. As in EIT, the reconstruction of absolute conductivity values is very difficult because of the method's sensitivity to numerical errors and noise. To overcome this problem, image reconstruction in EIT is often done differentially. Analogously, this concept has been adopted for MIT. Two different methods for differential imaging are applicable. The first one is state-differential, for example when the conductivity change between inspiration and expiration in the lung regions is being detected. The second one is frequency-differential, which is of high interest in motionless organs like the brain, where a state-differential method cannot be applied. An equation for frequency-differential MIT was derived taking into consideration the frequency dependence of the sensitivity matrix. This formula is valid if we can assume that only small conductivity changes occur. In this way, the non-linear inverse problem of MIT can be approximated by a linear one (depending only on the frequency), similar to in EIT. Keeping this limitation in mind, the conductivity changes between one or more reference frequencies and several measurement frequencies were reconstructed, yielding normalized conductivity spectra. Due to the differential character of the method, these spectra do not provide absolute conductivities but preserve the shape of the spectrum. The validity of the method was tested with artificial data generated with a spherical perturbation within a conducting cylinder as well as for real measurement data. The measurement data were obtained from a potato immersed in saline. The resulting spectra were compared with reference measurements and the preservation of the shape of the spectra was analyzed.

Image reconstruction in electrical impedance tomography using the sensitivity theorem is generally based on the assumption that the initial conductivity distribution of the body being imaged is uniform. The technique of image reconstruction using this method is described and reconstructed images are presented. Improvements in image quality and accuracy are demonstrated when accurate a priori `anatomical' information, in the form of a model of the distribution of conductivity within the region to be imaged, is used to construct the sensitivity matrix. In practice correct a priori information is not available, for example the conductivity values within the various anatomical regions will not be known. An iterative algorithm is presented which allows the conductivity parameters of the a priori model to be determined during reconstruction. Keywords: electrical impedance tomography, image reconstruction, a priori information

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

Impact of the Radial Basis Function Spread Factor onto Image Reconstruction in Electrical Impedance Tomography

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.

Image reconstruction in electrical impedance tomography is, generally, an ill-posed nonlinear inverse problem. Regularization methods are widely used to ensure a stable solution. In this paper we present a case study, which uses a novel electrical impedance tomography method for reconstruction of layered biological tissues with piece-wise continuous plane-stratified profiles. The algorithm implements the recently proposed reconstruction scheme for piecewise constant conductivity profiles, based on an improved Prony method in conjunction with Bessel transform. It is shown that the proposed algorithm is capable of successfully reconstructing piece-wise continuous conductivity profiles with moderate (WKB) slop. This reconstruction procedure, which calculate both the locations and the conductivities, repetitively provides inhomogeneous depth discretization, i.e., the depths grid is not equispaced. Incorporation of this specific inhomogeneous grid, the main result of our investigation, in the widely used MLS reconstruction procedure results in a stable reconstruction, whereas, utilization of equispaced depth grid leads to inaccurate reconstruction. The effectiveness and the significance of inhomogeneous discretization in EIT reconstruction procedure is further demonstrated and verified via numerical simulations.

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.

Resistivity reconstruction of practical phantoms is studied with Projection Error Propagation-based Regularization (PEPR) method to improve the image quality in Electrical Impedance Tomography (EIT). PEPR method calculates the regularization parameter as a function of the projection error produced due to the mismatch between experimental measurements and the calculated data. The regularization parameter in the reconstruction algorithm automatically adjusts its magnitude depending on the noise level present in measured data as well as the ill-posedness of the Hessian matrix. Resistivity images are reconstructed from the practical phantom data using the Electrical Impedance Diffuse Optical Reconstruction Software (EIDORS). The resistivity images reconstructed with PEPR method are compared with the Single step Tikhonov Regularization (STR) and Modified Levenberg Regularization (LMR) techniques. The results show that, the PEPR technique reduces the reconstruction errors in each iteration and improves the reconstructed images with better contrast to noise ratio (CNR), percentage of contrast recovery (PCR), coefficient of contrast (COC) and diametric resistivity profile (DRP).KeywordsElectrical Impedance Tomography (EIT)Projection Error Propagation-based Regularization (PEPR)Practical PhantomsBoundary potentialsEIDORSResistivity reconstructionSTRLMR

Comparison of statistical inversion with iteratively regularized Gauss Newton method for image reconstruction in electrical impedance tomography

Interval Simulated Annealing applied to Electrical Impedance Tomography image reconstruction with fast objective function evaluation

In electrical impedance tomography (EIT), the internal resistivity distribution of the unknown object is computed using the boundary voltage data induced by different current patterns with various reconstruction algorithms. The paper presents an image reconstruction algorithm based on a genetic algorithm (GA) via a two-step approach for the solution of the EIT inverse problem, in particular for the reconstruction of static images. Computer simulations with the 32 channels synthetic data show that the spatial resolution of reconstructed images by the proposed scheme is improved compared to that of the modified Newton-Raphson algorithm at the expense of increased computational burden.