Shape optimizing diffeomorphisms for medical image analysis

An important question, historically, is how structure and function relate. The field of medical image analysis addresses this question by using imaging devices such MRI or CT to gain in vivo structural and functional measurements. Statistical analysis of the data requires smoothly mapping the images of an individual into a common (or atlas) coordinate system. Novel mathematics and principles for this mapping process (labeled Diffeomorphometry) are the subject of this dissertation. Diffeomorphometry is a rigorous mathematical framework for statistical analysis of shape and shape coordinate systems derived from images. Diffeomorphometry is characterized by computing diffeomorphisms and their inverses in the Lagrangian frame, representing diffeomorphisms by their initial conditions, and combining initial conditions to evaluate the statistics and neighborhood relationships of shapes. This approach addresses many of the basic issues in image registration, including how to generate geodesics and geodesic distances between shape objects, how to represent shape and topology differences, how to register images symmetrically and how to define unbiased coordinate systems for medical or pharmacological studies. These general techniques are useful in a wide array of image processing applications. We apply these techniques to analyze atrophy as it correlates with cognitive decline in frontotemporal dementia, to reverse engineer the evolution between chimpanzee and human cortex and to fuse histological data with MRI. Other applications include matching dynamic series of pulmonary data