Methods in statistical shape analysis for landmark-based three-dimensional data, with applications to an orthodontic study

The field of statistical shape analysis is aimed at describing and comparing the shapes of objects. Areas of application are often in fields related to medicine or biology, with shapes of interest that are often complex and non-geometric. Shape analysis methods allow clinicians and researchers to analyze shape differences and shape changes within individuals, across groups, and over time. These can occur due to growth, varying treatment techniques, or natural differences between groups.;Because of some statistical issues that arise in the standard shape analysis setting, an alternative approach will also be applied. This approach, borrowed from computational topology, is called persistent homology. It will be applied to the existing shape analysis framework to detect clusters or subgroups of similarly-shaped objects.;This thesis will discuss some methods that have been developed in the field of landmark-based statistical shape analysis, and apply these methods to an example three-dimensional data set taken from the field of orthodontics. Landmark-based methods apply to shapes that have been represented as a set of landmarks, or points, of interest (as opposed to shapes that are represented by boundary information, or an entire pixelated/voxelated image). Shape analysis methods involve transforming these landmark configurations into some shape space (or approximation to it), in order to define distances between shapes, and perform statistical analyses.