Geodesic estimation for large deformation anatomical shape averaging and interpolation

The goal of this research is to promote variational methods for anatomical averaging that operate within the space of the underlying image registration problem. This approach is effective when using the large deformation viscous framework, where linear averaging is not valid, or in the elastic case. The theory behind this novel atlas building algorithm is similar to the traditional pairwise registration problem, but with single image forces replaced by average forces. These group forces drive an average transport ordinary differential equation allowing one to estimate the geodesic that moves an image toward the mean shape configuration. This model gives large deformation atlases that are optimal with respect to the shape manifold as defined by the data and the image registration assumptions. We use the techniques in the large deformation context here, but they also pertain to small deformation atlas construction. Furthermore, a natural, inherently inverse consistent image registration is gained for free, as is a tool for constant arc length geodesic shape interpolation. The geodesic atlas creation algorithm is quantitatively compared to the Euclidean anatomical average to elucidate the need for optimized atlases. The procedures generate improved average representations of highly variable anatomy from distinct populations.