Methods for the Electrical Impedance Tomography Inverse Problem: Deep Learning and Regularization with Wavelets

Electrical impedance tomography, also known as EIT, is a type of diffusive imaging modality that is non-invasive, radiation-free, and cost-effective for recovering electrical properties within a closed domain from surface measurements. The process involves injecting electrical current into a set of electrodes to measure the voltage on the smooth surface of the domain. The recovered EIT images show how well different materials or tissues within the domain conduct or impede electrical flow, which is helpful in detecting and locating anomalies. For the EIT inverse problem, it is challenging to recover reliable and resolvable electrical conductivity images since it is highly nonlinear and severely ill-posed, especially when the data is corrupted with noise. To address this issue, we propose (1) a wavelet-based modified Gauss-Newton (WGN) method that uses wavelets as a form of regularization and parameter reduction. In (1), we enforce regularization through the use of wavelet coefficients by projecting the original formulation to the wavelet domain and then only retaining the wavelet coefficients of highest power. The projected wavelet formulation is of a smaller dimension and, therefore, shows promise in improving the ill-posedness of the EIT inverse problem. Different wavelet families are implemented to capture localized features, smoothness, and irregularities within the domain. In addition, we also propose (2) a novel deep learning algorithm to solve the EIT inverse problem. In (2), we develop a deep neural network (DNN) with multiple transposed convolutional layers and activation functions to recover the EIT images. The DNN is first trained on a large set of EIT images and data, and then we recover EIT images in real-time from the trained DNN. We compare the image reconstructions from the DNN with a benchmark algorithm. For model validation, we employed a set of synthetic examples with various anomalies to test the performance and efficacy of both the DNN and WGN method. The results from both methods show promise in improving EIT image reconstructions

Review on electrical impedance tomography: Artificial intelligence methods and its applications

© 2019 by the authors. Electrical impedance tomography (EIT) has been a hot topic among researchers for the last 30 years. It is a new imaging method and has evolved over the last few decades. By injecting a small amount of current, the electrical properties of tissues are determined and measurements of the resulting voltages are taken. By using a reconstructing algorithm these voltages then transformed into a tomographic image. EIT contains no identified threats and as compared to magnetic resonance imaging (MRI) and computed tomography (CT) scans (imaging techniques), it is cheaper in cost as well. In this paper, a comprehensive review of efforts and advancements undertaken and achieved in recent work to improve this technology and the role of artificial intelligence to solve this non-linear, ill-posed problem are presented. In addition, a review of EIT clinical based applications has also been presented

A Wavelet-Based Multiresolution Reconstruction Method for Fluorescent Molecular Tomography

Image reconstruction of fluorescent molecular tomography (FMT) often involves repeatedly solving large-dimensional matrix equations, which are computationally expensive, especially for the case where there are large deviations in the optical properties between the target and the reference medium. In this paper, a wavelet-based multiresolution reconstruction approach is proposed for the FMT reconstruction in combination with a parallel forward computing strategy, in which both the forward and the inverse problems of FMT are solved in the wavelet domain. Simulation results demonstrate that the proposed approach can significantly speed up the reconstruction process and improve the image quality of FMT

2-D and 3-D tools for electrical imaging in environmental applications

The work presented in this thesis is focused on the characterization of contaminated sites via the application of non-invasive methods, such as electrical resistivity and electromagnetic techniques. With the aim of improving the relationships between geophysical parameters and hydro-geological features of the subsoil, some innovative methods are proposed. In particular, being electrical resistivity tomography an ill-posed and under-determined problem, the solution is computed by an over-parameterization of the model element with respect of the available measured data. The need to improve the resolution of resistivity imaging technique without stressing the intrinsic under-determination of the geo-electrical problem, and to provide more robust results in integrated characterization approaches, guided the work presented in this thesis to works on both “static” 2-D and 3-D problems, and on the inversion of time-lapse data. In addition, joint characterization of contaminated site has been carried out together with chemico-physical and biological analysis. Grid manipulation techniques have been explored and adapted to the inversion of geo-electrical data, in order to solve inversion of resistivity data set trying to improve space resolution without decaying the robustness of the inversion process. Furthermore, results obtained by the testing of a novel code for the inversion of complex resistivity data are presented. The software was implemented on the basis of the Primal-Dual method, which allows flexibility in choosing to minimize the L1- or the L2- norm on both the data and regularization term of the objective function. Strategies related to the inversion of electrical resistivity and IP data have been developed with the aim of improving the interpretability of the results both in space and in time. These activity are part of a common effort for achieving an integrated characterization of contaminated sites, jointly with chemico-physical analysis of soils and groundwater and with measures of presence and degradation activity of hydrocarbon-degrading biomass

Transformer Meets Boundary Value Inverse Problems

A Transformer-based deep direct sampling method is proposed for a class of boundary value inverse problems. A real-time reconstruction is achieved by evaluating the learned inverse operator between carefully designed data and the reconstructed images. An effort is made to give a specific example to a fundamental question: whether and how one can benefit from the theoretical structure of a mathematical problem to develop task-oriented and structure-conforming deep neural networks? Specifically, inspired by direct sampling methods for inverse problems, the 1D boundary data in different frequencies are preprocessed by a partial differential equation-based feature map to yield 2D harmonic extensions as different input channels. Then, by introducing learnable non-local kernels, the direct sampling is recast to a modified attention mechanism. The proposed method is then applied to electrical impedance tomography, a well-known severely ill-posed nonlinear inverse problem. The new method achieves superior accuracy over its predecessors and contemporary operator learners, as well as shows robustness with respect to noise. This research shall strengthen the insights that the attention mechanism, despite being invented for natural language processing tasks, offers great flexibility to be modified in conformity with the a priori mathematical knowledge, which ultimately leads to the design of more physics-compatible neural architectures

Biomedical Sensing and Imaging

This book mainly deals with recent advances in biomedical sensing and imaging. More recently, wearable/smart biosensors and devices, which facilitate diagnostics in a non-clinical setting, have become a hot topic. Combined with machine learning and artificial intelligence, they could revolutionize the biomedical diagnostic field. The aim of this book is to provide a research forum in biomedical sensing and imaging and extend the scientific frontier of this very important and significant biomedical endeavor