Bayesian methods for inverse problems

Inverse problems are an important class of problems that appear in many practical disciplines, in which we infer causes from incomplete or corrupt observations of the effects. Inverse problems can be challenging to solve and the solution is often ambiguous. This ambiguity is typically reduced by the application of prior information, which is classically of a generic form such as a sparsity or energy prior. In some inverse problems, there is the possibility of simultaneously solving the problem while having the ability to choose the observations to make. So an additional point of interest is making these choices optimally.

Deep learning can be applied to inverse problems, although the methods are often inflexible. Typically deep learning models are trained on a set of solution-observation pairs for a specific inverse problem setting. Unfortunately, deep learning models do not offer justifications for observation optimisation due to being unexplainable, which raises reliability concerns.

Moreover, re-training is needed if the observation model is changed.

The first contribution of this thesis is adaptive informed sensing, which is an inverse problem solving method that exploits the power of deep learning models to solve inverse problems and optimally choose observations. Adaptive informed sensing is based on information theory and makes use of conditional generation to separate the deep model, in the form of a prior, from the solution process itself. The method addresses the explainability and inflexibility issues of common deep learning approaches.

The second contribution is the investigation of methods to accelerate conditional generation for inverse problem solution. We present experiments on proposed conditional generative models that can deal with noisy observations of any size, allowing a variable sized observation vector. We also present challenges and insights for future improvements.

The final contribution is a demonstration of the use of our proposed framework for the acceleration of magnetic resonance imaging (MRI). We present a method for generating observation trajectories for MRI that take into consideration machine constraints, and use our informed sensing to accelerate the reconstruction by optimising the choice of trajectories online.