Solving Dynamic Inverse Problems with Dynamic Inverse Problems

Abstract

This thesis studies the use of Physics-Informed Neural Networks/Neural Fields for dynamic inverse imaging problems. These untrained neural networks constitute a novel continuous parametrisation of the quantity of interest. Therefore, their performance needs to be compared against the classical grid-based parametrisation of images. Dynamic imaging refers to the scanning of a moving object in a non-invasive manner using a specialised device. Due to motion and the scanner's sampling speed, measurements are highly undersampled at each time step, resulting in an ill-posed problem. Such a situation arises in seismic imaging for the study of the subsurface, atmospheric and oceanic tomography, where data is collected from satellites, etc. This thesis focuses on medical imaging, in particular, dynamic computed tomography and magnetic resonance imaging.

In the introduction, we discuss the basics of imaging problems, motivate the relevance and challenges of the undersampled dynamic setting, and discuss the drawbacks of discrete voxelated images. Also, a brief introduction to neural fields is provided, motivating their application to dynamic inverse problems.

Posteriorly, chapters 2 and 3 serve as a background for this thesis. In chapter 2, we introduce the maths behind computed tomography and magnetic resonance imaging and establish how these experiments result in an inverse problem. After that, we dive into variational regularisation, the optimisation framework employed to find the solution. These ingredients are then used to explain dynamic imaging in the highly undersampled regime for the two considered methodologies. Chapter 3 introduces the basics of deep learning. In this chapter, we discuss neural fields and physics-informed neural networks in more detail, and how these untrained networks are suitable for static and dynamic imaging.

Dynamic computed tomography is addressed in chapter 4, where a physics-based prior in terms of a partial differential equation and motion is imposed as a regularisation. The problem is posed as a joint image reconstruction and motion estimation, and is compared against its voxelated counterpart. Through numerical experiments on several phantoms, we validate the method and compare it against grid-based representations. Chapter 5 studies 2D cine phase-contrast MRI, a modality of dynamic MRI that allows studying the blood flow through major vessels. The method is compared against a locally low-rank regularised grid-based representation, particularly suitable for dynamic MRI. Overall, our results show that neural fields outperform the grid-based methods, making them suitable for dynamic imaging.

We finish with chapter 6, where we discuss the main conclusions and potential future work to follow from the research carried out in this thesis.

Date of Award	25 Mar 2026
Original language	English
Awarding Institution	University of Bath
Supervisor	Matthias Ehrhardt (Supervisor) & Eike Mueller (Supervisor)