Mapping is a central problem in medical imaging and computer graphics. Most methods for this task apply only to two-dimensional (2D) surfaces. The neglected task of volumetric mapping, a natural extension relevant to shapes extracted from medical imaging, simulation, and volume rendering present unique challenges that do not appear in the two-dimensional case. In this thesis, we propose methods for mapping volumes represented as tetrahedral meshes. We are motivated by problems using magnetic resonance imaging (MRI) to examine placental health and function. We first develop a standardized representation of the placenta in MRI to enable local signal analysis. We then propose a symmetric mapping algorithm to find correspondences across a diverse set of geometric shapes. Finally, we extend this method to propose a joint shape-and-image registration framework to track placental signal changes over time. The combination of these works can be used to assess localized placental function through MRI, necessary to develop biomarkers of fetal health. We conclude by discussing the potential of this work in future clinical research studies to improve fetal-maternal health.
Thesis Committee: Polina Golland, Justin Solomon, Esra Abaci Turk, Juan Eugenio Iglesias