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.

Steerable Conditional Diffusion for Out-of-Distribution Adaptation in Imaging Inverse Problems

Denoising diffusion models have emerged as the go-to framework for solving inverse problems in imaging. A critical concern regarding these models is their performance on out-of-distribution (OOD) tasks, which remains an under-explored challenge. Realistic reconstructions inconsistent with the measured data can be generated, hallucinating image features that are uniquely present in the training dataset. To simultaneously enforce data-consistency and leverage data-driven priors, we introduce a novel sampling framework called Steerable Conditional Diffusion. This framework adapts the denoising network specifically to the available measured data. Utilising our proposed method, we achieve substantial enhancements in OOD performance across diverse imaging modalities, advancing the robust deployment of denoising diffusion models in real-world applications.

Score-Based Generative Models for PET Image Reconstruction

Score-based generative models have demonstrated highly promising results for medical image reconstruction tasks in magnetic resonance imaging or computed tomography. However, their application to Positron Emission Tomography (PET) is still largely unexplored. PET image reconstruction involves a variety of challenges, including Poisson noise with high variance and a wide dynamic range. To address these challenges, we propose several PET-specific adaptations of score-based generative models. The proposed framework is developed for both 2D and 3D PET. In addition, we provide an extension to guided reconstruction using magnetic resonance images. We validate the approach through extensive 2D and 3D $\textit{in-silico}$ experiments with a model trained on patient-realistic data without lesions, and evaluate on data without lesions as well as out-of-distribution data with lesions. This demonstrates the proposed method's robustness and significant potential for improved PET reconstruction.

Solving Elliptic Optimal Control Problems using Physics Informed Neural Networks

In this work, we present and analyze a numerical solver for optimal control problems (without / with box constraint) for linear and semilinear second-order elliptic problems. The approach is based on a coupled system derived from the first-order optimality system of the optimal control problem, and applies physics informed neural networks (PINNs) to solve the coupled system. We present an error analysis of the numerical scheme, and provide $L^2(\Omega)$ error bounds on the state, control and adjoint state in terms of deep neural network parameters (e.g., depth, width, and parameter bounds) and the number of sampling points in the domain and on the boundary. The main tools in the analysis include offset Rademacher complexity and boundedness and Lipschitz continuity of neural network functions. We present several numerical examples to illustrate the approach and compare it with three existing approaches.

On the Approximation of Bi-Lipschitz Maps by Invertible Neural Networks

Invertible neural networks (INNs) represent an important class of deep neural network architectures that have been widely used in several applications. The universal approximation properties of INNs have also been established recently. However, the approximation rate of INNs is largely missing. In this work, we provide an analysis of the capacity of a class of coupling-based INNs to approximate bi-Lipschitz continuous mappings on a compact domain, and the result shows that it can well approximate both forward and inverse maps simultaneously. Furthermore, we develop an approach for approximating bi-Lipschitz maps on infinite-dimensional spaces that simultaneously approximate the forward and inverse maps, by combining model reduction with principal component analysis and INNs for approximating the reduced map, and we analyze the overall approximation error of the approach. Preliminary numerical results show the feasibility of the approach for approximating the solution operator for parameterized second-order elliptic problems.

Electrical Impedance Tomography with Deep Calderón Method

Electrical impedance tomography (EIT) is a noninvasive medical imaging modality utilizing the current-density/voltage data measured on the surface of the subject. Calder\'on's method is a relatively recent EIT imaging algorithm that is non-iterative, fast, and capable of reconstructing complex-valued electric impedances. However, due to the regularization via low-pass filtering and linearization, the reconstructed images suffer from severe blurring and underestimation of the exact conductivity values. In this work, we develop an enhanced version of Calder\'on's method, using convolution neural networks (i.e., U-net) via a postprocessing step. Specifically, we learn a U-net to postprocess the EIT images generated by Calder\'on's method so as to have better resolutions and more accurate estimates of conductivity values. We simulate chest configurations with which we generate the current-density/voltage boundary measurements and the corresponding reconstructed images by Calder\'on's method. With the paired training data, we learn the neural network and evaluate its performance on real tank measurement data. The experimental results indicate that the proposed approach indeed provides a fast and direct (complex-valued) impedance tomography imaging technique, and substantially improves the capability of the standard Calder\'on's method.

