| The work in this dissertation is motivated by applications in the analysis of imaging data, with an emphasis on diffusion tensor imaging (DTI), a modality of MRI used to noninvasively map the structure of the brain in living subjects. In the DTI model, the local movement of water molecules within a small region of the brain is summarized by a 3-by-3 symmetric positive-definite (SPD) matrix, called a diffusion tensor. Diffusion tensors can be uniquely associated with three-dimensional ellipsoids which, when plotted, provide an image of the brain. We are interested in analyzing diffusion tensor data on the eigen-decomposition space because the eigenvalues and eigenvectors of a diffusion tensor describe the shape and orientation of its corresponding ellipsoid, respectively. One of the major contributions of this dissertation is the creation of the first statistical estimation framework for SPD matrices using the eigen-decomposition-based scaling-rotation (SR) geometric framework from Jung et al (2015). In chapter 3, we define the set of sample scaling-rotation means of a sample of SPD matrices, propose a procedure for approximating the sample SR mean set, provide conditions under which this procedure will provide a unique solution, and provide conditions guaranteeing consistency and a Central Limit Theorem for the sample SR mean set. Our procedure for approximating the sample SR mean can also be extended to compute a weighted SR mean, which can be useful for smoothing DTI data or interpolation to improve image resolution. In chapter 4, we present moment-based hypothesis tests concerning the eigenvalue multiplicity pattern of the mean of a sample of diffusion tensors which can be used to classify the mean as one of four possible shapes: isotropic, prolate, oblate, or triaxial. The derivations of these test procedures lead to the creation of consistent estimators of the eigenvalues of the mean diffusion tensor. In the final chapter, we present a mixture distribution framework which can be used to model the variability of SPD matrices on the eigen-decomposition space, and an accompanying likelihood-based estimation procedure which can be used for estimation of parameters of interest or inference via likelihood ratio tests. |
