A Probabilistic Framework for Statistical Shape Models and Atlas Construction: Application to Neuroimaging

Accurate and reliable registration of shapes and multi-dimensional point sets describing the morphology/physiology of anatomical structures is a pre-requisite for constructing statistical shape models (SSMs) and atlases. Such statistical descriptions of variability across populations (regarding shape or other morphological/physiological quantities) are based on homologous correspondences across the multiple samples that comprise the training data. The notion of exact correspondence can be ambiguous when these data contain noise and outliers, missing data, or significant and abnormal variations due to pathology. But, these phenomena are common in medical image-derived data, due, for example, to inconsistencies in image quality and acquisition protocols, presence of motion artefacts, differences in pre-processing steps, and inherent variability across patient populations and demographics. This thesis therefore focuses on formulating a unified probabilistic framework for the registration of shapes and so-called \textit{generalised point sets}, which is robust to the anomalies and variations described. Statistical analysis of shapes across large cohorts demands automatic generation of training sets (image segmentations delineating the structure of interest), as manual and semi-supervised approaches can be prohibitively time consuming. However, automated segmentation and landmarking of images often result in shapes with high levels of outliers and missing data. Consequently, a robust method for registration and correspondence estimation is required. A probabilistic group-wise registration framework for point-based representations of shapes, based on Student’s t-mixture model (TMM) and a multi-resolution extension to the same (mrTMM), are formulated to this end. The frameworks exploit the inherent robustness of Student’s t-distributions to outliers, which is lacking in existing Gaussian mixture model (GMM)-based approaches. The registration accuracy of the proposed approaches was quantitatively evaluated and shown to outperform the state-of-the-art, using synthetic and clinical data. A corresponding improvement in the quality of SSMs generated subsequently was also shown, particularly for data sets containing high levels of noise. In general, the proposed approach requires fewer user specified parameters than existing methods, whilst affording much improved robustness to outliers. Registration of generalised point sets, which combine disparate features such as spatial positions, directional/axial data, and scalar-valued quantities, was studied next. A hybrid mixture model (HMM), combining different types of probability distributions, was formulated to facilitate the joint registration and clustering of multi-dimensional point sets of this nature. Two variants of the HMM were developed for modelling: (1) axial data; and (2) directional data. The former, based on a combination of Student’s t, Watson and Gaussian distributions, was used to register hybrid point sets comprising magnetic resonance diffusion tensor image (DTI)-derived quantities, such as voxel spatial positions (defining a region/structure of interest), associated fibre orientations, and scalar measures reflecting tissue anisotropy. The latter meanwhile, formulated using a combination of Student’s t and Von-Mises-Fisher distributions, is used for the registration of shapes represented as hybrid point sets comprising spatial positions and associated surface normal vectors. The Watson-variant of the HMM facilitates statistical analysis and group-wise comparisons of DTI data across patient populations, presented as an exemplar application of the proposed approach. The Fisher-variant of the HMM on the other hand, was used to register hybrid representations of shapes, providing substantial improvements over point-based registration approaches in terms of anatomical validity in the estimated correspondences.