Doctor of Philosophy, Case Western Reserve University, 2017, Applied Mathematics

Image segmentation and registration play active roles in machine vision and medical image analysis of historical data. Individually, the two has seen important research contributions, and the joint treatment of the two problems has become an active area of research. In this thesis we will explore the joint problem of segmenting and registering a template image given a reference image. We formulate the joint problem through an energy functional that integrates two well studied approaches in segmentation and registration: Geodesic Active Contours and nonlinear elastic registration. In the registration regime, the domain is modeled as a St. Venant-Kirchhoff material. We minimize the potential energy of this elastic system using variational methods, and derive an evolution equation which we solve using implicit-explicit integration methods. The numerical discretization of the problem allows us to take advantage of the Fast Fourier Transform. In the segmentation regime, we will adopt an active contours based energy with a weighted total variation penalty on the segmenting front. This particular choice allows for fast solution using the dual formulation of the total variation. The weight of the total variation penalty is an edge stopping function which depends on the deforming template. This allows the segmenting front to accurately track spontaneous changes in the shape of objects embedded in the template image as it deforms.

Committee: Weihong Guo (Committee Chair); Daniela Calvetti (Committee Member); Julia Dobrotsoskya (Committee Member); Erkki Somersalo (Committee Member); Michael Lewicki (Committee Member) Subjects: Applied Mathematics; Biomedical Engineering