Conductivity Imaging from Internal Measurements with Mixed Least-Squares Deep Neural Networks

In this work we develop a novel approach using deep neural networks to reconstruct the conductivity distribution in elliptic problems from one internal measurement. The approach is based on a mixed reformulation of the governing equation and utilizes the standard least-squares objective to approximate the conductivity and flux simultaneously, with deep neural networks as ansatz functions. We provide a thorough analysis of the neural network approximations for both continuous and empirical losses, including rigorous error estimates that are explicit in terms of the noise level, various penalty parameters and neural network architectural parameters (depth, width and parameter bound). We also provide extensive numerical experiments in two- and multi-dimensions to illustrate distinct features of the approach, e.g., excellent stability with respect to data noise and capability of solving high-dimensional problems.

SVD-DIP: Overcoming the Overfitting Problem in DIP-based CT Reconstruction

The deep image prior (DIP) is a well-established unsupervised deep learning method for image reconstruction; yet it is far from being flawless. The DIP overfits to noise if not early stopped, or optimized via a regularized objective. We build on the regularized fine-tuning of a pretrained DIP, by adopting a novel strategy that restricts the learning to the adaptation of singular values. The proposed SVD-DIP uses ad hoc convolutional layers whose pretrained parameters are decomposed via the singular value decomposition. Optimizing the DIP then solely consists in the fine-tuning of the singular values, while keeping the left and right singular vectors fixed. We thoroughly validate the proposed method on real-measured $\mu$CT data of a lotus root as well as two medical datasets (LoDoPaB and Mayo). We report significantly improved stability of the DIP optimization, by overcoming the overfitting to noise.

Fast and Painless Image Reconstruction in Deep Image Prior Subspaces

The deep image prior (DIP) is a state-of-the-art unsupervised approach for solving linear inverse problems in imaging. We address two key issues that have held back practical deployment of the DIP: the long computing time needed to train a separate deep network per reconstruction, and the susceptibility to overfitting due to a lack of robust early stopping strategies in the unsupervised setting. To this end, we restrict DIP optimisation to a sparse linear subspace of the full parameter space. We construct the subspace from the principal eigenspace of a set of parameter vectors sampled at equally spaced intervals during DIP pre-training on synthetic task-agnostic data. The low-dimensionality of the resulting subspace reduces DIP's capacity to fit noise and allows the use of fast second order optimisation methods, e.g., natural gradient descent or L-BFGS. Experiments across tomographic tasks of different geometry, ill-posedness and stopping criteria consistently show that second order optimisation in a subspace is Pareto-optimal in terms of optimisation time to reconstruction fidelity trade-off.

Solving Elliptic Problems with Singular Sources using Singularity Splitting Deep Ritz Method

In this work, we develop an efficient solver based on deep neural networks for the Poisson equation with variable coefficients and singular sources expressed by the Dirac delta function $\delta(\mathbf{x})$. This class of problems covers general point sources, line sources and point-line combinations, and has a broad range of practical applications. The proposed approach is based on decomposing the true solution into a singular part that is known analytically using the fundamental solution of the Laplace equation and a regular part that satisfies a suitable elliptic PDE with smoother sources, and then solving for the regular part using the deep Ritz method. A path-following strategy is suggested to select the penalty parameter for penalizing the Dirichlet boundary condition. Extensive numerical experiments in two- and multi-dimensional spaces with point sources, line sources or their combinations are presented to illustrate the efficiency of the proposed approach, and a comparative study with several existing approaches is also given, which shows clearly its competitiveness for the specific class of problems. In addition, we briefly discuss the error analysis of the approach.