A mathematical and numerical framework for ultrasonically-induced Lorentz force electrical impedance tomography

We provide a mathematical analysis and a numerical framework for Lorentz force electrical conductivity imaging. Ultrasonic vibration of a tissue in the presence of a static magnetic field induces an electrical current by the Lorentz force. This current can be detected by electrodes placed around the tissue; it is proportional to the velocity of the ultrasonic pulse, but depends nonlinearly on the conductivity distribution. The imaging problem is to reconstruct the conductivity distribution from measurements of the induced current. To solve this nonlinear inverse problem, we first make use of a virtual potential to relate explicitly the current measurements to the conductivity distribution and the velocity of the ultrasonic pulse. Then, by applying a Wiener filter to the measured data, we reduce the problem to imaging the conductivity from an internal electric current density. We first introduce an optimal control method for solving such a problem. A new direct reconstruction scheme involving a partial differential equation is then proposed based on viscosity-type regularization to a transport equation satisfied by the current density field. We prove that solving such an equation yields the true conductivity distribution as the regularization parameter approaches zero. We also test both schemes numerically in the presence of measurement noise, quantify their stability and resolution, and compare their performance

Deep D-Bar: Real-Time Electrical Impedance Tomography Imaging With Deep Neural Networks

The mathematical problem for electrical impedance tomography (EIT) is a highly nonlinear ill-posed inverse problem requiring carefully designed reconstruction procedures to ensure reliable image generation. D-bar methods are based on a rigorous mathematical analysis and provide robust direct reconstructions by using a low-pass filtering of the associated nonlinear Fourier data. Similarly to low-pass filtering of linear Fourier data, only using low frequencies in the image recovery process results in blurred images lacking sharp features, such as clear organ boundaries. Convolutional neural networks provide a powerful framework for post-processing such convolved direct reconstructions. In this paper, we demonstrate that these CNN techniques lead to sharp and reliable reconstructions even for the highly nonlinear inverse problem of EIT. The network is trained on data sets of simulated examples and then applied to experimental data without the need to perform an additional transfer training. Results for absolute EIT images are presented using experimental EIT data from the ACT4 and KIT4 EIT systems

A fully non-linear optimization approach to acousto-electric tomography

This paper considers the non-linear inverse problem of reconstructing an electric conductivity distribution from the interior power density in a bounded domain. Applications include the novel tomographic method known as acousto-electric tomography, in which the measurement setup in Electrical Impedance Tomography is modulated by ultrasonic waves thus giving rise to a method potentially having both high contrast and high resolution. We formulate the inverse problem as a regularized non-linear optimization problem, show the existence of a minimizer, and derive optimality conditions. We propose a non-linear conjugate gradient scheme for finding a minimizer based on the optimality conditions. All our numerical experiments are done in two-dimensions. The experiments reveal new insight into the non-linear effects in the reconstruction. One of the interesting features we observe is that, depending on the choice of regularization, there is a trade-off between high resolution and high contrast in the reconstructed images. Our proposed non-linear optimization framework can be generalized to other hybrid imaging modalities

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

Regularized Shallow Image Prior for Electrical Impedance Tomography

Untrained Neural Network Prior (UNNP) based algorithms have gained increasing popularity in tomographic imaging, as they offer superior performance compared to hand-crafted priors and do not require training. UNNP-based methods usually rely on deep architectures which are known for their excellent feature extraction ability compared to shallow ones. Contrary to common UNNP-based approaches, we propose a regularized shallow image prior method that combines UNNP with hand-crafted prior for Electrical Impedance Tomography (EIT). Our approach employs a 3-layer Multi-Layer Perceptron (MLP) as the UNNP in regularizing 2D and 3D EIT inversion. We demonstrate the influence of two typical hand-crafted regularizations when representing the conductivity distribution with shallow MLPs. We show considerably improved EIT image quality compared to conventional regularization algorithms, especially in structure preservation. The results suggest that combining the shallow image prior and the hand-crafted regularization can achieve similar performance to the Deep Image Prior (DIP) but with less architectural dependency and complexity of the neural network

Tecniche Elettrotomografiche per la caratterizzazione dei tessuti biologici

Electrical impedance tomography (EIT) is an imaging modality wherein the spatial map of conductivity and permittivity inside a medium is obtained from a set of surface electrical measurements. Electrodes are brought into contact with the surface of the object being imaged and a set of currents are applied and the corresponding voltages are measured. These voltages and currents are then used to estimate the electrical properties of the object using an image reconstruction algorithm which relies on an accurate model of the electrical interaction. The process of property estimation, called inverse problem, is highly ill-posed and it requires a Regularization method. The objective of this Thesis was to develop a device for imaging using the EIT technique, which was convenient, noninvasive, easily programmable, portable and relatively cheap in contrast to many other diagnostic tool. In this direction a simple EIT system and its hardware and software parts are developed. The data processing was accomplished by utilizing the EIDORS toolkit, which was developed for application to this nonlinear and ill-posed inverse problem. Experiments have indicated that the EIT system can reconstruct resistive and capacitive images of good contrast despite errors in the measurement are not taken in account