Statistics on computational anatomy: From template estimation to geodesically controlled diffeomorphic active shapes

This thesis addresses two important problems in the domain of statistical analysis on anatomical shapes: template estimation and PCA representation of diffeomorphic deformation.;Geodesic shooting provides a new philosophy for statistical analysis on anatomical shapes. According to the momentum conservation law, given a template, a diffeomorphic trajectory is uniquely determined by the initial momentum. Thus, statistics on the nonlinear space of diffeomorphims can be studied via the linear tangent space.;We present a Bayesian model for template estimation for 3D images in CA. It is assumed that observed images are generated by deforming the template through Gaussian distributed random initial momenta. The template is modeled as a deformation from a given hypertemplate with initial momentum mu, which has a Gaussian prior. We employ a variant of EM (MAEM) procedure, and approximate the conditional expectation by Dirac measure. This leads to an image matching problem with a Jacobian weight term. We derive a weighted Euler-Lagrange equation and develop a numerical algorithm to solve it. The results of template estimation for hippocampus and cardiac data are presented.;The Bayesian framework and MAEM scheme have also been applied to template estimation for surfaces and have been validated with caudate, thalamus and hippocampus surface data, showing its effectiveness and convergence, and also experimentally proved to be robust to variations in the choice of the hypertemplate.;The other major topic of this thesis is Geodesically Controlled Diffeomorphic Active Shapes (GDAS). The motivation of GDAS is to learn shape variability from training samples and describe this variability by low dimensional data. For the "shape statistics", we map a template to each surface in the training set and obtain its corresponding initial momentum vector. We perform PCA on these initial momentum vectors and define in this way a subspace for shape deformation. Any new initial momentum is projected onto this subspace and is represented by a small number of PCA coefficients.;Applications of GDAS usually boil down to finding the optimal coefficients minimizing an energy term. The energy turns out to have a common pattern: it is a combination of a coefficient regulation term and a mismatch term. The latter is a functional of the deformed surface. We derive a general gradient descent procedure for GDAS optimization and apply this procedure in the context of 3D medical image segmentation and PCA based surface matching.