Shape Statistics via Skeletal Structures

Abstract

Statistical shape analysis has emerged as a crucial tool for medical researchers and clinicians to study medical objects such as brain subcortical structures. The insights gained from such analyses hold immense potential for diagnoses and enhancing our understanding of various diseases, particularly neurological disorders. This thesis explores three important areas of statistical shape analysis, which are detailed in three separate papers: “Statistical Analysis of Locally Parameterized Shapes,” “Fitting Discrete Swept Skeletal Structures to Slabular Objects,” and “The Mean Shape under the Relative Curvature Condition.” The innovative approaches discussed in these papers offer a fresh perspective for representing complex shapes, enabling more nuanced analysis and interpretation. Central to this work is the discussion surrounding the introduction of robust skeletal representations for establishing correspondences for a class of swept regions called slabular objects and providing proper mathematical methodologies supporting the statistical objectives such as hypothesis testing and classification. The proposed skeletal models are alignment-independent and invariant to the act of Euclidean similarity transformations of translation, rotations, and scaling. Damon’s criterion of the relative curvature condition (RCC) is an essential factor for valid swept skeletal structures. This work extensively discusses fitting skeletal models, defining shape space, and calculating the mean shape for such models following the RCC. The efficacy of the proposed methodology is underscored through rigorous examinations, both visually and statistically. These methodologies are specifically applied to medical contexts, focusing on analyzing subcortical structures. Synthetic and actual datasets serve for validation, facilitating a comprehensive comparison with existing skeletal representations. This work highlights the resilience and adaptability of innovative approaches, paving the way for further medical research and diagnostic endeavors.