Inverse problems naturally arise in many scientific settings, and the study of these problems has been crucial in the development of important technologies such as medical imaging. In inverse problems, the goal is to estimate an underlying ground truth u∗, typically an image, from corresponding measurements y, where u∗ and y are related by

y = N(A(u∗)) (1)

for some forward operator A and noise-generating process N (both of which are generally assumed to be known). Variational regularisation is a well-established approach that can be used to approximately solve inverse problems such as Problem (1). In this approach an image is reconstructed from measurements y by solving a minimisation problem such as

uˆ = argmin d(A(u),y) +αJ(u). (2)

While this approach has proven very successful, it generally requires the parts that make up the optimisation problem to be carefully chosen, and the optimisation problem may require considerable computational effort to solve. There is an active line of research into overcoming these issues using data-driven approaches, which aim to use multiple instances of data to inform a method that can be used on similar data. In this dissertation we investigate ways in which favourable properties of the variational regularisation approach can be combined with a data-driven approach to solving inverse problems.

In the first chapter of the dissertation, we propose a bilevel optimisation framework that can be used to optimise sampling patterns and regularisation parameters for variational image reconstruction in accelerated magnetic resonance imaging. We use this framework to learn sampling patterns that result in better image reconstructions than standard random variable density sampling patterns that sample with the same rate. In the second chapter of the dissertation, we study the use of group symmetries in learned reconstruction methods for inverse problems. We show that group invariance of a functional implies that the corresponding proximal operator satisfies a group equivariance property. Applying this idea to model proximal operators as roto-translationally equivariant in an unrolled iterative reconstruction method, we show that reconstruction performance is more robust when tested on images in orientations not seen during training (compared to similar methods that model proximal operators to just be translationally equivariant) and that good methods can be learned with less training data.

In the final chapter of the dissertation, we propose a ResNet-styled neural network architecture that is provably nonexpansive. This architecture can be thought of as composing discretisations of gradient flows along learnable convex potentials. Appealing to a classical result on the numerical integration of ODEs, we show that constraining the operator norms of the weight operators is sufficient to give nonexpansiveness, and additional analysis in the case that the numerical integrator is the forward Euler method shows that the neural network is an averaged operator. This guarantees that its fixed point iterations are convergent, and makes it a natural candidate for a learned denoiser in a Plug-and-Play approach to solving inverse problems