Probabilistic Multi - Label Representations for Anatomical Statistical Shape Analysis

Several sources of uncertainties in shape boundaries in medical images have motivated the use of probabilistic labeling approaches. Being able to perform statistical analysis on these probabilistic multi-shape representations is important in understanding normal and pathological geometrical variability of anatomical structures. By making use of methods for dealing with what is known as compositional data, we propose a new framework intrinsic to the unit simplex for statistical analysis of probabilistic multi-shape anatomy. In this framework, an isometric log-ratio transformation is used to isometrically and bijectively map the simplex to the Euclidean real space. As another contribution of this thesis, the label space multi-shape representation (of Malcolm et al. [49]) is extended to the barycentric label space, in which a proper invertible mapping between probability vectors and label space is proposed. Favorable properties of the proposed methods are demonstrated quantitatively and qualitatively on artificial objects and brain image data.