The Linearized Inverse Problem in Multifrequency Electrical Impedance Tomography

This paper provides an analysis of the linearized inverse problem in multifrequency electrical impedance tomography. We consider an isotropic conductivity distribution with a finite number of unknown inclusions with different frequency dependence, as is often seen in biological tissues. We discuss reconstruction methods for both fully known and partially known spectral profiles, and demonstrate in the latter case the successful employment of difference imaging. We also study the reconstruction with an imperfectly known boundary, and show that the multifrequency approach can eliminate modeling errors and recover almost all inclusions. In addition, we develop an efficient group sparse recovery algorithm for the robust solution of related linear inverse problems. Several numerical simulations are presented to illustrate and validate the approach.Comment: 25 pp, 11 figure

Enhanced image reconstruction of electrical impedance tomography using simultaneous algebraic reconstruction technique and K-means clustering

Electrical impedance tomography (EIT), as a non-ionizing tomography method, has been widely used in various fields of application, such as engineering and medical fields. This study applies an iterative process to reconstruct EIT images using the simultaneous algebraic reconstruction technique (SART) algorithm combined with K-means clustering. The reconstruction started with defining the finite element method (FEM) model and filtering the measurement data with a Butterworth low-pass filter. The next step is solving the inverse problem in the EIT case with the SART algorithm. The results of the SART algorithm approach were classified using the K-means clustering and thresholding. The reconstruction results were evaluated with the peak signal noise ratio (PSNR), structural similarity indices (SSIM), and normalized root mean square error (NRMSE). They were compared with the one-step gauss-newton (GN) and total variation regularization based on iteratively reweighted least-squares (TV-IRLS) methods. The evaluation shows that the average PSNR and SSIM of the proposed reconstruction method are the highest of the other methods, each being 24.24 and 0.94; meanwhile, the average NRMSE value is the lowest, which is 0.04. The performance evaluation also shows that the proposed method is faster than the other methods

(An overview of) Synergistic reconstruction for multimodality/multichannel imaging methods

Imaging is omnipresent in modern society with imaging devices based on a zoo of physical principles, probing a specimen across different wavelengths, energies and time. Recent years have seen a change in the imaging landscape with more and more imaging devices combining that which previously was used separately. Motivated by these hardware developments, an ever increasing set of mathematical ideas is appearing regarding how data from different imaging modalities or channels can be synergistically combined in the image reconstruction process, exploiting structural and/or functional correlations between the multiple images. Here we review these developments, give pointers to important challenges and provide an outlook as to how the field may develop in the forthcoming years. This article is part of the theme issue 'Synergistic tomographic image reconstruction: part 1'

Applied Harmonic Analysis and Data Science (hybrid meeting)

Data science has become a field of major importance for science and technology nowadays and poses a large variety of challenging mathematical questions. The area of applied harmonic analysis has a significant impact on such problems by providing methodologies both for theoretical questions and for a wide range of applications in signal and image processing and machine learning. Building on the success of three previous workshops on applied harmonic analysis in 2012, 2015 and 2018, this workshop focused on several exciting novel directions such as mathematical theory of deep learning, but also reported progress on long-standing open problems in the field

A Bayesian conjugate gradient method (with Discussion)

A fundamental task in numerical computation is the solution of large linear systems. The conjugate gradient method is an iterative method which offers rapid convergence to the solution, particularly when an effective preconditioner is employed. However, for more challenging systems a substantial error can be present even after many iterations have been performed. The estimates obtained in this case are of little value unless further information can be provided about the numerical error. In this paper we propose a novel statistical model for this numerical error set in a Bayesian framework. Our approach is a strict generalisation of the conjugate gradient method, which is recovered as the posterior mean for a particular choice of prior. The estimates obtained are analysed with Krylov subspace methods and a contraction result for the posterior is presented. The method is then analysed in a simulation study as well as being applied to a challenging problem in medical imaging

Parallel algorithms for three dimensional electrical impedance tomography

This thesis is concerned with Electrical Impedance Tomography (EIT), an imaging technique in which pictures of the electrical impedance within a volume are formed from current and voltage measurements made on the surface of the volume. The focus of the thesis is the mathematical and numerical aspects of reconstructing the impedance image from the measured data (the reconstruction problem). The reconstruction problem is mathematically difficult and most reconstruction algorithms are computationally intensive. Many of the potential applications of EIT in medical diagnosis and industrial process control depend upon rapid reconstruction of images. The aim of this investigation is to find algorithms and numerical techniques that lead to fast reconstruction while respecting the real mathematical difficulties involved. A general framework for Newton based reconstruction algorithms is developed which describes a large number of the reconstruction algorithms used by other investigators. Optimal experiments are defined in terms of current drive and voltage measurement patterns and it is shown that adaptive current reconstruction algorithms are a special case of their use. This leads to a new reconstruction algorithm using optimal experiments which is considerably faster than other methods of the Newton type. A tomograph is tested to measure the magnitude of the major sources of error in the data used for image reconstruction. An investigation into the numerical stability of reconstruction algorithms identifies the resulting uncertainty in the impedance image. A new data collection strategy and a numerical forward model are developed which minimise the effects of, previously, major sources of error. A reconstruction program is written for a range of Multiple Instruction Multiple Data, (MIMD), distributed memory, parallel computers. These machines promise high computational power for low cost and so look promising as components in medical tomographs. The performance of several reconstruction algorithms on these computers is analysed in detail