Volumetric Mapping for Medical Imaging and Geometry Processing

Abstract

Mapping is the task of computing a transformation from a source shape to a target domain. It 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 (3D) mapping, a natural extension relevant to shapes extracted from medical imaging, simulation, and volume rendering presents unique challenges that do not appear in the 2D case. In this thesis, we propose methods for mapping volumes represented as tetrahedral meshes. We are motivated by problems in medical imaging using magnetic resonance imaging (MRI) of the placenta to study placental health and function. This application presents a challenging problem setting relevant for mapping tasks in computer graphics and geometry processing as a whole. We propose an automatic segmentation method to extract placental shapes from MRI. To alleviate interpretation issues of placental MRI, we propose a volumetric parameterization to map the placenta to a standardized representation and enable visualization of local anatomy and function. To tackle the more general problem of computing a map between shapes, we propose a method to compute symmetric correspondences between an arbitrary class of highly dissimilar geometric shapes. Finally, we combine our proposed approaches to develop a combined shape-and-image mapping framework to find dense correspondences of placental shapes. 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.