Image registration methods underpin many analysis techniques in neuroimaging. They are essential in group studies when images of different individuals or different modalities need to be brought into a common reference frame. This thesis explores the potential of brain connectivity- driven alignment and develops surface registration techniques for magnetic resonance imaging (MRI), which is a noninvasive neuroimaging tool for probing function and structure of the human brain. The first part of this work develops a novel surface registration framework, based on free mesh deformations, which aligns cortical and subcortical surfaces by matching structural connectivity patterns derived using probabilistic tractography (diffusion-weighted MRI). Structural, i.e. white matter, connectivity is a good predictor of functional specialisation and structural connectivity-driven registration can therefore be expected to enhance the alignment of functionally homologous areas across subjects. The second part validates developed methods for cortical surfaces. Resting State Networks are used in an innovative way to delineate several functionally distinct regions, which were then used to quantify connectivity-driven registration performance by measuring the inter- subject overlap before and after registration. Consequently, the proposed method is assessed using an independent imaging modality and the results are compared to results from state-of-the-art cortical geometry-driven surface registration methods. A connectivity-driven registration pipeline is also developed for, and applied to, the surfaces of subcortical structures such as the thalamus. It is carefully validated on a set of artificial test examples and compared to another novel surface registration paradigm based on spherical wavelets. The proposed registration pipeline is then used to explore the differences in the alignment of two groups of subjects, healthy controls and Alzheimer's disease patients, to a common template. Finally, we propose how functional connectivity can be used instead of structural connectivity for driving registrations, as well as how the surface-based framework can be extended to a volumetric one. Apart from providing the benefits such as the improved functional alignment, we hope that the research conducted in this thesis will also represent the basis for the development of templates of structural and functional brain connectivity.

Sequence alignment is one of the most intensely studied problems in bioinformatics, and is an important step in a wide range of analyses. An issue that has gained much attention in recent years is the fact that downstream analyses are often highly sensitive to the specific choice of alignment. One way to address this is to jointly sample alignments along with other parameters of interest. In order to extend the range of applicability of this approach, the first chapter of this thesis introduces a probabilistic evolutionary model for protein structures on a phylogenetic tree; since protein structures typically diverge much more slowly than sequences, this allows for more reliable detection of remote homologies, improving the accuracy of the resulting alignments and trees, and reducing sensitivity of the results to the choice of dataset. In order to carry out inference under such a model, a number of new Markov chain Monte Carlo approaches are developed, allowing for more efficient convergence and mixing on the high-dimensional parameter space. The second part of the thesis presents a directed acyclic graph (DAG)-based approach for representing a collection of sampled alignments. This DAG representation allows the initial collection of samples to be used to generate a larger set of alignments under the same approximate distribution, enabling posterior alignment probabilities to be estimated reliably from a reasonable number of samples. If desired, summary alignments can then be generated as maximum-weight paths through the DAG, under various types of loss or scoring functions. The acyclic nature of the graph also permits various other types of algorithms to be easily adapted to operate on the entire set of alignments in the DAG. In the final part of this work, methodology is introduced for alignment-DAG-based sequence annotation using hidden Markov models, and RNA secondary structure prediction using stochastic context-free grammars. Results on test datasets indicate that the additional information contained within the DAG allows for improved predictions, resulting in substantial gains over simply analysing a set of alignments one by one.