Nonsmooth Nonconvex Variational Reconstruction for Electrical Impedance Tomography

Electrical Impedance Tomography is an imaging technique that aims to reconstruct the inner conductivity distribution of a medium starting from a set of measured voltages registered by a series of electrodes that are positioned on the surface of the medium. Such technique was used for the first time in geological studies in 1930 and then applied to industrial procedures. The first clinical use of EIT dates back to 1987. In 2018 EIT was validated in tissue engineering as a noninvasive and label-free imaging and monitoring tool for cell distribution (cell growth, differentiation and tissue formation) in 3D scaffolds. EIT problem can be split into a Forward and an Inverse problem. The aim of Forward EIT is to define the set of measured voltages starting from a known conductivity distribution. If the forward problem is characterized by a nonlinear mapping, called Forward Operator, from the conductivity distribution to the measured voltages, inverse EIT consists of inverting the Forward Operator. This leads to an ill-posed problem which requires regularization, either in the model or in the numerical method that is applied to define the solution. The inverse problem is modelled as a Nonlinear Least Squares problem, where one seeks to minimize the mismatch beetween the measured voltages and the ones generated by the reconstructed conductivity. Reconstruction techniques require the introduction of a regularization term which forces the reconstructed data to stick to certain prior information. In this dissertation, some state-of-the-art regularization methods are analyzed and compared via EIDORS, a specific software for EIT problems. The aim is to reconstruct the variation in conductivity within a 2D section of a 3D scaffold. Furthermore a variational formulation on a 2D mesh for a space-variant regularization is proposed, based on a combination of high order and nonconvex operators, which respectively seek to recover piecewise inhomogeneous and piecewise linear regions

Improvement of image quality of time-domain diffuse optical tomography with lp sparsity regularization

An lp (0 < p ≤ 1) sparsity regularization is applied to time-domain diffuse optical tomography with a gradient-based nonlinear optimization scheme to improve the spatial resolution and the robustness to noise. The expression of the lp sparsity regularization is reformulated as a differentiable function of a parameter to avoid the difficulty in calculating its gradient in the optimization process. The regularization parameter is selected by the L-curve method. Numerical experiments show that the lp sparsity regularization improves the spatial resolution and recovers the difference in the absorption coefficients between two targets, although a target with a small absorption coefficient may disappear due to the strong effect of the lp sparsity regularization when the value of p is too small. The lp sparsity regularization with small p values strongly localizes the target, and the reconstructed region of the target becomes smaller as the value of p decreases. A phantom experiment validates the numerical simulations

Electrical Impedance Tomography: A Fair Comparative Study on Deep Learning and Analytic-based Approaches

Electrical Impedance Tomography (EIT) is a powerful imaging technique with diverse applications, e.g., medical diagnosis, industrial monitoring, and environmental studies. The EIT inverse problem is about inferring the internal conductivity distribution of an object from measurements taken on its boundary. It is severely ill-posed, necessitating advanced computational methods for accurate image reconstructions. Recent years have witnessed significant progress, driven by innovations in analytic-based approaches and deep learning. This review explores techniques for solving the EIT inverse problem, focusing on the interplay between contemporary deep learning-based strategies and classical analytic-based methods. Four state-of-the-art deep learning algorithms are rigorously examined, harnessing the representational capabilities of deep neural networks to reconstruct intricate conductivity distributions. In parallel, two analytic-based methods, rooted in mathematical formulations and regularisation techniques, are dissected for their strengths and limitations. These methodologies are evaluated through various numerical experiments, encompassing diverse scenarios that reflect real-world complexities. A suite of performance metrics is employed to assess the efficacy of these methods. These metrics collectively provide a nuanced understanding of the methods' ability to capture essential features and delineate complex conductivity patterns. One novel feature of the study is the incorporation of variable conductivity scenarios, introducing a level of heterogeneity that mimics textured inclusions. This departure from uniform conductivity assumptions mimics realistic scenarios where tissues or materials exhibit spatially varying electrical properties. Exploring how each method responds to such variable conductivity scenarios opens avenues for understanding their robustness and adaptability

Model Augmented Deep Neural Networks for Medical Image Reconstruction Problems

Solving an ill-posed inverse problem is difficult because it doesn\u27t have a unique solution. In practice, for some important inverse problems, the conventional methods, e.g. ordinary least squares and iterative methods, cannot provide a good estimate. For example, for single image super-resolution and CT reconstruction, the results of these conventional methods cannot satisfy the requirements of these applications. While having more computational resources and high-quality data, researchers try to use machine-learning-based methods, especially deep learning to solve these ill-posed problems. In this dissertation, a model augmented recursive neural network is proposed as a general inverse problem method to solve these difficult problems. In the dissertation, experiments show the satisfactory performance of the proposed method for single image super-resolution, CT reconstruction, and metal artifact reduction