Geometric methods for image registration and analysis

Registration and analysis of neuro-imaging data presents a challenging problem due to the complex folding patterns in the human brain. Specifically, the cortical surface of the human brain can be modeled as a highly convoluted 2D surface. Since it is non-flat, the non-Euclidean geometry of the cortex needs to be accounted for while performing registration and subsequent signal processing of anatomical and functional signals on the cortex. Techniques from differential geometry offer a powerful set of tools to deal with the convoluted nature of the cortex. We present a method based on p-harmonic mapping for performing cortical surface parameterization. A 2D coordinate system induced by the flat mapping is then used to compute the surface metric and discretize derivatives in the surface geometry. For performing inter-subject cortical registration based on sulcal landmarks, we generalize thin-plate splines to non-flat surfaces by using covariant derivatives. We also present an FEM based method for simultaneous parameterization and registration of sulcal landmarks based on elastic energy minimization. The manual effort required for selecting the sulcal landmarks can be minimized if we choose an optimal set of such landmarks. We present a method for optimally selecting a subset of any size from a set of candidate sulcal landmarks and also predict the associated registration error for that subset using conditional distributions. Surface signals from individual brains can be brought to a common atlas surface by using these surface based registration techniques.;Isotropic and anisotropic diffusion filtering methods are formulated for processing of the cortical data. This is performed by using parameterization-based methods which use covariant diffusion operators in the flat space. When the surface data is a point-set on the cortex, we propose a method to quantify its mean and variance with respect to the surface geometry.;The registration techniques presented for surface alignment are extended to volumes to perform full surface and volume registration. This is done by using volumetric harmonic mappings that extend the surface point correspondence to the cortical brain volume. Finally, the volumetric registration is refined by using inverse-consistent linear elastic intensity registration. This set of methods presents a unified framework for registration and analysis of brain signals for inter-subject neuroanatomical studies.